The Universe of Discourse

Wed, 20 Sep 2017

Gompertz' law for wooden utility poles

Gompertz' law says that the human death rate increases exponentially with age. That is, if your chance of dying during this year is !!x!!, then your chance of dying during next year is !!cx!! for some constant !!c>1!!. The death rate doubles every 8 years, so the constant !!c!! is empirically around !!2^{1/8} \approx 1.09!!. This is of course mathematically incoherent, since it predicts that sufficiently old people will have a mortality rate greater than 100%. But a number of things are both true and mathematically incoherent, and this is one of them. (Zipf's law is another.)

The Gravity and Levity blog has a superb article about this from 2009 that reasons backwards from Gompertz' law to rule out certain theories of mortality, such as the theory that death is due to the random whims of a fickle god. (If death were entirely random, and if you had a 50% chance of making it to age 70, then you would have a 25% chance of living to 140, and a 12.5% chance of living to 210, which we know is not the case.)

Gravity and Levity says:

Surprisingly enough, the Gompertz law holds across a large number of countries, time periods, and even different species.

To this list I will add wooden utility poles.

A couple of weeks ago Toph asked me why there were so many old rusty staples embedded in the utility poles near our house, and this is easy to explain: people staple up their yard sale posters and lost-cat flyers, and then the posters and flyers go away and leave behind the staples. (I once went out with a pliers and extracted a few dozen staples from one pole; it was very satisfying but ultimately ineffective.) If new flyer is stapled up each week, that is 52 staples per year, and 1040 in twenty years. If we agree that 20 years is the absolute minimum plausible lifetime of a pole, we should not be surprised if typical poles have hundreds or thousands of staples each.

But this morning I got to wondering what is the expected lifetime of a wooden utility pole? I guessed it was probably in the range of 40 to 70 years. And happily, because of the Wonders of the Internet, I could look it up right then and there, on the way to the trolley stop, and spend my commute time reading about it.

It was not hard to find an authoritative sounding and widely-cited 2012 study by electric utility consultants Quanta Technology.

Summary: Most poles die because of fungal rot, so pole lifetime varies widely depending on the local climate. An unmaintained pole will last 50–60 years in a cold or dry climate and 30-40 years in a hot wet climate. Well-maintained poles will last around twice as long.

Anyway, Gompertz' law holds for wooden utility poles also. According to the study:

Failure and breakdown rates for wood poles are thought to increase exponentially with deterioration and advancing time in service.

The Quanta study presents this chart, taken from the (then forthcoming) 2012 book Aging Power Delivery Infrastructures:

The solid line is the pole failure rate for a particular unnamed utility company in a median climate. The failure rate with increasing age clearly increases exponentially, as Gompertz' law dictates, doubling every 12½ years or so: Around 1 in 200 poles fails at age 50, around 1 in 100 of the remaining poles fails at age 62.5, and around 1 in 50 of the remaining poles fails at age 75.

(The dashed and dotted lines represent poles that are removed from service for other reasons.)

From Gompertz' law itself and a minimum of data, we can extrapolate the maximum human lifespan. The death rate for 65-year-old women is around 1%, and since it doubles every 8 years or so, we find that 50% of women are dead by age 88, and all but the most outlying outliers are dead by age 120. And indeed, the human longevity record is currently attributed to Jeanne Calment, who died in 1997 at the age of 122½.

Similarly we can extrapolate the maximum service time for a wooden utility pole. Half of them make it to 90 years, but if you have a large installed base of 110-year-old poles you will be replacing about one-seventh of them every year and it might make more sense to rip them all out at once and start over. At a rate of one yard sale per week, a 110-year-old pole will have accumulated 5,720 staples.

The Quanta study does not address deterioration of utility poles due to the accumulation of rusty staples.

[Other articles in category /tech] permanent link

Mon, 28 Aug 2017

Miscellanea about 24 puzzles

This is a collection of leftover miscellanea about twenty-four puzzles. In case you forgot what that is:

The puzzle «4 6 7 9 ⇒ 24» means that one should take the numbers 4, 6, 7, and 9, and combine them with the usual arithmetic operations of addition, subtraction, multiplication, and division, to make the number 24. In this case the unique solution is $$6\times\frac{7 + 9}{4}.$$ When the target number is 24, as it often is, we omit it and just write «4 6 7 9».

Prior articles on this topic:

How many puzzles have solutions?

For each value of !!T!!, there are 715 puzzles «a b c d ⇒ T». (I discussed this digression in two more earlier articles: [1] [2].) When the target !!T = 24!!, 466 of the 715 puzzles have solutions. Is this typical? Many solutions of «a b c d» puzzles end with a multiplication of 6 and 4, or of 8 and 3, or sometimes of 12 and 2—so many that one quickly learns to look for these types of solutions right away. When !!T=23!!, there won't be any solutions of this type, and we might expect that relatively few puzzles with prime targets have solutions.

This turns out to be the case:

ALT attribute suggestions welcome!

The x-axis is the target number !!T!!, with 0 at the left, 300 at right, and vertical guide lines every 25. The y axis is the number of solvable puzzles out of the maximum possible of 715, with 0 at the bottom, 715 at the top, and horizontal guide lines every 100.

Dots representing prime number targets are colored black. Dots for numbers with two prime factors (4, 6, 9, 10, 14, 15, 21, 22, etc.) are red; dots with three, four, five, six, and seven prime factors are orange, yellow, green, blue, and purple respectively.

Two countervailing trends are obvious: Puzzles with smaller targets have more solutions, and puzzles with highly-composite targets have more solutions. No target number larger than 24 has as many as 466 solvable puzzles.

These are only trends, not hard rules. For example, there are 156 solvable puzzles with the target 126 (4 prime factors) but only 93 with target 128 (7 prime factors). Why? (I don't know. Maybe because there is some correlation with the number of different prime factors? But 72, 144, and 216 have many solutions, and only two different prime factors.)

The smallest target you can't hit is 417. The following numbers 418 and 419 are also impossible. But there are 8 sets of four digits that can be used to make 416 and 23 sets that can be used to make 420. The largest target that can be hit is obviously !!6561 = 9⁴!!; the largest target with two solutions is !!2916 = 4·9·9·9 = 6·6·9·9!!.

(The raw data are available here).

There is a lot more to discover here. For example, from looking at the chart, it seems that the locally-best target numbers often have the form !!2^n3^m!!. What would we see if we colored the dots according to their largest prime factor instead of according to their number of prime factors? (I tried doing this, and it didn't look like much, but maybe it could have been done better.)

Making zero

As the chart shows, 705 of the 715 puzzles of the type «a b c d ⇒ 0», are solvable. This suggests an interesting inverse puzzle that Toph and I enjoyed: find four digits !!a,b,c, d!! that cannot be used to make zero. (The answers).

Identifying interesting or difficult problems

(Caution: this section contains spoilers for many of the most interesting puzzles.)

I spent quite a while trying to get the computer to rank puzzles by difficulty, with indifferent success.


Seven puzzles require the use of fractions. One of these is the notorious «3 3 8 8» that I mentioned before. This is probably the single hardest of this type. The other six are:

    «1 3 4 6»
    «1 4 5 6»
    «1 5 5 5»
    «1 6 6 8»
    «3 3 7 7»
    «4 4 7 7»

(Solutions to these (formatted image); solutions to these (plain text))

«1 5 5 5» is somewhat easier than the others, but they all follow pretty much the same pattern. The last two are pleasantly symmetrical.

Negative numbers

No puzzles require the use of negative intermediate values. This surprised me at first, but it is not hard to see why. Subexpressions with negative intermediate values can always be rewritten to have positive intermediate values instead.

For instance, !!3 × (9 + (3 - 4))!! can be rewritten as !!3 × (9 - (4 - 3))!! and !!(5 - 8)×(1 -9)!! can be rewritten as !!(8 - 5)×(9 -1)!!.

A digression about tree shapes

In one of the earlier articles I asserted that there are only two possible shapes for the expression trees of a puzzle solution:

(((a # b) # c) # d) ((a # b) # (c # d))
Form A Form B

(Pink square nodes contain operators and green round nodes contain numbers.)

Lindsey Kuper pointed out that there are five possible shapes, not two. Of course, I was aware of this (it is a Catalan number), so what did I mean when I said there were only two? It's because I had the idea that any tree that wasn't already in one of those two forms could be put into form A by using transformations like the ones in the previous section.

For example, the expression !!(4×((1+2)÷3))!! isn't in either form, but we can commute the × to get the equivalent !!((1+2)÷3)×4!!, which has form A. Sometimes one uses the associative laws, for example to turn !!a ÷ (b × c)!! into !!(a ÷ b) ÷ c!!.

But I was mistaken; not every expression can be put into either of these forms. The expression !!(8×(9-(2·3))!! is an example.

Unusual intermediate values

The most interesting thing I tried was to look for puzzles whose solutions require unusual intermediate numbers.

For example, the puzzle «3 4 4 4» looks easy (the other puzzles with just 3s and 4s are all pretty easy) but it is rather tricky because its only solution goes through the unusual intermediate number 28: !!4 × (3 + 4) - 4!!.

I ranked puzzles as follows: each possible intermediate number appears in a certain number of puzzle solutions; this is the score for that intermediate number. (Lower scores are better, because they represent rarer intermediate numbers.) The score for a single expression is the score of its rarest intermediate value. So for example !!4 × (3 + 4) - 4!! has the intermediate values 7 and 28. 7 is extremely common, and 28 is quite unusual, appearing in only 151 solution expressions, so !!4 × (3 + 4) - 4!! receives a fairly low score of 151 because of the intermediate 28.

Then each puzzle received a difficulty score which was the score of its easiest solution expression. For example, «2 2 3 8» has two solutions, one (!!(8+3)×2+2!!) involving the quite unusual intermediate value 22, which has a very good score of only 79. But this puzzle doesn't count as difficult because it also admits the obvious solution !!8·3·\frac22!! and this is the solution that gives it its extremely bad score of 1768.

Under this ranking, the best-scoring twenty-four puzzles, and their scores, were:

      «1 2 7 7» 3
    * «4 4 7 7» 12
    * «1 4 5 6» 13
    * «3 3 7 7» 14
    * «1 5 5 5» 15
      «5 6 6 9» 23
      «2 5 7 9» 24
      «2 2 5 8» 25
      «2 5 8 8» 45
      «5 8 8 8» 45
      «2 2 2 9» 47
    * «1 3 4 6» 59
    * «1 6 6 8» 59
      «2 4 4 9» 151
      «3 4 4 4» 151
    * «3 3 8 8» 152
      «6 8 8 9» 152
      «2 2 2 7» 155
      «2 2 5 7» 155
      «2 3 7 7» 155
      «2 4 7 7» 155
      «2 5 5 7» 155
      «2 5 7 7» 156
      «4 4 8 9» 162

(Something is not quite right here. I think «2 5 7 7» and «2 5 5 7» should have the same score, and I don't know why they don't. But I don't care enough to do it over.)

Most of these are at least a little bit interesting. The seven puzzles that require the use of fractions appear; I have marked them with stars. The top item is «1 2 7 7», whose only solution goes through the extremely rare intermediate number 49. The next items require fractions, and the one after that is «5 6 6 9», which I found difficult. So I think there's some value in this procedure.

