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Sun, 02 Apr 2017 A Unix system administrator of my acquaintance once got curious about
what people were putting into The root filesystem quickly filled up and the admin had to be called
back from dinner to fix it. But he found that he couldn't fix it: to
create a Unix device file you use the Having no other system of the same type with an intact device file to
check, he was forced to restore [Other articles in category /Unix] permanent link Sun, 05 Mar 2017
I said I had found an unusually difficult puzzle of this type, which is to make 2,5,6,6 total to 17. This is rather difficult. (I will reveal the solution later in this article.) Several people independently wrote to advise me that it is even more difficult to make 3,3,8,8 total to 24. They were right; it is amazingly difficult. After a couple of weeks I finally gave up and asked the computer, and when I saw the answer I didn't feel bad that I hadn't gotten it myself. (The solution is here if you want to give up without writing a program.) From now on I will abbreviate the two puzzles of the previous paragraph as «2 5 6 6 ⇒ 17» and «3 3 8 8 ⇒ 24», and others similarly. The article also inspired a number of people to write their own solvers and send them to me, and comparing them was interesting. My solver followed the tree search technique that I described in chapter 5 of Higher-Order Perl, and which has become so familiar to me that by now I can implement it without thinking about it very hard:
This is precisely a breadth-first search. To make it into depth-first
search, replace the queue with a stack. To make a heuristically
directed search, replace In my solver, each node contains a list of available expressions, annotated with its numerical value. Initially, the expressions are single numbers and the values are the same, say
Whether you represent expressions as strings or as something more structured depends on what you need to do with them at the end. If you just need to print them out, strings are good enough and are easy to handle. A node represents a successful search if it contains only a single expression and if the expression's value is the target sum, say 24:
From a node, the search should proceed by selecting two of the expressions, removing them from the node, selecting a legal operation, combining the two expressions into a single expression, and inserting the result back into the node. For example, from the initial node shown above, the search might continue by subtracting the fourth expression from the second:
or by multiplying the second and the third:
When the program encounters that first node it will construct both of these, and many others, and put them all into the queue to be investigated later. From
the search might proceed by dividing the first expression by the third:
Then perhaps by subtracting the first from the second:
From here there is no way to proceed, so when this node is removed from the queue, nothing is added to replace it. Had it been a winner, it would have been printed out, but since !!-\frac{35}3!! is not the target value of 24, it is silently discarded. To solve a puzzle of the «a b c d ⇒ t» sort requires examining a few thousand nodes. On modern hardware this takes approximately zero seconds. The actual code for my solver is a lot of Perl gobbledygook that may not be of general interest so I will provide a link for people who are interested in deciphering it. It also represents my second attempt: I lost the code that I described in the earlier article and had to rewrite it. It is rather bigger than I would have liked. Stuff goes wrongPeople showed me a lot of programs to solve this, and many didn't work. There are a few hard cases that several of them get wrong. FractionsSome puzzles require that some subexpressions have fractional values. Many of the programs people showed me used integer arithmetic (sometimes implicitly and unintentionally) and failed to solve those puzzles. We can detect this by asking for a solution to «2 5 6 6 ⇒ 17», which requires a fraction. The solution is !!6×(2+(5÷6))!!. A program using integer arithmetic will calculate !!5÷6 = 0!! and fail to recognize the solution. Several people on Twitter made this mistake and then mistakenly claimed that there was no solution at all. Usually it was possible to correct their programs by changing
to
or something like that. Some people also surprised me by claiming that I had lied when I stated that the puzzle could be solved without any “underhanded tricks”, and that the use of intermediate fractions was itself an underhanded trick. Your Honor, I plead not guilty. I originally described the puzzle this way:
The objectors are implicitly claiming that when you combine 5 and 6 with the “ordinary arithmetic operation” of division, you get something other than !!\frac56!!. This is an indefensible claim. I wasn't even trying to be tricky! It never occurred to me that fractions were something that some people would consider underhanded, and now that it has been suggested, I reject the suggestion. Folks, the result of division can be a fraction. Fractions are not some sort of obscure mathematical pettifoggery. They have been with us for at least 3,500 years now, so it is time everyone got used to them. Floating-point errorSome programs used floating-point arithmetic to deal with the fractions and then fell foul of floating-point error. I will defer discussion of this to a future article. I've complained about floating-point numbers on this blog before. ( 1 2 3 4 5 ) God, how I loathe them. Expression constructionA more subtle error that several programs made was to assume that all expressions can be constructed by combining a previous expression with a single input number. For example, to solve «2 3 5 7 ⇒ 24», you multiply 3 by 7 to get 21, then add 5 to get 26, then subtract 2 to get 24. But not every puzzle can be solved this way. Consider «2 3 5 7 ⇒ 41». You start by multiplying 2 by 3 to get 6, but if you try to combine the 6 with either 5 or 7 at this point you will lose. The only solution is to put the 6 aside and multiply 5 by 7 to get 35. Then add the 6 and the 35 to get 41. Another way to put this is that an unordered binary tree with 4 leaves can take two different shapes. (Imagine filling the green circles with numbers and the pink squares with operators.) The right-hand type of structure is sometimes necessary, as with «2 3 5 7 ⇒ 41». But several of the proposed solutions produced only expressions with structures like that on the left. Here's Sebastian Fischer's otherwise very elegant Haskell solution, in its entirety:
You can see the problem in the last line. Often the way these programs worked was to generate every possible permutation of the inputs and then apply the operators to the input lists stackwise: pop the first two values, combine them, push the result, and repeat. Here's a relevant excerpt from a program by Tim Dierks, this time in Python:
Here the expression structure is implicit, but the current result is always made by combining one of the input numbers with the old result. I have seen many people get caught by this and similar traps in the
past. I once posed the problem of enumerating all the strings of
balanced parentheses of a given length,
and several people assumed that all such strings have the form Division by zeroA less common error exhibited by some programs was a failure to properly deal with division by zero. «2 5 6 6 ⇒ 17» has a solution, and if a program dies while checking !!2+(5÷(6-6))!! and doesn't find the solution, that's a bug. Programs that workedIngo Blechschmidt (Haskell)Ingo Blechschmidt showed me a solution in
Haskell. The code is quite short.
