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Thu, 16 Feb 2017

Automatically checking for syntax errors with Git's pre-commit hook

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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.

    #!/bin/bash

    function typeof () {
        filename=$1
        case $filename in
            *.pl | *.pm) echo perl; exit ;;
        esac

        line1=$(head -1 $1)
        case $line1 in '#!'*perl )
            echo perl; exit ;;
        esac
    }

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 typeof function decides which set of checks to apply to each file, and the minimal demonstration version here can do that based on filename or by looking for the #!...perl line in the first line of the file contents. I expect that this function will expand to include other file types; for example

               *.py ) echo python; exit ;;

is an obvious next step.

    if [ ! -z $COMMIT_OK ]; then
        exit 0;
    fi

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 COMMIT_OK=1 git commit …. There is actually a --no-verify flag to git-commit that will skip the hook entirely, but I am unlikely to remember it.

(I am also unlikely to remember COMMIT_OK=1. But I know from experience that I will guess that I might have put an escape hatch into the hook. I will also guess that there might be a flag to git-commit that does what I want, but that will seem less likely to be true, so I will look in the hook program first. This will be a good move because my hook is much shorter than the git-commit man page. So I will want the escape hatch, I will look for it in the best place, and I will find it. That is worth two lines of code. Sometimes I feel like the guy in Memento. I have not yet resorted to tattooing COMMIT_OK=1 on my chest.)

    exec 1>&2

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 echo and such like. All the output this hook produces is diagnostic, so it is appropriate for it to go to standard error.

    allOK=true
    badFiles=
    for file in $(git diff --cached --name-only | sort) ; do

allOK is true if every file so far has passed its checks. badFiles is a list of files that failed their checks. the git diff --cached --name-only function interrogates the Git index for a list of the files that have been staged for commit.

        type=$(typeof "$file")

This invokes the typeof function from above to decide the type of the current file.

        BAD=false

When a check discovers that the current file is bad, it will signal this by setting BAD to true.

        echo
        echo "##  Checking file $file (type $type)"
        case $type in
            perl )
                perl -cw $file || BAD=true
                [ -x $file ] || { echo "File is not executable"; BAD=true; }
                ;;
            * )
                echo "Unknown file type: $file; no checks"
                ;;
        esac

This is the actual checking. To check Python files, we would add a python) … ;; block here. The * ) case is a catchall. The perl checks run perl -cw, which does syntax checking without executing the program. It then checks to make sure the file is executable, which I am sure is a mistake, because these checks are run for .pm files, which are not normally supposed to be executable. But I wanted to test it with more than one kind of check.

        if $BAD; then
            allOK=false;
            badFiles="$badFiles;$file"
        fi
    done

If the current file was bad, the allOK flag is set false, and the commit will be aborted. The current filename is appended to badFiles for a later report. Bash has array variables but I don't remember how they work and the manual made it sound gross. Already I regret not writing this in a real language.

After the modified files have been checked, the hook exits successfully if they were all okay, and prints a summary if not:

    if $allOK; then
        exit 0;
    else
        echo ''
        echo '## Aborting commit.  Failed checks:'
        for file in $(echo $badFiles | tr ';' ' '); do
            echo "    $file"
        done
        exit 1;
    fi

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 .pm, but that will be an easy fix.

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 git add -p to add the part I am ready to commit, but then the commit hook fails, even though the commit would be correct, because the incomplete changes are still in the working tree.

I did a little tinkering with git stash save -k to try to stash the unstaged changes before running the checks, something like this:

        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:

  • The hook could copy each file to a temporary file and then run the check on the temporary file. But then the diagnostics emitted by the checks would contain the wrong filenames.

  • It could move each file out of the way, check out the currently-staged version of the file, check that, and then restore the working tree version. (It can skip this process for files where the staged and working versions are identical.) This is not too complicated, but if it messes up it could catastrophically destroy the unstaged changes in the working tree.

  • Check out the entire repository and modified index into a fresh working tree and check that, then discard the temporary working tree. This is probably too expensive.

  • This one is kind of weird. It could temporarily commit the current index (using --no-verify), stash the working tree changes, and check the files. When the checks are finished, it would unstash the working tree changes, use git-reset --soft to undo the temporary commit, and proceed with the real commit if appropriate.

  • Come to think of it, this last one suggests a much better version of the same thing: instead of a pre-commit hook, use a post-commit hook. The post-commit hook will stash any leftover working tree changes, check the committed versions of the files, unstash the changes, and, if the checks failed, undo the commit with git-reset --soft.

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! ]


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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:

What can I learn from this? Most obviously, that I should have tested my code before I checked it in.

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:

    perl -cw the-script

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 perl -cw check. ]


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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 DateTime::Duration:

    my $duration = $end_time->subtract_datetime($start_time);
    my ( $hours, $minutes, $seconds ) =
    $duration->in_units( 'hours', 'minutes', 'seconds' );

    log_info "it took $hours hours $minutes minutes and $seconds seconds to run"

This looks plausible, but because DateTime::Duration is shit, it didn't work. Typical output:

    it took 0 hours 263 minutes and 19 seconds to run

I could explain to you why it does this, but it's not worth your time.

I got tired of seeing 0 hours 263 minutes show up in my cron email every morning, so I went to fix it. Here's what I changed it to:

    my $duration = $end_time->subtract_datetime_absolute($start_time)->seconds;
    my ( $hours, $minutes, $minutes ) = (int(duration/3600), int($duration/60)%60, $duration%3600);

I was at some pains to get that first line right, because getting DateTime to produce a useful time interval value is a tricky proposition. I did get the first line right. But the second line is just simple arithmetic, I have written it several times before, so I dashed it off, and it contains a syntax error, that duration/3600 is missing its dollar sign, which caused the cron job to crash the next day.