But is there enough value? I'm not sure. The last item on the list, «4 4 8 9», goes through the unusual number 36. Nevertheless I don't think it is a hard puzzle.

(I can also imagine that someone might see the answer to «5 6 6 9» right off, but find «4 4 8 9» difficult. The whole exercise is subjective.)

Solutions with unusual tree shapes

I thought about looking for solutions that involved unusual sequences of operations. Division is much less common than the other three operations.

To get it right, one needs to normalize the form of expressions, so that the shapes !!(a + b) + (c + d)!! and !!a + (b + (c + d))!! aren't counted separately. The Ezpr library can help here. But I didn't go that far because the preliminary results weren't encouraging.

There are very few expressions totaling 24 that have the form !!(a÷b)÷(c÷d)!!. But if someone gives you a puzzle with a solution in that form, then !!(a×d)÷(b×c)!! and !!(a×d) ÷ (b÷c)!! are also solutions, and one or another is usually very easy to see. For example, the puzzle «1 3 8 9» has the solution !!(8÷1)÷(3÷9)!!, which has an unusual form. But this is an easy puzzle; someone with even a little experience will find the solution !!8 × \frac93 × 1!! immediately.

Similarly there are relatively few solutions of the form !!a÷((b-c)÷d)!!, but they can all be transformed into !!a×d÷(b-c)!! which is not usually hard to find. Consider $$\frac 8{\left(\frac{6 - 4}6\right)}.$$ This is pretty weird-looking, but when you're trying to solve it one of the first things you might notice is the 8, and then you would try to turn the rest of the digits into a 3 by solving «4 6 6 ⇒ 3», at which point it wouldn't take long to think of !!\frac6{6-4}!!. Or, coming at it from the other direction, you might see the sixes and start looking for a way to make «4 6 8 ⇒ 4», and it wouldn't take long to think of !!\frac8{6-4}!!.

Ezpr shape

Ezprs (see previous article) correspond more closely than abstract syntax trees do with our intuitive notion of how expressions ought to work, so looking at the shape of the Ezpr version of a solution might give better results than looking at the shape of the expression tree. For example, one might look at the number of nodes in the Ezpr or the depth of the Ezpr.


When trying to solve one of these puzzles, there are a few things I always try first. After adding up the four numbers, I then look for ways to make !!8·3, 6·4,!! or !!12·2!!; if that doesn't work I start branching out looking for something of the type !!ab\pm c!!.

Suppose we take a list of all solvable puzzles, and remove all the very easy ones: the puzzles where one of the inputs is zero, or where one of the inputs is 1 and there is a solution of the form !!E×1!!.

Then take the remainder and mark them as “easy” if they have solutions of the form !!a+b+c+d, 8·3, 6·4,!! or !!12·2!!. Also eliminate puzzles with solutions of the type !!E + (c - c)!! or !!E×\left(\frac cc\right)!!.

How many are eliminated in this way? Perhaps most? The remaining puzzles ought to have at least intermediate difficulty, and perhaps examining just those will suggest a way to separate them further into two or three ranks of difficulty.

I give up

But by this time I have solved so many twenty-four puzzles that I am no longer sure which ones are hard and which ones are easy. I suspect that I have seen and tried to solve most of the 466 solvable puzzles; certainly more than half. So my brain is no longer a reliable gauge of which puzzles are hard and which are easy.

Perhaps looking at puzzles with five inputs would work better for me now. These tend to be easy, because you have more to work with. But there are 2002 puzzles and probably some of them are hard.

Close, but no cigar

What's the closest you can get to 24 without hitting it exactly? The best I could do was !!5·5 - \frac89!!. Then I asked the computer, which confirmed that this is optimal, although I felt foolish when I saw the simpler solutions that are equally good: !!6·4 \pm\frac 19!!.

The paired solutions $$5 × \left(4 + \frac79\right) < 24 < 7 × \left(4 - \frac59\right)$$ are very handsome.

Phone app

The search program that tells us when a puzzle has solutions is only useful if we can take it with us in the car and ask it about license plates. A phone app is wanted. I built one with Code Studio.

Code Studio is great. It has a nice web interface, and beginners can write programs by dragging blocks around. It looks very much like MIT's scratch project, which is much better-known. But Code Studio is a much better tool than Scratch. In Scratch, once you reach the limits of what it can do, you are stuck, and there is no escape. In Code Studio when you drag around those blocks you are actually writing JavaScript underneath, and you can click a button and see and edit the underlying JavaScript code you have written.

Suppose you need to convert A to 1 and B to 2 and so on. Scratch does not provide an ord function, so with Scratch you are pretty much out of luck; your only choice is to write a 26-way if-else tree, which means dragging around something like 104 stupid blocks. In Code Studio, you can drop down the the JavaScript level and type in ord to use the standard ord function. Then if you go back to blocks, the ord will look like any other built-in function block.

In Scratch, if you want to use a data structure other than an array, you are out of luck, because that is all there is. In Code Studio, you can drop down to the JavaScript level and use or build any data structure available in JavaScript.

In Scratch, if you want to initialize the program with bulk data, say a precomputed table of the solutions of the 466 twenty-four puzzles, you are out of luck. In Code Studio, you can upload a CSV file with up to 1,000 records, which then becomes available to your program as a data structure.

In summary, you spend a lot of your time in Scratch working around the limitations of Scratch, and what you learn doing that is of very limited applicability. Code Studio is real programming and if it doesn't do exactly what you want out of the box, you can get what you want by learning a little more JavaScript, which is likely to be useful in other contexts for a long time to come.

Once you finish your Code Studio app, you can click a button to send the URL to someone via SMS. They can follow the link in their phone's web browser and then use the app.

Code Studio is what Scratch should have been. Check it out.


Thanks to everyone who contributed to this article, including:

  • my daughters Toph and Katara
  • Shreevatsa R.
  • Dr. Lindsey Kuper
  • Darius Bacon
  • everyone else who emailed me

[Other articles in category /math] permanent link

Mon, 21 Aug 2017

Recognizing when two arithmetic expressions are essentially the same

[ Warning: The math formatting in the RSS / Atom feed for this article is badly mutilated. I suggest you read the article on my blog. ]

In this article, I discuss “twenty-four puzzles”. The puzzle «4 6 7 9 ⇒ 24» means that one should take the numbers 4, 6, 7, and 9, and combine them with the usual arithmetic operations of addition, subtraction, multiplication, and division, to make the number 24. In this case the unique solution is !!6·\frac{7 + 9}{4}!!.

When the target number after the is 24, as it often is, we omit it and just write «4 6 7 9». Every example in this article has target number 24.

This is a continuation of my previous articles on this topic:

My first cut at writing a solver for twenty-four puzzles was a straightforward search program. It had a couple of hacks in it to cut down the search space by recognizing that !!a+E!! and !!E+a!! are the same, but other than that there was nothing special about it and I've discussed it before.

It would quickly and accurately report whether any particular twenty-four puzzle was solvable, but as it turned out that wasn't quite good enough. The original motivation for the program was this: Toph and I play this game in the car. Pennsylvania license plates have three letters and four digits, and if we see a license plate FBV 2259 we try to solve «2 2 5 9». Sometimes we can't find a solution and then we wonder: it is because there isn't one, or is it because we just didn't get it yet? So the searcher turned into a phone app, which would tell us whether there was solution, so we'd know whether to give up or keep searching.

But this wasn't quite good enough either, because after we would find that first solution, say !!2·(5 + 9 - 2)!!, we would wonder: are there any more? And here the program was useless: it would cheerfully report that there were three, so we would rack our brains to find another, fail, ask the program to tell us the answer, and discover to our disgust that the three solutions it had in mind were:

$$ 2 \cdot (5 + (9 - 2)) \\ 2 \cdot (9 + (5 - 2)) \\ 2 \cdot ((5 + 9) - 2) $$

The computer thinks these are different, because it uses different data structures to represent them. It represents them with an abstract syntax tree, which means that each expression is either a single constant, or is a structure comprising an operator and its two operand expressions—always exactly two. The computer understands the three expressions above as having these structures:

It's not hard to imagine that the computer could be taught to understand that the first two trees are equivalent. Getting it to recognize that the third one is also equivalent seems somewhat more difficult.

Commutativity and associativity

I would like the computer to understand that these three expressions should be considered “the same”. But what does “the same” mean? This problem is of a kind I particularly like: we want the computer to do something, but we're not exactly sure what that something is. Some questions are easy to ask but hard to answer, but this is the opposite: the real problem is to decide what question we want to ask. Fun!

Certainly some of the question should involve commutativity and associativity of addition and multiplication. If the only difference between two expressions is that one has !!a + b!! where the other has !!b + a!!, they should be considered the same; similarly !!a + (b + c)!! is the same expression as !!(a + b) + c!! and as !!(b + a) + c!! and !!b + (a + c)!! and so forth.

The «2 2 5 9» example above shows that commutativity and associativity are not limited to addition and multiplication. There are commutative and associative properties of subtraction also! For example, $$a+(b-c) = (a+b)-c$$ and $$(a+b)-c = (a-c)+b.$$ There ought to be names for these laws but as far as I know there aren't. (Sure, it's just commutativity and associativity of addition in disguise, but nobody explaining these laws to school kids ever seems to point out that subtraction can enter into it. They just observe that !!(a-b)-c ≠ a-(b-c)!!, say “subtraction isn't associative”, and leave it at that.)

Closely related to these identities are operator inversion identities like !!a-(b+c) = (a-b)-c!!, !!a-(b-c) = (a-b)+c!!, and their multiplicative analogues. I don't know names for these algebraic laws either.

One way to deal with all of this would to build a complicated comparison function for abstract syntax trees that tried to transform one tree into another by applying these identities. A better approach is to recognize that the data structure is over-specified. If we want the computer to understand that !!(a + b) + c!! and !!a + (b + c)!! are the same expression, we are swimming upstream by using a data structure that was specifically designed to capture the difference between these expressions.

Instead, I invented a data structure, called an Ezpr (“Ez-pur”), that can represent expressions, but in a somewhat more natural way than abstract syntax trees do, and in a way that makes commutativity and associativity transparent.

An Ezpr has a simplest form, called its “canonical” or “normal” form. Two Ezprs represent essentially the same mathematical expression if they have the same canonical form. To decide if two abstract syntax trees are the same, the computer converts them to Ezprs, simplifies them, and checks to see if resulting canonical forms are identical.

The Ezpr

Since associativity doesn't matter, we don't want to represent it. When we (humans) think about adding up a long column of numbers, we don't think about associativity because we don't add them pairwise. Instead we use an addition algorithm that adds them all at once in a big pile. We don't treat addition as a binary operation; we normally treat it as an operator that adds up the numbers in a list. The Ezpr makes this explicit: its addition operator is applied to a list of subexpressions, not to a pair. Both !!a + (b + c)!! and !!(a + b) + c!! are represented as the Ezpr

    SUM [ a b c - ]

which just says that we are adding up !!a!!, !!b!!, and !!c!!. (The - sign is just punctuation; ignore it for now.)

Similarly the Ezpr MUL [ a b c ÷ ] represents the product of !!a!!, !!b!!, and !!c!!. (Please ignore the ÷ sign for the time being.)