M. Blechschmidt's program defines a synthetic expression type and an
evaluator for it. It defines a function By “synthetic expression type” I mean this:
Probably 80% of the Haskell programs ever written have something like this in them somewhere. This approach has a lot of boilerplate. For example, M. Blechschmidt's program then continues:
Having made up our own synonyms for the arithmetic operators ( I spent a while trying to shorten the code by using a less artificial expression type:
but I was disappointed; I was only able to cut it down by 18%, from 34 lines to 28. I hope to discuss this in a future article. By the way, “Blechschmidt” is German for “tinsmith”. Shreevatsa R. (Python)Shreevatsa R. showed me a solution in Python. It generates every possible expression and prints it out with its value. If you want to filter the voluminous output for a particular target value, you do that later. Shreevatsa wrote up an extensive blog article about this which also includes a discussion about eliminating duplicate expressions from the output. This is a very interesting topic, and I have a lot to say about it, so I will discuss it in a future article. Jeff Fowler (Ruby)Jeff Fowler of the Recurse Center wrote a compact solution in Ruby that he described as “hot garbage”. Did I say something earlier about Perl gobbledygook? It's nice that Ruby is able to match Perl's level of gobbledygookitude. This one seems to get everything right, but it fails mysteriously if I replace the floating-point constants with integer constants. He did provide a version that was not “egregiously minified” but I don't have it handy. Lindsey Kuper (Scheme)Lindsey Kuper wrote a series of solutions in the Racket dialect of Scheme, and discussed them on her blog along with some other people’s work. M. Kuper's first draft was 92 lines long (counting whitespace) and when I saw it I said “Gosh, that is way too much code” and tried writing my own in Scheme. It was about the same size. (My Perl solution is also not significantly smaller.) Martin Janecke (PHP)I saved the best for last. Martin Janecke showed me an almost flawless solution in PHP that uses a completely different approach than anyone else's program. Instead of writing a lot of code for generating permutations of the input, M. Janecke just hardcoded them:
Then three nested loops generate the selections of operators:
Expressions are constructed from templates:
(I don't think those templates are all necessary, but hey, whatever.)
Finally, another set of nested loops matches each ordering of the
input numbers with each selection of operators, uses
If loving this is wrong, I don't want to be right. It certainly satisfies Larry Wall's criterion of solving the problem before your boss fires you. The same approach is possible in most reasonable languages, and some unreasonable ones, but not in Haskell, which was specifically constructed to make this approach as difficult as possible. M. Janecke wrote up a blog article about this, in
German. He says “It's not an elegant
program and PHP is probably not an obvious choice for arithmetic
puzzles, but I think it works.” Indeed it does. Note that the use of
ThanksThanks to everyone who discussed this with me. In addition to the people above, thanks to Stephen Tu, Smylers, Michael Malis, Kyle Littler, Jesse Chen, Darius Bacon, Michael Robert Arntzenius, and anyone else I forgot. (If I forgot you and you want me to add you to this list, please drop me a note.) Coming upI 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
and that was in July 2016, so don't hold your breath. [Other articles in category /math] permanent link Thu, 23 Feb 2017
Miscellaneous notes on anagram scoring
My article on finding the best anagram in English was well-received, and I got a number of interesting comments about it.
[Other articles in category /lang] permanent link Wed, 22 Feb 2017
Moore's law beats a better algorithm
Yesterday I wrote about the project I did in the early 1990s to find the best anagrams. The idea is to give pair of anagram words a score, which is the number of chunks into which you have to divide one word in order to rearrange the chunks to form the other word. This was motivated by the observation that while “cholecysto-duodeno-stomy” and “duodeno-cholecysto-stomy” are very long words that are anagrams of one another, they are not interesting because they require so few chunks that the anagram is obvious. A shorter but much more interesting example is “aspired / diapers”, where the letters get all mixed up. I wrote:
I wrote about the brute-force search yesterday. Today I am going to discuss the clever algorithm. The plan is to convert a pair of anagrams into a graph that expresses
the constraints on how the letters can move around when one turns into
the other. Shown below is the graph for comparing
The “2,4” node at the top means that the letters Usually the graph is much less complicated than this. For simple cases it is empty and the maximal independent set is trivial. This one has two maximal independent sets, one (3,1; 5,5; 6,6; 7,7) corresponding to the obvious minimal splitting: and the other (2,4; 3,5; 5,1; 6,2) to this other equally-good splitting: In an earlier draft of yesterday's post, I wrote:
I was going to leave it at that, but then I did do it over again, and this time around I implemented the “good” algorithm. It was not that hard. The code is on GitHub if you would like to see it. To solve the maximal independent set instances, I used a guided brute-force search. Maximal independent set is NP-complete, and so the best known algorithm for it runs in exponential time. But the instances in which we are interested here are small enough that this doesn't matter. The example graph above has 8 nodes, so one needs to check at most 256 possible sets to see which is the maximal independent set. I collated together all the dictionaries I had handy. (I didn't know yet about SCOWL.) These totaled 275,954 words, which is somewhat more than Webster's Second by itself. One of the new dictionaries did contain crumpets so the result does include “spectrum / crumpets”. The old scored anagram list that I made in the 1990s contained 23,521 pairs. The new one contains 38,333. Unfortunately most of the new stuff is of poor quality, as one would expect. Most of the new words that were missing from my dictionary the first time around are obscure. Perhaps some people would enjoy discovering that that “basiparachromatin” and “Marsipobranchiata” are anagrams, but I find it of very limited appeal. But the new stuff is not all junk. It includes:
which I think are pretty good. I wasn't sure how long the old program had taken to run back in the early nineties, but I was sure it had been at least a couple of hours. The new program processes the 275,954 inputs in about 3.5 seconds. I wished I knew how much of this was due to Moore's law and how much to the improved algorithm, but as I said, the old code was long lost. But then just as I was finishing up the article, I found the old brute-force code that I thought I had lost! I ran it on the same input, and instead of 3.5 seconds it took just over 4 seconds. So almost all of the gain since the 1990s was from Moore's law, and hardly any was from the “improved” algorithm. I had written in the earlier article:
which turned out to be completely true, since implementing the maximal independent set algorithm took me a couple of hours. (Although most of that was building out a graph library because I didn't want to look for one on CPAN.) But hey, at least the new program is only twice as much code! [ Addendum: The program had a minor bug: it would disregard
capitalization when deciding if two words were anagrams, but then
compute the scores with capitals and lowercase letters distinct. So
for example [ Addendum 20170223: More about this ] [Other articles in category /lang] permanent link Tue, 21 Feb 2017
I found the best anagram in English
I planned to publish this last week sometime but then I wrote a line of code with three errors and that took over the blog. A few years ago I mentioned in passing that in the 1990s I had constructed a listing of all the anagrams in Webster's Second International dictionary. (The Webster's headword list was available online.) This was easy to do, even at the time, when the word list itself, at 2.5 megabytes, was a file of significant size. Perl and its cousins were not yet common; in those days I used Awk. But the task is not very different in any reasonable language:
The key technique is to reduce each word to a normal form so that
two words have the same normal form if and only if they are anagrams
of one another. In this case we do this by sorting the letters into
alphabetical order, so that both megalodon and moonglade become
Then we insert the words into a (hash | associative array | dictionary), keyed by their normal forms, and two or more words are anagrams if they fall into the same hash bucket. (There is some discussion of this technique in Higher-Order Perl pages 218–219 and elsewhere.) (The thing you do not want to do is to compute every permutation of the letters of each word, looking for permutations that appear in the word list. That is akin to sorting a list by computing every permutation of the list and looking for the one that is sorted. I wouldn't have mentioned this, but someone on StackExchange actually asked this question.) Anyway, I digress. This article is about how I was unhappy with the results of the simple procedure above. From the Webster's Second list, which contains about 234,000 words, it finds about 14,000 anagram sets (some with more than two words), consisting of 46,351 pairs of anagrams. The list starts with
and ends with
which exemplify the problems with this simple approach: many of the 46,351 anagrams are obvious, uninteresting or even trivial. There must be good ones in the list, but how to find them? I looked in the list to find the longest anagrams, but they were also disappointing:
(Webster's Second contains a large amount of scientific and medical jargon. A cholecystoduodenostomy is a surgical operation to create a channel between the gall bladder (cholecysto-) and the duodenum (duodeno-). A duodenocholecystostomy is the same thing.) This example made clear at least one of the problems with boring anagrams: it's not that they are too short, it's that they are too simple. Cholecystoduodenostomy and duodenocholecystostomy are 22 letters long, but the anagrammatic relation between them is obvious: chop cholecystoduodenostomy into three parts:
and rearrange the first two:
and there you have it. This gave me the idea to score a pair of anagrams according to how many chunks one had to be cut into in order to rearrange it to make the other one. On this plan, the “cholecystoduodenostomy / duodenocholecystostomy” pair would score 3, just barely above the minimum possible score of 2. Something even a tiny bit more interesting, say “abler / blare” would score higher, in this case 4. Even if this strategy didn't lead me directly to the most interesting anagrams, it would be a big step in the right direction, allowing me to eliminate the least interesting. This rule would judge both “aal / ala” and “zolotink / zolotnik” as being uninteresting (scores 2 and 4 respectively), which is a good outcome. Note that some other boring-anagram problems can be seen as special cases of this one. For example, short anagrams never need to be cut into many parts: no four-letter anagrams can score higher than 4. The trivial anagramming of a word to itself always scores 1, and nontrivial anagrams always score more than this. So what we need to do is: for each anagram pair, say
One could do this with a clever algorithm, if one were available. There is a clever algorithm, based on finding maximal independent sets in a certain graph. (More about this tomorrow.) I did not find this algorithm at the time; nor did I try. Instead, I used a brute-force search. Or rather, I used a very small amount of cleverness to reduce the search space, and then used brute-force search to search the reduced space. Let's consider a example, scoring the anagram “abscise / scabies”.
You do not have to consider every possible permutation of
and the second gives us because the To fully analyze Assigning scores in this way produced a scored anagram list which began
and ended
and somewhere in the middle was
all poor scores. But sorted by score, there were treasures at the end, and the clear winner was
14 cinematographer megachiropteran
I declare this the single best anagram in English. It is 15 letters
long, and the only letters that stay together are the And there is no serious competition. There was another 14-pointer, but both its words are Webster's Second jargon that nobody knows:
There are no score 13 pairs, and the score 12 pairs are all obscure. So this is the winner, and a deserving winner it is. I think there is something in the list to make everyone happy. If you are the type of person who enjoys anagrams, the list rewards casual browsing. A few examples:
“Clitoridean / directional” has been one of my favorites for years. But my favorite of all, although it scores only 6, is
I think I might love it just because the word yttrious is so delightful. (What a debt we owe to Ytterby, Sweden!) I also rather like
which shows that even some of the low-scorers can be worth looking at. Clearly my chunk score is not the end of the story, because “notaries / senorita” should score better than “abets / baste” (which is boring) or “Acephali / Phacelia” (whatever those are), also 5-pointers. The length of the words should be worth something, and the familiarity of the words should be worth even more. Here are the results: In former times there was a restaurant in Philadelphia named “Soupmaster”. My best unassisted anagram discovery was noticing that this is an anagram of “mousetraps”. [ Addendum 20170222: There is a followup article comparing the two algorithms I wrote for computing scores. ] [ Addendum 20170222: An earlier version of this article mentioned the putative 11-pointer “endometritria / intermediator”. The word “endometritria” seemed pretty strange, and I did look into it before I published the article, but not carefully enough. When Philip Cohen wrote to me to question it, I investigated more carefully, and discovered that it had been an error in an early WordNet release, corrected (to “endometria”) in version 1.6. I didn't remember that I had used WordNet's word lists, but I am not surprised to discover that I did. ] [ Addendum 20170223: More about this ] [Other articles in category /lang] permanent link Thu, 16 Feb 2017
Automatically checking for syntax errors with Git's pre-commit hook
Previous related article Over the past couple of days I've written about how I committed a syntax error on a cron script, and a co-worker had to fix it on Saturday morning. I observed that I should have remembered to check the script for syntax errors before committing it, and several people wrote to point out to me that this is the sort of thing one should automate. (By the way, please don't try to contact me on Twitter. It won't work. I have been on Twitter Vacation for months and have no current plans to return.) Git has a “pre-commit hook” feature, which means that you can set up a program that will be run every time you attempt a commit, and which can abort the commit if it doesn't like what it sees. This is the natural place to put an automatic syntax check. Some people suggested that it should be part of the CI system, or even the deployment system, but I don't control those, and anyway it is much better to catch this sort of thing as early as possible. I decided to try to implement a pre-commit hook to check syntax. Unlike some of the git hooks, the pre-commit hook is very simple to use. It gets run when you try to make a commit, and the commit is aborted if the hook exits with a nonzero status. I made one mistake right off the bat: I wrote the hook in Bourne shell, even though I swore years ago to stop writing shell scripts. Everything that I want to write in shell should be written in Perl instead or in some equivalently good language like Python. But the sample pre-commit hook was written in shell and when I saw it I went into automatic shell scripting mode and now I have yet another shell script that will have to be replaced with Perl when it gets bigger. I wish I would stop doing this. Here is the hook, which, I should say up front, I have not yet tried in day-to-day use. The complete and current version is on github.