To handle commutativity, we want those [ a b c ] lists to be bags. Perl doesn't have a built-in bag object, so instead I used arrays and required that the array elements be in sorted order. (Exactly which sorted order doesn't really matter.)

Subtraction and division

This doesn't yet handle subtraction and division, and the way I chose to handle them is the only part of this that I think is at all clever. A SUM object has not one but two bags, one for the positive and one for the negative part of the expression. An expression like !!a - b + c - d!! is represented by the Ezpr:

SUM [ a c - b d ]

and this is also the representation of !!a + c - b - d!!, of !!c + a - d - b!!, of !!c - d+ a-b!!, and of any other expression of the idea that we are adding up !!a!! and !!c!! and then deducting !!b!! and !!d!!. The - sign separates the terms that are added from those that are subtracted.

Either of the two bags may be empty, so for example !!a + b!! is just SUM [ a b - ].

Division is handled similarly. Here conventional mathematical notation does a little bit better than in the sum case: MUL [ a c ÷ b d ] is usually written as !!\frac{ac}{bd}!!.

Ezprs handle the associativity and commutativity of subtraction and division quite well. I pointed out earlier that subtraction has an associative law !!(a + b) - c = a + (b - c)!! even though it's not usually called that. No code is required to understand that those two expressions are equal if they are represented as Ezprs, because they are represented by completely identical structures:

        SUM [ a b - c ]

Similarly there is a commutative law for subtraction: !!a + b - c = a - c + b!! and once again that same Ezpr does for both.

Ezpr laws

Ezprs are more flexible than binary trees. A binary tree can represent the expressions !!(a+b)+c!! and !!a+(b+c)!! but not the expression !!a+b+c!!. Ezprs can represent all three and it's easy to transform between them. Just as there are rules for building expressions out of simpler expressions, there are a few rules for combining and manipulating Ezprs.

Lifting and flattening

The most important transformation is lifting, which is the Ezpr version of the associative law. In the canonical form of an Ezpr, a SUM node may not have subexpressions that are also SUM nodes. If you have

  SUM [ a SUM [ b c - ] - … ]

you should lift the terms from the inner sum into the outer one:

  SUM [ a b c - … ]

effectively transforming !!a+(b+c)!! into !!a+b+c!!. More generally, in

   SUM [ a SUM [ b - c ]
       - d SUM [ e - f ] ]

we lift the terms from the inner Ezprs into the outer one:

   SUM [ a b f - c d e ]

This effectively transforms !!a + (b - c) - d - (e - f))!! to !!a + b + f - c - d - e!!.

Similarly, when a MUL node contains another MUL, we can flatten the structure.

Say we are converting the expression !!7 ÷ (3 ÷ (6 × 4))!! to an Ezpr. The conversion function is recursive and the naïve version computes this Ezpr:

      MUL [ 7 ÷ MUL [ 3 ÷ MUL [ 6 4 ÷ ] ] ]

But then at the bottom level we have a MUL inside a MUL, so the 4 and 6 in the innermost MUL are lifted upward:

      MUL [ 7 ÷ MUL [ 3 ÷ 6 4 ] ]

which represents !!\frac7{\frac{3}{6\cdot 4}}!!. Then again we have a MUL inside a MUL, and again the subexpressions of the innermost MUL can be lifted:

      MUL [ 7 6 4 ÷ 3 ]

which we can imagine as !!\frac{7·6·4}3!!.

The lifting only occurs when the sub-node has the same type as its parent; we may not lift terms out of a MUL into a SUM or vice versa.

Trivial nodes

The Ezpr SUM [ a - ] says we are adding up just one thing, !!a!!, and so it can be eliminated and replaced with just !!a!!. Similarly SUM [ - a ] can be replaced with the constant !!-a!!, if !!a!! is a constant. MUL can be handled similarly.

An even simpler case is SUM [ - ] which can be replaced by the constant 0; MUL [ ÷ ] can be replaced with 1. These sometimes arise as a result of cancellation.


Consider the puzzle «3 3 4 6». My first solver found 49 solutions to this puzzle. One is !!(3 - 3) + (4 × 6)!!. Another is !!(4 + (3 - 3)) × 6!!. A third is !!4 × (6 + (3 - 3))!!.

I think these are all the same: the solution is to multiply the 4 by the 6, and to get rid of the threes by subtracting them to make a zero term. The zero term can be added onto the rest of expression or to any of its subexpressions—there are ten ways to do this—and it doesn't really matter where.

This is easily explained in terms of Ezprs: If the same subexpression appears in both of a node's bags, we can drop it. For example, the expression !!(4 + (3 -3)) × 6!! starts out as

    MUL [ 6 SUM [ 3 4 - 3 ] ÷ ]

but the duplicate threes in SUM [ 3 4 - 3 ] can be canceled, to leave

    MUL [ 6 SUM [ 4 - ] ÷ ]

The sum is now trivial, as described in the previous section, so can be eliminated and replaced with just 4:

    MUL [ 6 4 ÷ ]

This Ezpr records the essential feature of each of the three solutions to «3 3 4 6» that I mentioned: they all are multiplying the 6 by the 4, and then doing something else unimportant to get rid of the threes.

Another solution to the same puzzle is !!(6 ÷ 3) × (4 × 3)!!. Mathematically we would write this as !!\frac63·4·3!! and we can see this is just !!6×4!! again, with the threes gotten rid of by multiplication and division, instead of by addition and subtraction. When converted to an Ezpr, this expression becomes:

    MUL [ 6 4 3 ÷ 3 ]

and the matching threes in the two bags are cancelled, again leaving

    MUL [ 6 4 ÷ ]

In fact there aren't 49 solutions to this puzzle. There is only one, with 49 trivial variations.

Identity elements

In the preceding example, many of the trivial variations on the !!4×6!! solution involved multiplying some subexpression by !!\frac 33!!. When one of the input numbers in the puzzle is a 1, one can similarly obtain a lot of useless variations by choosing where to multiply the 1.

Consider «1 3 3 5»: We can make 24 from !!3 × (3 + 5)!!. We then have to get rid of the 1, but we can do that by multiplying it onto any of the five subexpressions of !!3 × (3 + 5)!!:

$$ 1 × (3 × (3 + 5)) \\ (1 × 3) × (3 + 5) \\ 3 × (1 × (3 + 5)) \\ 3 × ((1 × 3) + 5) \\ 3 × (3 + (1×5)) $$

These should not be considered different solutions. Whenever we see any 1's in either of the bags of a MUL node, we should eliminate them. The first expression above, !!1 × (3 × (3 + 5))!!, is converted to the Ezpr

 MUL [ 1 3 SUM [ 3 5 - ] ÷ ]

but then the 1 is eliminated from the MUL node leaving

 MUL [ 3 SUM [ 3 5 - ] :- ]

The fourth expression, !!3 × ((1 × 3) + 5)!!, is initially converted to the Ezpr

 MUL [ 3 SUM [ 5 MUL [ 1 3 ÷ ] - ] ÷ ]

When the 1 is eliminated from the inner MUL, this leaves a trivial MUL [ 3 ÷ ] which is then replaced with just 3, leaving:

 MUL [ 3 SUM [ 5 3 - ] ÷ ]

which is the same Ezpr as before.

Zero terms in the bags of a SUM node can similarly be dropped.

Multiplication by zero

One final case is that MUL [ 0 … ÷ … ] can just be simplified to 0.

The question about what to do when there is a zero in the denominator is a bit of a puzzle. In the presence of division by zero, some of our simplification rules are questionable. For example, when we have MUL [ a ÷ MUL [ b ÷ c ] ], the lifting rule says we can simplify this to MUL [ a c ÷ b ]—that is, that !!\frac a{\frac bc} = \frac{ac}b!!. This is correct, except that when !!b=0!! or !!c=0!! it may be nonsense, depending on what else is going on. But since zero denominators never arise in the solution of these puzzles, there is no issue in this application.


The Ezpr module is around 200 lines of Perl code, including everything: the function that converts abstract syntax trees to Ezprs, functions to convert Ezprs to various notations (both MUL [ 4 ÷ SUM [ 3 - 2 ] ] and 4 ÷ (3 - 2)), and the two versions of the normalization process described in the previous section. The normalizer itself is about 35 lines.

Associativity is taken care of by the Ezpr structure itself, and commutativity is not too difficult; as I mentioned, it would have been trivial if Perl had a built-in bag structure. I find it much easier to reason about transformations of Ezprs than abstract syntax trees. Many operations are much simpler; for example the negation of SUM [ A - B ] is simply SUM [ B - A ]. Pretty-printing is also easier because the Ezpr better captures the way we write and think about expressions.

It took me a while to get the normalization tuned properly, but the results have been quite successful, at least for this problem domain. The current puzzle-solving program reports the number of distinct solutions to each puzzle. When it reports two different solutions, they are really different; when it fails to support the exact solution that Toph or I found, it reports one essentially the same. (There are some small exceptions, which I will discuss below.)

Since there is no specification for “essentially the same” there is no hope of automated testing. But we have been using the app for several months looking for mistakes, and we have not found any. If the normalizer failed to recognize that two expressions were essentially similar, we would be very likely to notice: we would be solving some puzzle, be unable to find the last of the solutions that the program claimed to exist, and then when we gave up and saw what it was we would realize that it was essentially the same as one of the solutions we had found. I am pretty confident that there are no errors of this type, but see “Arguable points” below.

A harder error to detect is whether the computer has erroneously conflated two essentially dissimilar expressions. To detect this we would have to notice that an expression was missing from the computer's solution list. I am less confident that nothing like this has occurred, but as the months have gone by I feel better and better about it.

I consider the problem of “how many solutions does this puzzle really have to have?” been satisfactorily solved. There are some edge cases, but I think we have identified them.

Code for my solver is on Github. The Ezpr code is in the Ezpr package in the file. This code is all in the public domain.

Some examples

The original program claims to find 35 different solutions to «4 6 6 6». The revised program recognizes that these are of only two types:

!!4 × 6 × 6 ÷ 6!!MUL [ 4 6 - ]
!!(6 - 4) × (6 + 6)!!MUL [ SUM [ 6 - 4 ] SUM [ 6 6 - ] ÷ ]

Some of the variant forms of the first of those include:

$$ 6 × (4 + (6 - 6)) \\ 6 + ((4 × 6) - 6) \\ (6 - 6) + (4 × 6) \\ (6 ÷ 6) × (4 × 6) \\ 6 ÷ ((6 ÷ 4) ÷ 6) \\ 6 ÷ (6 ÷ (4 × 6)) \\ 6 × (6 × (4 ÷ 6)) \\ (6 × 6) ÷ (6 ÷ 4) \\ 6 ÷ ((6 ÷ 6) ÷ 4) \\ 6 × (6 - (6 - 4)) \\ 6 × (6 ÷ (6 ÷ 4)) \\ \ldots

In an even more extreme case, the original program finds 80 distinct expressions that solve «1 1 4 6», all of which are trivial variations on !!4·6!!.

Of the 715 puzzles, 466 (65%) have solutions; for 175 of these the solution is unique. There are 3 puzzles with 8 solutions each («2 2 4 8», «2 3 6 9», and «2 4 6 8»), one with 9 solutions («2 3 4 6»), and one with 10 solutions («2 4 4 8»).