Some of the sample programs people showed me decided which files
needed to be checked based only on the filename. This is not good
enough. My most important Perl programs have filenames with no
extension. This *.py ) echo python; exit ;; is an obvious next step.
This block is an escape hatch. One day I will want to bypass the hook
and make a commit without performing the checks, and then I can
(I am also unlikely to remember
This redirects the standard output of all subsequent commands to go to
standard error instead. It makes it more convenient to issue error
messages with
This invokes the
When a check discovers that the current file is bad, it will signal
this by setting
This is the actual checking. To check Python files, we would add a
If the current file was bad, the After the modified files have been checked, the hook exits successfully if they were all okay, and prints a summary if not:
This hook might be useful, but I don't know yet; as I said, I haven't
really tried it. But I can see ahead of time that it has a couple of
drawbacks. Of course it needs to be built out with more checks. A
minor bug is that I'd like to apply that is-executable check to Perl
files that do not end in But it does have one serious problem I don't know how to fix yet. The hook checks the versions of the files that are in the working tree, but not the versions that are actually staged for the commit! The most obvious problem this might cause is that I might try to commit some files, and then the hook properly fails because the files are broken. Then I fix the files, but forget to add the fixes to the index. But because the hook is looking at the fixed versions in the working tree, the checks pass, and the broken files are committed! A similar sort of problem, but going the other way, is that I might
make several changes to some file, use I did a little tinkering with git stash save -k "pre-commit stash" || exit 2 trap "git stash pop" EXIT but I wasn't able to get anything to work reliably. Stashing a modified index has never worked properly for me, perhaps because there is something I don't understand. Maybe I will get it to work in the future. Or maybe I will try a different method; I can think of several offhand:
Right now the last one looks much the best but perhaps there's something straightforward that I didn't think of yet. [ Thanks to Adam Sjøgren, Jeffrey McClelland, and Jack Vickeridge for discussing this with me. Jeffrey McClelland also suggested that syntax checks could be profitably incorporated as a post-receive hook, which is run on the remote side when new commits are pushed to a remote. I said above that running the checks in the CI process seems too late, but the post-receive hook is earlier and might be just the thing. ] [ Addendum: Daniel Holz wrote to tell me that the Yelp pre-commit frameworkhandles the worrisome case of unstaged working tree changes. The strategy is different from the ones I suggested above. If I'm reading this correctly, it records the unstaged changes in a patch file, which it sticks somewhere, and then checks out the index. If all the checks succeed, it completes the commit and then tries to apply the patch to restore the working tree changes. The checks in Yelp's framework might modify the staged files, and if they do, the patch might not apply; in this case it rolls back the whole commit. Thank you M. Holtz! ] [Other articles in category /prog] permanent link Wed, 15 Feb 2017
More thoughts on a line of code with three errors
Yesterday I wrote, in great irritation, about a line of code I had written that contained three errors. I said:
Afterward, I felt that this was inane, and that the matter required a little more reflection. We do not test every single line of every program we write; in most applications that would be prohibitively expensive, and in this case it would have been excessive. The change I was making was in the format of the diagnostic that the program emitted as it finished to report how long it had taken to run. This is not an essential feature. If the program does its job properly, it is of no real concern if it incorrectly reports how long it took to run. Two of my errors were in the construction of the message. The third, however, was a syntax error that prevented the program from running at all. Having reflected on it a little more, I have decided that I am only really upset about the last one, which necessitated an emergency Saturday-morning repair by a co-worker. It was quite acceptable not to notice ahead of time that the report would be wrong, to notice it the following day, and to fix it then. I would have said “oops” and quietly corrected the code without feeling like an ass. The third problem, however, was serious. And I could have prevented it with a truly minimal amount of effort, just by running:
This would have diagnosed the syntax error, and avoided the main problem at hardly any cost. I think I usually remember to do something like this. Had I done it this time, the modified script would have gone into production, would have run correctly, and then I could have fixed the broken timing calculation on Monday. In the previous article I showed the test program that I wrote to test the time calculation after the program produced the wrong output. I think it was reasonable to postpone writing this until after program ran and produced the wrong output. (The program's behavior in all other respects was correct and unmodified; it was only its report about its running time that was incorrect.) To have written the test ahead of time might be an excess of caution. There has to be a tradeoff between cautious preparation and risk. Here I put everything on the side of risk, even though a tiny amount of caution would have eliminated most of the risk. In my haste, I made a bad trade. [ Addendum 20170216: I am looking into automating the [Other articles in category /prog] permanent link Tue, 14 Feb 2017
How I got three errors into one line of code
At work we had this script that was trying to report how long it had
taken to run, and it was using
This looks plausible, but because
I could explain to you why it does this, but it's not worth your time. I got tired of seeing
I was at some pains to get that first line right, because getting
A co-worker got there before I did and fixed it for me. While he was
there he also fixed the I came in this morning and looked at the cron mail and it said
so I went back to fix the third error, which is that
What can I learn from this? Most obviously, that I should have tested my code before I checked it in. Back in 2013 I wrote:
This was a “just write the tests, fool!” moment if ever there was one. Madame Experience runs an expensive school, but fools will learn in no other. I am not completely incorrigible. I did at least test the fixed code before I checked that in. The test program looks like this:
It was not necessary to commit the test program, but it was necessary to write it and to run it. By the way, the test program failed the first two times I ran it. Three errors in one line isn't even a personal worst. In 2012 I posted here about getting four errors into a one-line program. [ Addendum 20170215: I have some further thoughts on this. ] [Other articles in category /oops] permanent link Tue, 07 Feb 2017
How many 24 puzzles are there?
[ Note: The tables in this article are important, and look unusually crappy if you read this blog through an aggregator. The properly-formatted version on my blog may be easier to follow. ] A few months ago I wrote about puzzles of the following type: take four digits, say 1, 2, 7, 7, and, using only +, -, ×, and ÷, combine them to make the number 24. Since then I have been accumulating more and more material about these puzzles, which will eventually appear here. But meantime here is a delightful tangent. In the course of investigating this I wrote programs to enumerate the solutions of all possible puzzles, and these programs were always much faster than I expected at first. It appears as if there are 10,000 possible puzzles, from «0,0,0,0» through «9,9,9,9». But a moment's thought shows that there are considerably fewer, because, for example, the puzzles «7,2,7,1», «1,2,7,7», «7,7,2,1», and «2,7,7,1» are all the same puzzle. How many puzzles are there really? A back-of-the-envelope estimate is that only about 1 in 24 puzzles is really distinct (because there are typically 24 ways to rearrange the elements of a puzzle) and so there ought to be around !!\frac{10000}{24} \approx 417!! puzzles. This is an undercount, because there are fewer duplicates of many puzzles; for example there are not 24 variations of «1,2,7,7», but only 12. The actual number of puzzles turns out to be 715, which I think is not an obvious thing to guess. Let's write !!S(d,n)!! for the set of sequences of length !!n!! containing up to !!d!! different symbols, with the duplicates removed: when two sequences are the same except for the order of their symbols, we will consider them the same sequence. Or more concretely, we may imagine that the symbols are sorted into nondecreasing order, so that !!S(d,n)!! is the set of nondecreasing sequences of length !!n!! of !!d!! different symbols. Let's also write !!C(d,n)!! for the number of elements of !!S(d,n)!!. Then !!S(10, 4)!! is the set of puzzles where input is four digits. The claim that there are !!715!! such puzzles is just that !!C(10,4) = 715!!. A tabulation of !!C(\cdot,\cdot)!! reveals that it is closely related to binomial coefficients, and indeed that $$C(d,n)=\binom{n+d-1}{d-1}.\tag{$\heartsuit$}$$ so that the surprising !!715!! is actually !!\binom{13}{9}!!. This is not hard to prove by induction, because !!C(\cdot,\cdot)!! is easily shown to obey the same recurrence as !!\binom\cdot\cdot!!: $$C(d,n) = C(d-1,n) + C(d,n-1).\tag{$\spadesuit$}$$ To see this, observe that an element of !!C(d,n)!! either begins with a zero or with some other symbol. If it begins with a zero, there are !!C(d,n-1)!! ways to choose the remaining !!n-1!! symbols in the sequence. But if it begins with one of the other !!d-1!! symbols it cannot contain any zeroes, and what we really have is a length-!!n!! sequence of the symbols !!1\ldots (d-1)!!, of which there are !!C(d-1, n)!!.