The 10 solutions for «2 4 4 8» are as follows:

!!4 × 8 - 2 × 4 !!SUM [ MUL [ 4 8 ÷ ] - MUL [ 2 4 ÷ ] ]
!!4 × (2 + 8 - 4) !!MUL [ 4 SUM [ 2 8 - 4 ] ÷ ]
!!(8 - 4) × (2 + 4) !!MUL [ SUM [ 8 - 4 ] SUM [ 2 4 - ] ÷ ]
!!4 × (4 + 8) ÷ 2 !!MUL [ 4 SUM [ 4 8 - ] ÷ 2 ]
!!(4 - 2) × (4 + 8) !!MUL [ SUM [ 4 - 2 ] SUM [ 4 8 - ] ÷ ]
!!8 × (2 + 4/4) !!MUL [ 8 SUM [ 1 2 - ] ÷ ]
!!2 × 4 × 4 - 8 !!SUM [ MUL [ 2 4 4 ÷ ] - 8 ]
!!8 + 2 × (4 + 4) !!SUM [ 8 MUL [ 2 SUM [ 4 4 - ] ÷ ] - ]
!!4 + 4 + 2 × 8 !!SUM [ 4 4 MUL [ 2 8 ÷ ] - ]
!!4 × (8 - 4/2) !!MUL [ 4 SUM [ 8 - MUL [ 4 ÷ 2 ] ] ÷ ]

A complete listing of every essentially different solution to every «a b c d» puzzle is available here. There are 1,063 solutions in all.

Arguable points

There are a few places where we have not completely pinned down what it means for two solutions to be essentially the same; I think there is room for genuine disagreement.

  1. Any solution involving !!2×2!! can be changed into a slightly different solution involving !!2+2!! instead. These expressions are arithmetically different but numerically equal. For example, I mentioned earlier that «2 2 4 8» has 8 solutions. But two of these are !! 8 + 4 × (2 + 2)!! and !! 8 + 4 × 2 × 2!!. I am willing to accept these as essentially different. Toph, however, disagrees.

  2. A similar but more complex situation arises in connection with «1 2 3 7». Consider !!3×7+3!!, which equals 24. To get a solution to «1 2 3 7», we can replace either of the threes in !!3×7+3!! with !!(1+2)!!, obtaining !!((1 + 2) × 7) + 3!! or !! (3×7)+(1 +2)!!. My program considers these to be different solutions. Toph is unsure.

It would be pretty easy to adjust the normalization process to handle these the other way if the user wanted that.

Some interesting puzzles

«1 2 7 7» has only one solution, quite unusual. (Spoiler) «2 2 6 7» has two solutions, both somewhat unusual. (Spoiler)

Somewhat similar to «1 2 7 7» is «3 9 9 9» which also has an unusual solution. But it has two other solutions that are less surprising. (Spoiler)

«1 3 8 9» has an easy solution but also a quite tricky solution. (Spoiler)

One of my neighbors has the license plate JJZ 4631. «4 6 3 1» is one of the more difficult puzzles.

What took so long?

Back in March, I wrote:

I have enough material for at least three or four more articles about this that I hope to publish here in the coming weeks.

But the previous article on this subject ended similarly, saying

I hope to write a longer article about solvers in the next week or so.

and that was in July 2016, so don't hold your breath.

And here we are, five months later!

This article was a huge pain to write. Sometimes I sit down to write something and all that comes out is dreck. I sat down to write this one at least three or four times and it never worked. The tortured Git history bears witness. In the end I had to abandon all my earlier drafts and start over from scratch, writing a fresh outline in an empty file.

But perseverance paid off! WOOOOO.

[ Addendum 20170825: I completely forgot that Shreevatsa R. wrote a very interesting article on the same topic as this one, in July of last year soon after I published my first article in this series. ]

[ Addendum 20170829: A previous version of this article used the notations SUM [ … # … ] and MUL [ … # … ], which I said I didn't like. Zellyn Hunter has persuaded me to replace these with SUM [ … - … ] and MUL [ … ÷ … ]. Thank you M. Hunter! ]

[ Yet more on this topic! ]

[Other articles in category /math] permanent link

Tue, 08 Aug 2017

That time I met Erdős

I should have written about this sooner, by now it has been so long that I have forgotten most of the details.

I first encountered Paul Erdős in the middle 1980s at a talk by János Pach about almost-universal graphs. Consider graphs with a countably infinite set of vertices. Is there a "universal" graph !!G!! such that, for any finite or countable graph !!H!!, there is a copy of !!H!! inside of !!G!!? (Formally, this means that there is an injection from the vertices of !!H!! to the vertices of !!G!! that preserves adjacency.) The answer is yes; it is quite easy to construct such a !!G!! and in fact nearly all random graphs have this property.

But then the questions become more interesting. Let !!K_\omega!! be the complete graph on a countably infinite set of vertices. Say that !!G!! is “almost universal” if it includes a copy of !!H!! for every finite or countable graph !!H!! except those that contain a copy of !!K_\omega!!. Is there an almost universal graph? Perhaps surprisingly, no! (Sketch of proof.)

I enjoyed the talk, and afterward in the lobby I got to meet Ron Graham and Joel Spencer and talk to them about their Ramsey theory book, which I had been reading, and about a problem I was working on. Graham encouraged me to write up my results on the problem and submit them to Mathematics Magazine, but I unfortunately never got around to this. Graham was there babysitting Erdős, who was one of Pách's collaborators, but I did not actually talk to Erdős at that time. I think I didn't recognize him. I don't know why I was able to recognize Graham.

I find the almost-universal graph thing very interesting. It is still an open research area. But none of this was what I was planning to talk about. I will return to the point. A couple of years later Erdős was to speak at the University of Pennsylvania. He had a stock speech for general audiences that I saw him give more than once. Most of the talk would be a description of a lot of interesting problems, the bounties he offered for their solutions, and the progress that had been made on them so far. He would intersperse the discussions with the sort of Erdősism that he was noted for: referring to the U.S. and the U.S.S.R. as “Sam” and “Joe” respectively; his ever-growing series of styles (Paul Erdős, P.G.O.M., A.D., etc.) and so on.

One remark I remember in particular concerned the $3000 bounty he offered for proving what is sometimes known as the Erdős-Túran conjecture: if !!S!! is a subset of the natural numbers, and if !!\sum_{n\in S}\frac 1n!! diverges, then !!S!! contains arbitrarily long arithmetic progressions. (A special case of this is that the primes contain arbitrarily long arithmetic progressions, which was proved in 2004 by Green and Tao, but which at the time was a long-standing conjecture.) Although the $3000 was at the time the largest bounty ever offered by Erdős, he said it was really a bad joke, because to solve the problem would require so much effort that the per-hour payment would be minuscule.

I made a special trip down to Philadelphia to attend the talk, with the intention of visiting my girlfriend at Bryn Mawr afterward. I arrived at the Penn math building early and wandered around the halls to kill time before the talk. And as I passed by an office with an open door, I saw Erdős sitting in the antechamber on a small sofa. So I sat down beside him and started telling him about my favorite graph theory problem.

Many people, preparing to give a talk to a large roomful of strangers, would have found this annoying and intrusive. Some people might not want to talk about graph theory with a passing stranger. But most people are not Paul Erdős, and I think what I did was probably just the right thing; what you don't do is sit next to Erdős and then ask how his flight was and what he thinks of recent politics. We talked about my problem, and to my great regret I don't remember any of the mathematical details of what he said. But he did not know the answer offhand, he was not able solve it instantly, and he did say it was interesting. So! I had a conversation with Erdős about graph theory that was not a waste of his time, and I think I can count that as one of my lifetime accomplishments.

After a little while it was time to go down to the auditorium for the the talk, and afterward one of the organizers saw me, perhaps recognized me from the sofa, and invited me to the guest dinner, which I eagerly accepted. At the dinner, I was thrilled because I secured a seat next to Erdős! But this was a beginner mistake: he fell asleep almost immediately and slept through dinner, which, I learned later, was completely typical.

[Other articles in category /math] permanent link

Sun, 06 Aug 2017

How Shazam works

Yesterday I discussed an interesting failure on the part of Shazam, a phone app that can recognize music by listening to it. I said I had no idea how it worked, but I did not let that stop me from pulling the following vague speculation out of my butt:

I imagine that it does some signal processing to remove background noise, accumulates digests of short sections of the audio data, and then matches these digests against a database of similar digests, compiled in advance from a corpus of recordings.

Julia Evans provided me with the following reference: “An Industrial-Strength Audio Search Algorithm” by Avery Li-Chun Wang of Shazam Entertainment, Ltd. Unfortunately the paper has no date, but on internal evidence it seems to be from around 2002–2006.

M. Evans summarizes the algorithm as follows:

  1. find the strongest frequencies in the music and times at which those frequencies happen
  2. look at pairs !!(freq_1, time_1, freq_2, time_2)!! and turn those into pairs into hashes (by subtracting !!time_1!! from !!time_2!!)
  3. look up those hashes in your database

She continues:

so basically Shazam will only recognize identical recordings of the same piece of music—if it's a different performance the timestamps the frequencies happen at will likely be different and so the hashes won't match

Thanks Julia!

Moving upwards from the link Julia gave me, I found a folder of papers maintained by Dan Ellis, formerly of the Columbia University Electrical Engineering department, founder of Columbia's LabROSA, the Laboratory for the Recognition and Organization of Speech and Audio, and now a Google research scientist.

In the previous article, I asked about research on machine identification of composers or musical genre. Some of M. Ellis’s LabROSA research is closely related to this. See for example:

There is a lot of interesting-looking material available there for free. Check it out.

(Is there a word for when someone gives you a URL like http://host/a/b/c/d.html and you start prying into http://host/a/b/c/ and http://host/a/b/ hoping for more goodies? If not, does anyone have a suggestion?)

[Other articles in category /tech] permanent link

Sat, 05 Aug 2017

Another example of a machine perception failure

IEEE Spectrum has yet another article about fooling computer vision algorithms with subtle changes that humans don't even notice. For more details and references to the literature, see this excellent article by Andrej Karpathy. Here is a frequently-reprinted example:

The classifier is 57.7% confident that the left-hand image is a panda. When the image is perturbed—by less than one part in 140—with the seemingly-random pattern of colored dots to produce the seemingly identical image on the right, the classifier identifies it as a gibbon with 99.3% confidence.

(Illustration from Goodfellow, Shlens, and Szegedy, “Explaining and Harnessing Adversarial Examples”, International Conference on Learning Representations 2015.)

Here's an interesting complementary example that surprised me recently. I have the Shazam app on my phone. When activated, the app tries to listen for music, and then it tries to tell you what the music was. If I'm in the pharmacy and the background music is something I like but don't recognize, I can ask Shazam what it is, and it will tell me. Magic!

Earlier this year I was in the car listening to the radio and I tried this, and it failed. I ran it again, and it failed again. I pulled over to the side of the road, activated the app, and held the phone's microphone up to the car's speaker so that Shazam could hear clearly. Shazam was totally stumped.

So I resumed driving and paid careful attention when the piece ended so that I wouldn't miss when the announcer said what it was. It had been Mendelssohn's fourth symphony.

Shazam can easily identify Mendelssohn's fourth symphony, as I confirmed later. In fact, it can identify it much better than a human can—in some ways. When I tested it, it immediately recognized not only the piece, but the exact recording I used for the test: it was the 1985 recording by the London Symphony Orchestra, conducted by Claudio Abbado.

Why had Shazam failed to recognize the piece on the radio? Too much background noise? Poor Internet connectivity? Nope. It was because the piece was being performed live by the Detroit Symphony Orchestra and as far as Shazam was concerned, it had never heard it before. For a human familiar with Mendelssohn's fourth symphony, this would be of no import. This person would recognize Mendelssohn's fourth symphony whenever it was played by any halfway-competent orchestra.

But Shazam doesn't hear the way people do. I don't know what it does (really I have no idea), but I imagine that it does some signal processing to remove background noise, accumulates digests of short sections of the audio data, and then matches these digests against a database of similar digests, compiled in advance from a corpus of recordings. The Detroit Orchestra's live performance hadn't been in the corpus, so there was no match in the database.

Shazam's corpus has probably a couple of dozen recordings of Mendelssohn's fourth symphony, but it has no idea that all these recordings are of the same piece, or that they sound very similar, because to Shazam they don't sound similar at all. I imagine it doesn't even have a notion of whether two pieces in the corpus sound similar, because it knows them only as distillations of short snatches, and it never compares corpus recordings with one another. Whatever Shazam is doing is completely different from what people do. One might say it hears the sound but not the music, just as the classifier from the Goodfellow paper sees the image but not the panda.

I wonder about a different example. When I hear an unfamiliar piece on the radio, I can often guess who wrote it. “Aha,” I say. “This is obviously Dvořák.” And then more often than not I am right, and even when I am not right, I am usually very close. (For some reasonable meaning of “close” that might be impossible to explain to Shazam.) In one particularly surprising case, I did this with Daft Punk, at that time having heard exactly two Daft Punk songs in my life. Upon hearing this third one, I said to myself “Huh, this sounds just like those Daft Punk songs.” I not claiming a lot of credit for this; Daft Punk has a very distinctive sound. I bring it up just to suggest that whatever magic Shazam is using probably can't do this even a little bit.

Do any of my Gentle Readers know anything about research on the problem of getting a machine to identify the author or genre of music from listening to it?

[ Addendum 20170806: Julia Evans has provided a technical reference and a high-level summary of Shazam's algorithm. This also led me to a trove of related research. ]

[Other articles in category /tech] permanent link

Mon, 31 Jul 2017

Sabotaged by Polish orthography

This weekend my family was doing a bookstore event related to Fantastic Beasts and Where to Find Them. One of the movie's characters, Jacob Kowalski, dreams of becoming a baker, and arrives to a bank appointment with a suitcase full of Polish confections, including pączki, a sort of Polish jelly donut. My wife wanted to serve these at the event.

The little tail on the ą in pączki is a diacritical mark called an ogonek, which is Polish for “little tail”. If I understand correctly, this nasalizes the sound of the a so that it is more like /an/, and furthermore in modern Polish the value of this particular letter has changed so that pączki is pronounced something like “pawnch-kee”. (Polish “cz” is approximately like English “ch”.)

I was delegated to travel to Philadelphia's Polish neighborhood to obtain the pączki. This turned out to be more difficult than I expected. The first address I visited was simply wrong. When I did find the bakery I was looking for, it was sold out of pączki. The bakery across the street was closed, so I started walking down Allegheny Avenue looking for the next bakery.

Before I got there, though, I passed a storefront with a sign listing its goods and services in blue capital letters. One of the items was PACZKI. Properly, of course, this should be PĄCZKI but Poles often omit the ogonek, especially when buying blue letter decals in Philadelphia, where large blue ogoneks are often unavailable. But when I went in to ask I immediately realized that I had probably made a mistake. The store seemed to sell toiletries, paper goods, and souvenirs, with no baked goods in sight.

I asked anyway: “Your sign outside says you sell PĄCZKI?”

“No,” replied the storekeeper. “Pach-kee.”

I thought she was correcting my pronunciation. “But I thought the ogonek made it ‘pawnch-kee’?”

“No, not pawnch-kee. Pach-kee. For sending, to Poland.” She pointed at a box.

I had misunderstood the sign. It did not say PĄCZKI, but PACZKI, which I have since learned means “boxes”.

The storekeeper directed me to the deli across the street, where I was able to buy the pączki. I also bought some interesting-looking cold roast pork loin and asked what it was called. A customer told me it was “po-lend-witsa”, and from this I was able to pick out the price label on the deli case, which said “POLEDWICA”.

After my embarrassment about the boxes I was concerned that I didn't understand ogoneks as well as I thought I did. I pointed to the ‘E’. “Shouldn't there be an ogonek on the ‘E’ here?”

“Yes,” he said, and shrugged. They had left it off, just as I had (incorrectly) thought had happened on the PACZKI sign.

I think the only way to win this one would have been to understand enough of the items in blue capital letters to guess from context that it really was PACZKI and not PĄCZKI.

[ Addendum 20170803: A thirty-year-old mystery has been cleared up! When I was a teenager the news was full of the struggles of the Polish workers’ union Solidarity and its charismatic leader, Lech Walesa, later president of Poland. But his name was always pronounced ‘walensa’. Why? Last night I suddenly understood the mysterious ‘n’: the name was actually ‘Walęsa’! ]

[ (Well, not quite. That does explain the mystery ‘n’. But on looking it up, I find that the name is actually ‘Wałęsa’. The ‘W’ is more like English ‘v’ than like English ‘w’, and the ‘ł’ is apparently very much like English ‘w’. So the correct pronunciation of ‘Wałęsa’ is more like ‘va-wen-sa’ than ‘wa-len-sa’. Perhaps the people who pronounced the ę but not the W or the ł were just being pretentious.) ]

[ Addendum 20170803: Maciej Cegłowski says that “paczki” is more like “packages” than like “boxes”; Google translate suggests “parcels”. He would also like me to remind you that “paczki” and “pączki” are plural, the singulars being “paczka” and “pączek”, respectively. Alicja Raszkowska she loves my use of “ogoneks” (the English plural) in place of the Polish “ogonki”. ]

[Other articles in category /lang] permanent link

Fri, 28 Jul 2017


Neal Stephenson's Seveneves is very fat, so I bought it to read on a long trip this summer. I have mixed feelings about Stephenson, but there are a lot of things I like about his writing. A few years ago I wrote a long review of his “Baroque Cycle” in which I said:

People have been complaining for years that Stephenson's books are "too long". But it seems to me now that the real problem with his earlier books is that they were not long enough.

I am a fan of short books. Usually, I agree with the opinion of Jorge Luis Borges, who said “Writing long books is a laborious and impoverishing act of foolishness: expanding in five hundred pages an idea that could be perfectly explained in a few minutes.”

But Stephenson, I think, is one of very few exceptions who does better writing longer books than shorter ones. I said:

Stephenson at 600 pages is a semi-coherent rambler; to really get what he is saying, you have to turn him up to 2,700 pages.

I was interested to see how that bore out in Seveneves. The good news: Stephenson has learned how to write a good 600-page book. The bad news: Seveneves is 900 pages long.

Seveneves is in three parts of roughly equal size. The first two parts deal with an astronomical catastrophe (never explained) that destroys the moon and renders the earth uninhabitable, and with the efforts of humans to establish a space habitat that will outlive the catastrophe. These first two parts told a story with a beginning and an end. They contain a lot of geeky details about the technical aspects of setting up a space habitat, which I enjoyed. I would gladly read any number of 600-page Stephenson books about space technology, an area in which he is an expert. I said ten years ago that his article Mother Earth, Mother Board about undersea telecommunications cables was brilliant. Ten years on, I'm giving it a promotion: it's one of the best nonfiction essays I've ever read on any topic. If you are one of the people who consider the mass of technical detail in Stephenson's novels to be tedious bloat, I think you probably don't want to read Seveneves. But then, if that's you, you probably gave up on Stephenson a long time ago.

Anyway, the first two parts begin with the destruction of the moon, and end with the establishment of the human space colony. Along the way there are many challenges faced by some fairly interesting characters. Had Stephenson stopped there, I think nobody would have complained.

I realized partway through that he was not going to stop there and I was excited. “Aha!” I said. “The book is in four parts! The first two will deal with the establishment of the colony, and then the last two will take place thousands of years in the future, and deal with the resettlement of Earth.” I was pleased with Stephenson's daring: So many writers would have written just the first two parts, and would not been confident enough to go on. Stephenson has many flaws, but an excess of caution is not one of them, and I was looking forward to parts 3 and 4.

Then something went terribly wrong: He wrote part 3, but not part 4.

At the end of part 2, Seveneves takes all the characters and the world of the first two parts, wipes the blackboard clean and starts over. Which would be fine, if what followed was complete and well-developed. But it is only 300 pages long and Stephenson has never been able to write a 300-page story; Stephenson at 300 pages is a blatherer. The 300 pages contains a lot of implausible-seeming stuff about future technology. In 2006 I said that while I loved his long descriptions of real technologies, I found his descriptions of fanciful technology vacuous:

When they're fictitious technologies and imaginary processes, it's just wankery, a powerful exercise of imagination for no real purpose. Well, maybe the idea will work, and maybe it won't, and it is necessarily too vague to really give you a clear idea of what is going on.

Much of the appeal was gone for me. I can enjoy 600 pages of talk about how people in the 21st century would construct the cheapest possible space habitat. I cannot tolerate with that much material about how Stephenson imagines people in the 71st century might organize their flying cities.

And the plot is just awful. The new characters are one-dimensional, and they spend most of the third part literally doing nothing. They are assembled into a team of seven by a nebulous authority for some secret purpose; neither they nor we are told what it is. They go from place to place to investigate something or other, making several pointless stops and excursions and wondering, as I did, what was going on and when something was actually going to happen. Nothing happens for 250 pages, and then when finally something does happen there is not enough space left in the book to finish it up, and the novel ends in the air, as so many of Stephenson's novels do.

There were several ways this could have been fixed. The whole third part could have gotten the axe. Considered as a 600-page novel, I think the first two parts of Seveneves are excellent. I said before that “Stephenson at 600 pages is a semi-coherent rambler”. That is clearly no longer true.

Or the the third part could have been delayed for a year or two, after Stephenson had first expanded it from 300 to 900 pages and then trimmed it back down to 600 pages. The resulting novel of the 71st century could have been published separately, or the first two parts of Seveneves could have been held back until it was ready; it doesn't matter. In some alternate universe he wrote that second novel and it could be have been really good, even great. The character development might have been better. The mysterious project organizers might have been revealed. We might have gotten some wonderful fish-out-of-water moments with Sonar Taxlaw. (Sonar Taxlaw fanfic, please!) The book could have ended with the characters discovering out what actually happened to the moon back on page 1.

So that's my review: once again, people will say this book's great defect was that it was too long, but actually, the real problem is that it was too short.