Now we can observe that !!\binom74=\binom73!! (they are both 35) so that !!C(5,3) = C(4,4)!!. We might ask if there is a combinatorial proof of this fact, consisting of a natural bijection between !!S(5,3)!! and !!S(4,4)!!. Using the relation !!(\spadesuit)!! we have: $$ \begin{eqnarray} C(4,4) & = & C(3, 4) + & C(4,3) \\ C(5,3) & = & & C(4,3) + C(5,2) \\ \end{eqnarray}$$ so part of the bijection, at least, is clear: There are !!C(4,3)!! elements of !!S(4,4)!! that begin with a zero, and also !!C(4,3)!! elements of !!S(5, 3)!! that do not begin with a zero, so whatever the bijection is, it ought to match up these two subsets of size 20. This is perfectly straightforward; simply match up !!«0, a, b, c»!! (blue) with !!«a+1, b+1, c+1»!! (pink), as shown at right. But finding the other half of the bijection, between !!S(3,4)!! and !!S(5,2)!!, is not so straightforward. (Both have 15 elements, but we are looking for not just any bijection but for one that respects the structure of the elements.) We could apply the recurrence again, to obtain: $$ \begin{eqnarray} C(3,4) & = \color{darkred}{C(2, 4)} + \color{darkblue}{C(3,3)} \\ C(5,2) & = \color{darkblue}{C(4,2)} + \color{darkred}{C(5,1)} \end{eqnarray}$$ and since $$ \begin{eqnarray} \color{darkred}{C(2, 4)} & = \color{darkred}{C(5,1)} \\ \color{darkblue}{C(3,3)} & = \color{darkblue}{C(4,2)} \end{eqnarray}$$ we might expect the bijection to continue in that way, mapping !!\color{darkred}{S(2,4) \leftrightarrow S(5,1)}!! and !!\color{darkblue}{S(3,3) \leftrightarrow S(4,2)}!!. Indeed there is such a bijection, and it is very nice. To find the bijection we will take a detour through bitstrings. There is a natural bijection between !!S(d, n)!! and the bit strings that contain !!d-1!! zeroes and !!n!! ones. Rather than explain it with pseudocode, I will give some examples, which I think will make the point clear. Consider the sequence !!«1, 1, 3, 4»!!. Suppose you are trying to communicate this sequence to a computer. It will ask you the following questions, and you should give the corresponding answers:
At each stage the
computer asks about the identity of the next symbol. If the answer is
“yes” the computer has learned another symbol and moves on to the next
element of the sequence. If it is “no” the computer tries guessing a
different symbol. The “yes” answers become ones and “no”
answers become zeroes, so that the resulting bit string is It sometimes happens that the computer figures out all the elements of the sequence before using up its !!n+d-1!! questions; in this case we pad out the bit string with zeroes, or we can imagine that the computer asks some pointless questions to which the answer is “no”. For example, suppose the sequence is !!«0, 1, 1, 1»!!:
The bit string is We can reverse the process, simply taking over the role of the
computer. To find the sequence that corresponds to the bit string
We have recovered the sequence !!«1, 1, 2, 4»!! from the
bit string This correspondence establishes relation !!(\heartsuit)!! in a different way from before: since there is a natural bijection between !!S(d, n)!! and the bit strings with !!d-1!! zeroes and !!n!! ones, there are certainly !!\binom{n+d-1}{d-1}!! of them as !!(\heartsuit)!! says because there are !!n+d-1!! bits and we may choose any !!d-1!! to be the zeroes. We wanted to see why !!C(5,3) = C(4,4)!!. The detour above shows that there is a simple bijection between
on one hand, and between
on the other hand. And of course the bijection between the two sets of bit strings is completely obvious: just exchange the zeroes and the ones. The table below shows the complete bijection between !!S(4,4)!! and its descriptive bit strings (on the left in blue) and between !!S(5, 3)!! and its descriptive bit strings (on the right in pink) and that the two sets of bit strings are complementary. Furthermore the top portion of the table shows that the !!S(4,3)!! subsets of the two families correspond, as they should—although the correct correspondence is the reverse of the one that was displayed earlier in the article, not the suggested !!«0, a, b, c» \leftrightarrow «a+1, b+1, c+1»!! at all. Instead, in the correct table, the initial digit of the !!S(4,4)!! entry says how many zeroes appear in the !!S(5,3)!! entry, and vice versa; then the increment to the next digit says how many ones, and so forth.