I used to hope that Stephenson's editors would take him more firmly in hand and make him write books that started in one place and ended in another, but by now I have given up. It is too late. The books keep selling and at this point nobody is going to mess with success.

Having bought Seveneves because of its fatness, I then decided it was too fat to actually carry around on my trip. Instead I took Yoon-Ha Lee's Ninefox Gambit, which is not fat. But it didn't need to be fat, because instead it was so brilliant that when I finished reading the last page I turned back to the first page and started over, something I don't think I have done in the last thirty years.

I may have something more to say about Ninefox Gambit another time; it fits right into an unfinished article I was writing in 2012 about Stephenson's Anathem and Burgess’ A Clockwork Orange.

[ Addendum: It occurred to me on the bus that that putative four-part novel makes sense in another way. The Seven Eves themselves lie at the exact center of the four-part novel, bridging the transition between the first half and the second half, a structure that perfectly justifies the title's palindromic styling as “Seveneves”. Except no, part 4 is missing and the promised symmetry is spoiled. ]

[Other articles in category /book] permanent link

Mon, 19 Jun 2017

Git's rejected push error

On Saturday I posted an article explaining how remote branches and remote-tracking branches work in Git. That article is a prerequisite for this one. But here's the quick summary:

When dealing with a branch (say, master) copied from a remote repository (say, remote), there are three branches one must consider:
  1. The copy of master in the local repository
  2. The copy of master in the remote repository
  3. The local branch origin/master that records the last known position of the remote branch
Branch 3 is known as a “remote-tracking branch”. This is because it tracks the remote branch, not because it is itself a remote branch. Actually it is a local copy of the remote branch. From now on I will just call it a “tracking branch”.

The git-fetch command (green) copies branch (2) to (3).

The git-push command (red) copies branch (1) to (2), and incidentally updates (3) to match the new (2).

The diagram at right summarizes this.

We will consider the following typical workflow:

  1. Fetch the remote master branch and check it out.
  2. Do some work and commit it on the local master.
  3. Push the new work back to the remote.

But step 3 fails, saying something like:

    ! [rejected]        master -> master (fetch first)
    error: failed to push some refs to '../remote/'
    hint: Updates were rejected because the remote contains work that you do
    hint: not have locally. This is usually caused by another repository pushing
    hint: to the same ref. You may want to first integrate the remote changes
    hint: (e.g., 'git pull ...') before pushing again.
    hint: See the 'Note about fast-forwards' in 'git push --help' for details.

In older versions of Git the hint was a little shorter:

    hint: Updates were rejected because the tip of your current branch is behind
    hint: its remote counterpart. Merge the remote changes (e.g. 'git pull')
    hint: before pushing again.
    hint: See the 'Note about fast-forwards' in 'git push --help' for details.

Everyone at some point gets one of these messages, and in my experience it is one of the most confusing and distressing things for beginners. It cannot be avoided, worked around, or postponed; it must be understood and dealt with.

Not everyone gets a clear explanation. (Reading it over, the actual message seems reasonably clear, but I know many people find it long and frighting and ignore it. It is tough in cases like this to decide how to trade off making the message shorter (and perhaps thereby harder to understand) or longer (and frightening people away). There may be no good solution. But here we are, and I am going to try to explain it myself, with pictures.)

In a large project, the remote branch is always moving, as other people add to it, and they do this without your knowing about it. Immediately after you do the fetch in step 1 above, the tracking branch origin/master reflects the state of the remote branch. Ten seconds later, it may not; someone else may have come along and put some more commits on the remote branch in the interval. This is a fundamental reality that new Git users must internalize.

Typical workflow

We were trying to do this:

  1. Fetch the remote master branch and check it out.
  2. Do some work and commit it on the local master.
  3. Push the new work back to the remote.

and the failure occurred in step 3. Let's look at what each of these operations actually does.

1. Fetch the remote master branch and check it out.

git fetch origin master
git checkout master

The black circles at the top represent some commits that we want to fetch from the remote repository. The fetch copies them to the local repository, and the tracking branch origin/master points to the local copy. Then we check out master and the local branch master also points to the local copy.

Branch names like master or origin/master are called “refs”. At this moment all three refs refer to the same commit (although there are separate copies in the two repositories) and the three branches have identical contents.

2. Do some work and commit it on the local master.

git add …
git commit …

The blue dots on the local master branch are your new commits. This happens entirely inside your local repository and doesn't involve the remote one at all.

But unbeknownst to you, something else is happening where you can't see it. Your collaborators or co-workers are doing their own work in their own repositories, and some of them have published this work to the remote repository. These commits are represented by the red dots in the remote repository. They are there, but you don't know it yet because you haven't looked at the remote repository since they appeared.

3. Push the new work back to the remote.

git push origin master

Here we are trying to push our local master, which means that we are asking the remote repo to overwrite its master with our local one. If the remote repo agreed to this, the red commits would be lost (possibly forever!) and would be completely replaced by the blue commits. The error message that is the subject of this article is Git quite properly refusing to fulfill your request:

    ! [rejected]        master -> master (fetch first)
    error: failed to push some refs to '../remote/'
    hint: Updates were rejected because the remote contains work that you do
    hint: not have locally. This is usually caused by another repository pushing
    hint: to the same ref. You may want to first integrate the remote changes
    hint: (e.g., 'git pull ...') before pushing again.
    hint: See the 'Note about fast-forwards' in 'git push --help' for details.

Let's read through that slowly:

Updates were rejected because the remote contains work that you do not have locally.

This refers specifically to the red commits.

This is usually caused by another repository pushing to the same ref.

In this case, the other repository is your co-worker's repo, not shown in the diagram. They pushed to the same ref (master) before you did.

You may want to first integrate the remote changes (e.g., 'git pull ...') before pushing again.

This is a little vague. There are many ways one could conceivably “integrate the remote changes” and not all of them will solve the problem.

One alternative (which does not integrate the changes) is to use git push -f. The -f is for “force”, and instructs the remote repository that you really do want to discard the red commits in favor of the blue ones. Depending on who owns it and how it is configured, the remote repository may agree to this and discard the red commits, or it may refuse. (And if it does agree, the coworker whose commits you just destroyed may try to feed you poisoned lemonade, so use -f with caution.)

See the 'Note about fast-forwards' in 'git push --help' for details.

To “fast-forward” the remote ref means that your local branch is a direct forward extension of the remote branch, containing everything that the remote branch does, in exactly the same order. If this is the case, overwriting the remote branch with the local branch is perfectly safe. Nothing will be lost or changed, because the local branch contains everything the remote branch already had. The only change will be the addition of new commits at the end.

There are several ways to construct such a local branch, and choosing between them depends on many factors including personal preference, your familiarity with the Git tool set, and the repository owner's policies. Discussing all of this is outside the scope of the article, so I'll just use one as an example: We are going to rebase the blue commits onto the red ones.

4. Refresh the tracking branch.

git fetch origin master

The first thing to do is to copy the red commits into the local repo; we haven't even seen them yet. We do that as before, with git-fetch. This updates the tracking branch with a copy of the remote branch just as it did in step 1.

If instead of git fetch origin master we did git pull --rebase origin master, Git would do exactly the same fetch, and then automatically do a rebase as described in the next section. If we did git pull origin master without --rebase, it would do exactly the same fetch, and then instead of a rebase it would do a merge, which I am not planning to describe. The point to remember is that git pull is just a convenient way to combine the commands of this section and the next one, nothing more.

5. Rewrite the local changes.

git rebase origin/master

Now is the moment when we “integrate the remote changes” with our own changes. One way to do this is git rebase origin/master. This tells Git to try to construct new commits that are just like the blue ones, but instead of starting from the last black commit, the will start from the last red one. (For more details about how this works, see my talk slides about it.) There are many alternatives here to rebase, some quite elaborate, but that is a subject for another article, or several other articles.

If none of the files modified in the blue commits have also been modified in any of the red commits, there is no issue and everything proceeds automatically. And if some of the same files are modified, but only in non-overlapping portions, Git can automatically combine them. But if some of the files are modified in incompatible ways, the rebase process will stop in the middle and asks how to proceed, which is another subject for another article. This article will suppose that the rebase completed automatically. In this case the blue commits have been “rebased onto” the red commits, as in the diagram at right.

The diagram is a bit misleading here: it looks as though those black and red commits appear in two places in the local repository, once on the local master branch and once on the tracking branch. They don't. The two branches share those commits, which are stored only once.

Notice that the command is git rebase origin/master. This is different in form from git fetch origin master or git push origin master. Why a slash instead of a space? Because with git-fetch or git-push, we tell it the name of the remote repo, origin, and the name of the remote branch we want to fetch or push, master. But git-rebase operates locally and has no use for the name of a remote repo. Instead, we give it the name of the branch onto which we want to rebase the new commits. In this case, the target branch is the tracking branch origin/master.

6. Try the push again.

git push origin master

We try the exact same git push origin master that failed in step 3, and this time it succeeds, because this time the operation is a “fast-forward”. Before, our blue commits would have replaced the red commits. But our rewritten local branch does not have that problem: it includes the red commits in exactly the same places as they are already on the remote branch. When the remote repository replaces its master with the one we are pushing, it loses nothing, because the red commits are identical. All it needs to do is to add the blue commits onto the end and then move its master ref forward to point to the last blue commit instead of to the last red commit. This is a “fast-forward”.

At this point, the push is successful, and the git-push command also updates the tracking branch to reflect that the remote branch has moved forward. I did not show this in the illustration.

But wait, what if someone else had added yet more commits to the remote master while we were executing steps 4 and 5? Wouldn't our new push attempt fail just like the first one did? Yes, absolutely! We would have to repeat steps 4 and 5 and try a third time. It is possible, in principle, to be completely prevented from pushing commits to a remote repo because it is always changing so quickly that you never get caught up on its current state. Repeated push failures of this type are sign that the project is large enough that repository's owner needs to set up a more structured code release mechanism than “everyone lands stuff on master whenever they feel like it”.

An earlier draft of this article ended at this point with “That is all I have to say about this.” Ha!

Unavoidable problems

Everyone suffers through this issue at some point or another. It is tempting to wonder if Git couldn't somehow make it easier for people to deal with. I think the answer is no. Git has multiple, distributed repositories. To abandon that feature would be to go back to the dark ages of galley slaves, smallpox, and SVN. But if you have multiple distributed anythings, you must face the issue of how to synchronize them. This is intrinsic to distributed systems: two components receive different updates at the same time, and how do you reconcile them?

For reasons I have discussed before, it does not appear possible to automate the reconciliation in every case in a source code control system, because sometimes the reconciliation may require going over to a co-worker's desk and arguing for two hours, then calling in three managers and the CTO and making a strategic decision which then has to be approved by a representative of the legal department. The VCS is not going to do this for you.

I'm going to digress a bit and then come back to the main point. Twenty-five years ago I taught an introductory programming class in C. The previous curriculum had tried hard to defer pointers to the middle of the semester, as K&R does (chapter 7, I think). I decided this was a mistake. Pointers are everywhere in C and without them you can't call scanf or pass an array to a function (or access the command-line arguments or operate on strings or use most of the standard library or return anything that isn't a number…). Looking back a few years later I wrote:

Pointers are an essential part of [C's] solution to the data hiding problem, which is an essential issue. Therefore, they cannot be avoided, and in fact should be addressed as soon as possible. … They presented themselves in the earliest parts of the material not out of perversity, but because they were central to the topic.

I developed a new curriculum that began treating pointers early on, as early as possible, and which then came back to them repeatedly, each time elaborating on the idea. This was a big success. I am certain that it is the right way to do it.

(And I've been intending since 2006 to write an article about K&R's crappy discussion of pointers and how its deficiencies and omissions have been replicated down the years by generation after generation of C programmers.)

I think there's an important pedagogical principle here. A good teacher makes the subject as simple as possible, but no simpler. Many difficult issues, perhaps most, can be ignored, postponed, hidden, prevaricated, fudged, glossed over, or even solved. But some must be met head-on and dealt with, and for these I think the sooner they are met and dealt with, the better.

Push conflicts in Git, like pointers in C, are not minor or peripheral; they are an intrinsic and central issue. Almost everyone is going to run into push conflicts, not eventually, but right away. They are going to be completely stuck until they have dealt with it, so they had better be prepared to deal with it right away.

If I were to write a book about Git, this discussion would be in chapter 2. Dealing with merge conflicts would be in chapter 3. All the other stuff could wait.

That is all I have to say about this. Thank you for your kind attention, and thanks to Sumana Harihareswara and AJ Jordan for inspiration.

[Other articles in category /prog] permanent link

Sat, 17 Jun 2017

Git remote branches and Git's missing terminology

Beginning and even intermediate Git users have several common problem areas, and one of these is the relationship between remote and local branches. I think the basic confusion is that it seems like there ought to be two things, the remote branch and the local one, and you copy back and forth between them. But there are not two but three, and the Git documentation does not clearly point this out or adopt clear terminology to distinguish between the three.

Let's suppose we have a remote repository, which could be called anything, but is typically named origin. And we have a local repository which has no name; it's just the local repo. And let's suppose we're working on a branch named master, as one often does.

There are not two but three branches of interest, and they might all be pointing to different commits:

  1. The branch named master in the local repo. This is where we do our work and make our commits. This is the local branch. It is at the lower left in the diagram.

  2. The branch named master in the remote repo. This is the remote branch, at the top of the diagram. We cannot normally see this at all because it is (typically) on another computer and (typically) requires a network operation to interact with it. So instead, we mainly deal with…

  3. The branch named origin/master in the local repo. This is the tracking branch, at the lower right in the diagram.

    We never modify the tracking branch ourselves. It is automatically maintained for us by Git. Whenever Git communicates with the remote repo and learns something about the disposition of the remote master branch, it updates the local branch origin/master to reflect what it has learned.

I think this triangle diagram is the first thing one ought to see when starting to deal with remote repositories and with git-fetch and git-push.

The Git documentation often calls the tracking branch the “remote-tracking branch”. It is important to understand that the remote-tracking branch is a local branch in the local repository. It is called the “remote-tracking” branch because it tracks the state of the remote branch, not because it is itself remote. From now on I will just call it the “tracking branch”.

Now let's consider a typical workflow:

  1. We use git fetch origin master. This copies the remote branch master from the remote repo to the tracking branch origin/master in the local repo. This is the green arrow in the diagram.

    If other people have added commits to the remote master branch since our last fetch, now is when we find out what they are. We can compare the local branch master with the tracking branch origin/master to see what is new. We might use git log origin/master to see the new commits, or git diff origin/master to compare the new versions of the files with the ones we had before. These commands do not look at the remote branch! They look at the copy of the remote branch that Git retrieved for us. If a long time elapses between the fetch and the compare, the actual remote branch might be in a completely different place than when we fetched at it.

    (Maybe you use pull instead of fetch. But pull is exactly like fetch except that it does merge or rebase after the fetch completes. So the process is the same; it merely combines this step and the next step into one command. )

  2. We decide how to combine our local master with origin/master. We might use git merge origin/master to merge the two branches, or we might use git rebase origin/master to copy our new local commits onto the commits we just fetched. Or we could use git reset --hard origin/master to throw away our local commits (if any) and just take the ones on the tracking branch. There are a lot of things that could happen here, but the blue arrow in the diagram shows the general idea: we see new stuff in origin/master and update the local master to include that new stuff in some way.

  3. After doing some more work on the local master, we want to publish the new work. We use git push origin master. This is the red arrow in the diagram. It copies the local master to the remote master, updating the remote master in the process. If it is successful, it also updates the tracking branch origin/master to reflect the new position of the remote master.

In the last step, why is there no slash in git push origin master? Because origin/master is the name of the tracking branch, and the tracking branch is not involved. The push command gets two arguments: the name of the remote (origin) and the branch to push (master) and then it copies the local branch to the remote one of the same name.

Deleting a branch

How do we delete branches? For the local branch, it's easy: git branch -d master does it instantly.

For the tracking branch, we include the -r flag: git branch -d -r origin/master. This deletes the tracking branch, and has no effect whatever on the remote repo. This is a very unusual thing to do.

To delete the remote branch, we have to use git-push because that is the only way to affect the remote repo. We use git push origin :master. As is usual with a push, if this is successful Git also deletes the tracking branch origin/master.

This section has glossed over an important point: git branch -d master does not delete the master branch, It only deletes the ref, which is the name for the branch. The branch itself remains. If there are other refs that refer to it, it will remain as long as they do. If there are no other refs that point to it, it will be deleted in due course, but not immediately. Until the branch is actually deleted, its contents can be recovered.


Another way to delete a local ref (whether tracking or not) is just to go into the repository and remove it. The repository is usually in a subdirectory .git of your working tree, and if you cd .git/refs you can see where Git records the branch names and what they refer to. The master branch is nothing more nor less than a file heads/master in this directory, and its contents are the commit ID of the commit to which it refers. If you edit this commit ID, you have pointed the ref at a different commit. If you remove the file, the ref is gone. It is that simple.

Tracking branches are similar. The origin/master ref is in .git/refs/remotes/origin/master.

The remote master branch, of course, is not in your repository at all; it's in the remote repository.

Poking around in Git's repository is fun and rewarding. (If it worries you, make another clone of the repo, poke around in the clone, and throw it away when you are finished poking.) Tinkering with the refs is a good place to start Git repo hacking: create a couple of branches, move them around, examine them, delete them again, all without using git-branch. Git won't know the difference. Bonus fun activity: HEAD is defined by the file .git/HEAD. When you make a new commit, HEAD moves forward. How does that work?

There is a gitrepository-layout manual that says what else you can find in the repository.

Failed pushes

We're now in a good position to understand one of the most common problems that Git beginners face: they have committed some work, and they want to push it to the remote repository, but Git says

      ! [rejected]        master -> master (fetch first)
      error: failed to push some refs to 'remote'
      something something fast-forward, whatever that is

My article explaining this will appear here on Monday. (No, I really mean it.)

Terminology problems

I think one of the reasons this part of Git is so poorly understood is that there's a lack of good terminology in this area. There needs to be a way to say "the local branch named master” and “the branch named master in the remote named origin” without writing a five- or nine-word phrase every time. The name origin/master looks like it might be the second of these, but it isn't. The documentation uses the descriptive but somewhat confusing term “remote-tracking branch” to refer to it. I think abbreviating this to “tracking branch” would tend to clear things up more than otherwise.

I haven't though of a good solution to the rest of it yet. It's tempting to suggest that we should abbreviate “the branch named master in the remote named origin” to something like “origin:master” but I think that would be a disaster. It would be too easy to confuse with origin/master and also with the use of the colon in the refspec arguments to git-push. Maybe something like origin -> master that can't possibly be mistaken for part of a shell command and that looks different enough from origin/master to make clear that it's related but not the same thing.

Git piles yet another confusion on this:

    $ git checkout master 
    Branch master set up to track remote branch master from origin.

This sounds like it has something to with the remote-tracking branch, but it does not! It means that the local branch master has been associated with the remote origin so that fetches and pushes that pertain to it will default to using that remote.

I will think this over and try to come up with something that sucks a little less. Suggestions are welcome.

[Other articles in category /prog] permanent link

Thu, 15 Jun 2017

Base-4 fractions in Telugu

Rik Signes brought to my attention that since version 5.1 Unicode has contained the following excitingly-named characters:



I looked into this a little and found out what they are for. It makes a lot of sense! The details were provided by “Telugu Measures and Arithmetic Marks” by Nāgārjuna Venna.

Telugu is the third-most widely spoken language in India, spoken mostly in the southeast part of the country. Traditional Telugu units of measurement are often divided into four or eight subunits. For example, the tūmu is divided into four kuṁcamulu, the kuṁcamulu, into four mānikalu, and the mānikalu into four sōlalu.

These days they mainly use liters like everyone else. But the traditional measurements are mostly divided into fours, so amounts are written with a base-10 integer part and a base-4 fractional part. The characters above are the base-4 fractional digits.

To make the point clearer, I hope, let's imagine that we are using the Telugu system, but with the familar western-style symbols 0123456789 instead of the Telugu digits ౦౧౨౩౪౫౬౭౮౯. (The Telugu had theirs first of course.) And let's use 0-=Z as our base-four fractional digits, analogous to Telugu ౦౼౽౾. (As in Telugu, we'll use the same zero symbol for both the integer and the fractional parts.) Then to write the number of gallons (7.4805195) in a cubic foot, we say


which is 7 gallons plus one (-) quart plus three (Z) cups plus two (=) quarter-cups plus three (Z) tablespoons plus zero (0) drams, a total of 7660 drams almost exactly. Or we could just round off to 7.=, seven and a half gallons.

(For the benefit of readers who might be a bit rusty on the details of these traditional European measurements, I should mention that there are four drams in a tablespoon, four tablespoons in a quarter cup, four quarter cups in a cup, four cups in a quart, and four quarts in a gallon, so 4⁵ = 1024 drams in a gallon and 7.4805195·4⁵ = 7660.052 drams in a cubic foot. Note also that these are volume (fluid) drams, not mass drams, which are different.)

We can omit the decimal point (as the Telegu did) and write


and it is still clear where the integer part leaves off and the fraction begins, because we are using special symbols for the fractional part. But no, this isn't quite enough, because if we wrote 20ZZ= it might not be clear whether we meant 20.ZZ= or 2.0ZZ=.

So the system has an elaboration. In the odd positions, we don't use the 0-=Z symbols; we use Q|HN instead. And we don't write 7-Z=Z0, we write


This is always unambiguous: 20.ZZ= is actually written 20NZH and 2.0ZZ= is written 2QZN=, quite different.

This is all fanciful in English, but Telugu actually did this. Instead of 0-=Z they had ౦౼౽౾ as I mentioned before. And instead of Q|HN they had ౸౹౺౻. So if the Telugu were trying to write 7.4805195, where we had 7|ZHZQ they might have written ౭౹౾౺౾౸. Like us, they then appended an abbreviation for the unit of measurement. Instead of “gal.” for gallon they might have put ఘ (letter “gha”), so ౭౹౾౺౾౸ఘ. It's all reasonably straightforward, and also quite sensible. If you have ౭౹౾౺ tūmu, you can read off instantly that there are ౺ (two) sōlalu left over, just as you can see that $7.43 has three pennies left over.