Observe that since !!C(d,n) = \binom{n+d-1}{d-1} = \binom{n+d-1}{n} = C(n+1, d-1)!! we have in general that !!C(d,n) = C(n+1, d-1)!!, which may be surprising. One might have guessed that since !!C(5,3) = C(4,4)!!, the relation was !!C(d,n) = C(d+1, n-1)!! and that !!S(d,n)!! would have the same structure as !!S(d+1, n-1)!!, but it isn't so. The two arguments exchange roles. Following the same path, we can identify many similar ‘coincidences’. For example, there is a simple bijection between the original set of 715 puzzles, which was !!S(10,4)!!, and !!S(5,9)!!, the set of nondecreasing sequences of !!0\ldots 4!! of length !!9!!. [ Thanks to Bence Kodaj for a correction. ] [Other articles in category /math] permanent link Tue, 31 Jan 2017Below, One Liberty Place, the second-tallest building in my home city of Philadelphia. (Completed 1987, height 288 meters.) Below, Zhongtian International Mansion at Fortune Plaza, the tallest building in Ürümqi, capital city of Xinjiang in northwest China. (Completed 2007, height 230 meters.) [ Addendum: Perhaps I should mention that One Liberty Place is itself widely seen as a knockoff of the much more graceful and elegant Chrysler Building in New York City. (Completed 1930, height 319 meters.) ] [ Addendum: I brought this to the attention of GroJLart, the foulmouthed architecture blogger who knows everything, absolutely everything, about Philadelphia buildings, and he said “Thanks. I wrote an article on the same subject in 2011”. Of course. ] [Other articles in category /misc] permanent link Mon, 30 Jan 2017
Digit symbols in the Parshvanatha magic square
In last month's article about the magic square at the Parshvanatha temple, shown at right, I said:
Shreevatsa R. replied in detail, and his reply was so excellent that, finding no way to improve it by adding or taking away, I begged his permission to republish it without change, which he generously granted. Am sending this email to say:
The Parshvanatha temple is located in the current state of Madhya Pradesh. Here is the location of the temple within a map of the state: And here you can see that the above state of Madhya Pradesh (14 in the image below) is adjacent to the state of Gujarat (7): The states of India are (sort of) organized along linguistic lines, and neighbouring states often have overlap or similarities in their languages. So a priori it shouldn't be too surprising if the language is that of a neighbouring state. But, as you rightly say, the location of the Parshvanatha temple is actually quite far from the state (7) where Gujarat is spoken; it's closer to 27 in the above map (state named Uttar Pradesh). Well, the Parshvanatha temple is believed to have been built "during the reign of the Chandela king Dhanga", and the Chandela kings were feudatories (though just beginning to assert sovereignty at the time) of the Gurjara-Pratihara kings, and "Gurjara" is where the name of the language of "Gujarati" comes from. So it's possible that they used the "official" language of the reigning kings, as with colonies. In fact the green area of the Gurjara-Pratihara kings in this map covers the location of the Parshvanatha temple: But actually this is not a very convincing argument, because the link between Gurjara-Pratiharas and modern Gujarati is not too strong (at least I couldn't find it in a few minutes on Wikipedia :P) So moving on... Are the numerals really similar to Gujarati numerals? These are the numbers 1 to 16 from your blog post, ordered according to the usual order:
These are the numerals in a few current Indic scripts (as linked from your blog post): Look at the first two rows above. Perhaps because of my familiarity with Devanagari, I cannot really see any big difference between the Devanagari and Gujarati symbols except for the 9: the differences are as minor as variation between fonts. (To see how much the symbols can change because of font variation, one can go to Google Fonts' Devanagari page and Google Fonts' Gujarati page and click on one of the sample texts and enter "० १ २ ३ ४ ५ ६ ७ ८ ९" and "૦ ૧ ૨ ૩ ૪ ૫ ૬ ૭ ૮ ૯" respectively, then "Apply to all fonts". Some fonts are bad, though.) (In fact, even the Gurmukhi and Tibetan are somewhat recognizable, for someone who can read Devanagari.) So if we decide that the Parshvanatha temple's symbols are actually closer not to modern Gujarati but to modern Devanagari (e.g. the "3" has a tail in the temple symbols which is present in Devanagari but missing in Gujarati), then the mystery disappears: Devanagari is still the script used in the state of Madhya Pradesh (and Uttar Pradesh, etc: it's the script used for Hindi, Marathi, Nepali, Sanskrit, and many other languages). Finally, for the complete answer, we can turn to history. The Parshvanatha temple was built during 950 to 970 CE. Languages: Modern Gujarati dates from 1800, Middle Gujarati from ~1500 to 1800, Old Gujarati from ~1100 to 1500. So the temple is older than the earliest language called "Gujarati". (Similarly, modern Hindi is even more recent.) Turning to scripts instead: see under Brahmic scripts.
So at the time the temple was built, neither Gujarati script nor Devanagari proper existed. The article on the Gujarati script traces its origin to the Devanagari script, which itself is a descendant of Nagari script. At right are the symbols from the Nagari script, which I think are closer in many respects to the temple symbols. So overall, if we trace the numerals in (a subset of) the family tree of scripts:
we'll find that the symbols of the temple are somewhere between the "Nagari" and "Devanagari" forms. (Most of the temple digits are the same as in the "Nagari" example above, except for the 5 which is closer to the Devanagari form.) BTW, your post was about the numerals, but from being able to read modern Devanagari, I can also read some of the words above the square: the first line ends with ".. putra śrī devasarmma" (...पुत्र श्री देवसर्म्म) (Devasharma, son of...), and these words have the top bar which is missing in Gujarati script. [Other articles in category /lang] permanent link |