Notice that both sets of Telugu fraction digits are easy to remember: the digits for 3 have either three horizonal strokes ౾ or three vertical strokes ౻, and the others similarly.

I have an idea that the alternating vertical-horizontal system might have served as an error-detection mechanism: if a digit is omitted, you notice right away because the next symbol is wrong.

I find this delightful. A few years back I read all of The Number Concept: Its Origin and Development (1931) by Levi Leonard Conant, hoping to learn something really weird, and I was somewhat disappointed. Conant spends most of his book describing the number words and number systems used by dozens of cultures and almost all of them are based on ten, and a few on five or twenty. (“Any number system which passes the limit 10 is reasonably sure to have either a quinary, a decimal, or a vigesimal structure.”) But he does not mention Telugu!

[Other articles in category /math] permanent link

Wed, 07 Jun 2017

Annual self-evaluation time, woo-hoo!

It's annual performance evaluation time at my employer, ZipRecruiter, and as part of that I have to write a self-evaluation. I know many people dread these, and I used to dread them, but these days I like doing it. Instead of being a torture or even a chore, for the last couple of years I have come out of it feeling much better about my job and my performance than I went in.

I think that is about 20% because my company does it in a good way, 30% because it suits my personality, and 50% because I have learned how to handle it well. The first half of that might not help you much, but if you're an evaluation loather, you might be able to transfer some of the second half and make it a little less horrible for yourself.

How ZipRecruiter does self-evaluations

I will get this out of the way because it's quick. ZipRecruiter does these evaluations in a way that works well for me. They do not pester me with a million questions. They ask only four, which are roughly:

  1. What were your main accomplishments this year?
  2. Describe areas you feel require improvement.
  3. What do you hope to accomplish in the coming year?
  4. How can your manager help you?

I very much appreciate this minimalist approach. It gets right to the point, covers all the important issues and nothing more. None of these questions feels to me like a meaningless bureaucratism or a waste of time.

Answering the questions thoroughly takes (only) two or three hours, but would take less if I didn't write such detailed answers; I'm sure I could write an acceptable report in an hour. I can see going in that it will be a finite process.

Why this suits my personality well

If you have followed this blog for a while, you may have noticed that I like writing essays, particularly essays about things I have been working on or thinking about. ZipRecruiter's self-evaluation process invites me to write a blog post about my year's work. This is not everyone's cup of tea, but it is right up my alley. Tea alley. Hee hee.

My brain has problems

My big problems with writing a self-evaluation are first, that I have a very poor memory, and second, that I think of myself as a lazy slacker who spends a lot of time goofing off and who accomplishes very little. These combine badly at evaluation time.

In the past, I would try to remember what work I did in the previous year so I could write it up. My memory is poor, so I wouldn't remember most of what I had done, and then it was easy to come to the conclusion that I had not done very much, probably because I was a lazy slacker whose spent a lot of time goofing off. I would go through several iterations of this, until, tormented by guilt and self-hatred, I would write that into the self-evaluation. This is not a practice I would recommend.

If there were two projects, A and B, and I promptly finished A but B dragged on and was running late, which one would I be more likely to remember when the time came to write the self-evaluation report? B, of course. It was still on my mind because I spent so long thinking about it and because it was still in progress. But I had forgotten about A immediately after putting it to rest. Since I could remember only the unfinished projects, I would conclude that I was a lazy slacker who never finished anything, and write that into the self-evaluation. This is also a a practice I recommend you avoid.

The ticketing system is my bionic brain

The way I have been able to escape this horrible trap is by tracking every piece of work I do, every piece, as a ticket in our ticketing system. People often come to me and ask me to do stuff for them, and I either write up a ticket or I say “sure, write me a ticket”. If they ask why I insist on the ticket (they usually don't), I say it's because when self-evaluation time comes around I want to be able to take credit for working on their problem. Everyone seems to find this reasonable.

Then, when it's time to write the self-evaluation, the first thing I do is visit the ticket website, select all my tickets from the past year, and sort them into chronological order. I look over the list of ticket titles and make a list of stuff that might be worth mentioning on the evaluation. I will have forgotten about three-fourths of it. If I didn't have the list in the ticketing system, I would only remember the most recent one-fourth and conclude that I spent three-fourths of my time goofing off because I am a lazy slacker. Instead, there is this long list of the year's accomplishments, too many to actually include in the report.

Well, this is not rocket science. One is called upon to describe the year's accomplishments. Even people with better memory than me surely cannot remember all this without keeping records, can they? Anyway I surely cannot, so I must keep records and then consult them when the time comes. Put that way, it seems obvious. Why did it take so long to figure out? But there are a lot of happy little details that were not so obvious.

  • Instead of thinking “Why didn't I finish big project X? I must have been goofing off. What a lazy slacker I am” I think “holy cow, I resolved 67 tickets related to big project X! That is great progress! No wonder I got hardly anything else done last fall” and also “holy cow, X has 78 resolved tickets and 23 still open. It is huge! No wonder it is not finished yet.”

    Writing “I completed 67 tickets related to X” is a lot more concrete than “I worked hard on X”. If you are neurotic in the way I am, and concerned that you might be a lazy slacker, it feels much more persuasive. I have an idea that it sounds better to my boss also, particularly if he were to be called upon to discuss it with his manager. (“Under my leadership, Mark completed 67 tickets related to X!”) Andy Lester says that your job is to make your boss happy, and that means making it easy for him to do his job, which is to make his boss happy. So this is no small thing.

  • Instead of thinking “Gee, the CTO declared top priority initiative Y, and while everyone else was working hard on it I mostly ignored it because I am a lazy slacker” I might see that I have tagged 9 tickets “top priority initiative Y”. Then on the report, I proudly boast “I completed 9 tickets in support of the CTO's mandate, including (most important one) and (most impressive one).” This also comes under the heading of “make it easy for your boss to do his job”.

  • Instead of completely forgetting that I did project Z, I see the tickets and can put it in my report.

  • Instead of remembering awful project W, which dragged on for months, and thinking what a lazy slacker I was because I couldn't get it done, I have a progress record in the ticket and the details might suggest a different interpretation: Project W sucked, but I nevertheless pursued it doggedly to completion, even though it took months.

  • I might remember that I once helped Jones, but what did I help him with? Did I really spend much time on him? Without looking at the ticket list, I might not realize that I helped Jones every few weeks all year long. This sort of pattern is often apparent only in the retrospective summary. With the ticket system, instead of “oh, Jones sometimes asks me questions, I guess” I can see that supporting Jones was an ongoing thing and he kept coming back. This goes into the report: “I provided ongoing support to Jones, including (some cherry-picked example that makes me look especially good).”

  • One question (#2) on the report form is “Describe areas you feel require improvement”. If I wrote in last year's report that I would like to improve at doing X, I can look in the ticket system for specific evidence that I might have improved, even if I wasn't consciously trying to improve X at the time. Probably there is something somewhere that can at least be spun as an attempt to improve at X. And if it didn't actually improve X, I can still ask myself why it didn't and what might work instead, and put that in the report as something to try next time, which is question #3.

    Hey, look at that, I am evaluating my prior performance and making informed corrections. That might be a useful practice. Wouldn't it be great if I took time every so often to do that? Some sort of periodic self-evaluation perhaps?

  • Another question (#3) is “What would you like to do in the coming year?” If I wrote in last year's report said “I would like to do more of X” I can look for evidence that I did do that, and then write it into this year's report: “Last year I said I would try to do more of X, and I did.”

  • Even if I were having a bad year and got very little done—and this has happened—having a list of the stuff I did get done leaves me in a much better position to write the report than not having such a list.

None of this good stuff would be possible without an actual record of what I did. If there weren't a ticketing system, I would have to invent one or maybe tattoo it on my chest like the guy in Memento. Even aside from its use in writing annual self-evaluations, keeping a work diary is crucial for me, because without it I can't remember day-to-day what I am working on and what needs to happen next. And even for people with better memory than me, are they really going to remember all 317 things they did for work this year, or that 67 of them pertained to big project X? If they can that's great but I doubt it.

Keeping a work record is part of my job

I think it is important to maintain the correct attitude to this. It would be easy to imagine ticket management as unproductive time that I wasted instead of accomplishing something useful. This is wrong. The correct attitude is to consider ticket updates to be part of my work product: I produce code. I produce bug fixes. I produce documentation, reports, and support interactions. And I also produce ticket updates. This is part of my job and while I am doing it I am not goofing off, I am not procrastinating, I am doing my job and earning my salary. If I spent the whole day doing nothing but updating tickets, that would be a day well-spent.

Compare “I produce ticket updates” with “I produce unit tests”. The attitude for ticket updates is the same as for testing. When something happens in a project, I update the ticket, because keeping the tickets updated is part of the project, just like writing tests is. An organization that fails to support ticket updates is broken in the same way as one that fails to support test development.

My boss gets email every time I update a ticket. I don't know if he reads these, but he has the option to, and I don't need to worry as much that maybe he thinks I am a lazy slacker who is goofing off, because he is getting a stream of automatic notifications about what I am up to. I'm not my boss but if I were I would appreciate this very much.

Maybe some of this can help you?

There might be some use in this even for people who aren't already in the habit of writing self-absorbed blog posts.

If doing the annual self-evaluation makes you suffer, maybe it would help to practice writing some blog posts. You don't have to publish them or show anyone. Next time you finish a project, set aside thirty or sixty minutes to try to write up a little project report: What worked, what didn't, what are you pleased about, what was surprising, what was fun, what was annoying? I'm not certain this will help but it seems like this is a skill that might get easier with practice, and then when you have to write your annual self-evaluation it might be easier because you have more practice doing it. Also, you would have a little file of material to draw on and would not have to start from zero.

If your employer's process requires you to fill in some giant questionnaire, it might be easier to do if you go into it with answers to the four basic questions prepared ahead of time. (I imagine that it's even possible that if you address the four big questions and ignore everything on the giant questionnaire that doesn't pertain to them, everyone will be perfectly satisfied and nobody will notice the omissions.)

And keep a work diary! Tattoo it on your chest if you have to. If it seems daunting, realize that you don't have to do it all at once. Keeping a work diary of some sort is forgiving in the same way as writing unit tests:

  • It's not all-or-nothing, you don't have to test every piece of code to get any benefit from testing. If you write tests for 1% of the code, you get about 1% of the benefit, and you can ramp up.

  • If you break your test-writing streak you don't have to start over from zero. If you didn't write any tests for the code you wrote last week, that's a shame, but it doesn't affect the benefit you can get from writing a unit test for whatever you're working on today.

The work diary is similar. When time comes to write your evaluation, a small and incomplete record is better than no record at all. If you forget to write in the diary for a month, that's a shame, but it doesn't affect the benefit you can get from writing down today what you did today.

Our ticketing system

This isn't important, but I expect someone will want to know: At work we use FogBugz. Some of my co-workers dislike it but I have no major complaints. If you want to try it on your own, they have a seven-day free trial offer, after which you can sign up for a permanent free account that supports up to two users. I am experimenting with using a free tier account to manage my personal to-do list.

Coming soon

I wrote another 2,000 words about my specific practices for managing tickets. I hope it'll be along in a few days.

[Other articles in category /misc] permanent link