A co-worker got there before I did and fixed it for me. While he was there he also fixed the $hours, $minutes, $minutes that should have been $hours, $minutes, $seconds.

I came in this morning and looked at the cron mail and it said

    it took 4 hours 23 minutes and 1399 seconds to run

so I went back to fix the third error, which is that $duration%3600 should have been $duration%60. The thrice-corrected line has

    my ( $hours, $minutes, $seconds ) = (int($duration/3600), int($duration/60)%60, $duration%60);

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:

Usually I like to draw some larger lesson from this sort of thing. … “Just write the tests, fool!”

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:

    sub dur {
      my $duration = shift;
      my ($hours, $minutes, $seconds ) = (int($duration/3600), int($duration/60)%60, $duration%60);
      sprintf  "%d:%02d:%02d", $hours, $minutes, $seconds;
    }

    use Test::More;
    is(dur(0),  "0:00:00");
    is(dur(1),  "0:00:01");
    is(dur(59), "0:00:59");
    is(dur(60), "0:01:00");
    is(dur(62), "0:01:02");
    is(dur(122), "0:02:02");
    is(dur(3599), "0:59:59");
    is(dur(3600), "1:00:00");
    is(dur(10000), "2:46:40");
    done_testing();

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. ]


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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)!!.

0 0 0 0 1 1 1
0 0 0 1 1 1 2
0 0 0 2 1 1 3
0 0 0 3 1 1 4
0 0 1 1 1 2 2
0 0 1 2 1 2 3
0 0 1 3 1 2 4
0 0 2 2 1 3 3
0 0 2 3 1 3 4
0 0 3 3 1 4 4
0 1 1 1 2 2 2
0 1 1 2 2 2 3
0 1 1 3 2 2 4
0 1 2 2 2 3 3
0 1 2 3 2 3 4
0 1 3 3 2 4 4
0 2 2 2 3 3 3
0 2 2 3 3 3 4
0 2 3 3 3 4 4
0 3 3 3 4 4 4

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:

  • “Is the first symbol 0?” (“No”)
  • “Is the first symbol 1?” (“Yes”)
  • “Is the second symbol 1?” (“Yes”)
  • “Is the third symbol 1?” (“No”)
  • “Is the third symbol 2?” (“No”)
  • “Is the third symbol 3?” (“Yes”)
  • “Is the fourth symbol 3?” (“No”)
  • “Is the fourth symbol 4?” (“Yes”)

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 0 1 1 0 0 1 0 1.

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»!!:

  • “Is the first symbol 0?” (“Yes”)
  • “Is the second symbol 0?” (“No”)
  • “Is the second symbol 1?” (“Yes”)
  • “Is the third symbol 1?” (“Yes”)
  • “Is the fourth symbol 1?” (“Yes”)

The bit string is 1 0 1 1 1 0 0 0, where the final three 0 bits are the padding.

We can reverse the process, simply taking over the role of the computer. To find the sequence that corresponds to the bit string 0 1 1 0 1 0 0 1, we ask the questions ourselves and use the bits as the answers:

  • “Is the first symbol 0?” (“No”)
  • “Is the first symbol 1?” (“Yes”)
  • “Is the second symbol 1?” (“Yes”)
  • “Is the third symbol 1?” (“No”)
  • “Is the third symbol 2?” (“Yes”)
  • “Is the fourth symbol 2?” (“No”)
  • “Is the fourth symbol 3?” (“No”)
  • “Is the fourth symbol 4?” (“Yes”)

We have recovered the sequence !!«1, 1, 2, 4»!! from the bit string 0 1 1 0 1 0 0 1.

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

!!S(5,3)!! and the bit strings with 4 zeroes and 3 ones

on one hand, and between

!!S(4,4)!! and the bit strings with 3 zeroes and 4 ones

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.

!!S(4,4)!!(bits)(complement bits)!!S(5,3)!!
0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 4 4 4
0 0 0 1 1 1 1 0 1 0 0 0 0 0 1 0 1 1 3 4 4
0 0 0 2 1 1 1 0 0 1 0 0 0 0 1 1 0 1 3 3 4
0 0 0 3 1 1 1 0 0 0 1 0 0 0 1 1 1 0 3 3 3
0 0 1 1 1 1 0 1 1 0 0 0 0 1 0 0 1 1 2 4 4
0 0 1 2 1 1 0 1 0 1 0 0 0 1 0 1 0 1 2 3 4
0 0 1 3 1 1 0 1 0 0 1 0 0 1 0 1 1 0 2 3 3
0 0 2 2 1 1 0 0 1 1 0 0 0 1 1 0 0 1 2 2 4
0 0 2 3 1 1 0 0 1 0 1 0 0 1 1 0 1 0 2 2 3
0 0 3 3 1 1 0 0 0 1 1 0 0 1 1 1 0 0 2 2 2
0 1 1 1 1 0 1 1 1 0 0 0 1 0 0 0 1 1 1 4 4
0 1 1 2 1 0 1 1 0 1 0 0 1 0 0 1 0 1 1 3 4
0 1 1 3 1 0 1 1 0 0 1 0 1 0 0 1 1 0 1 3 3
0 1 2 2 1 0 1 0 1 1 0 0 1 0 1 0 0 1 1 2 4
0 1 2 3 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 2 3
0 1 3 3 1 0 1 0 0 1 1 0 1 0 1 1 0 0 1 2 2
0 2 2 2 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 4
0 2 2 3 1 0 0 1 1 0 1 0 1 1 0 0 1 0 1 1 3
0 2 3 3 1 0 0 1 0 1 1 0 1 1 0 1 0 0 1 1 2
0 3 3 3 1 0 0 0 1 1 1 0 1 1 1 0 0 0 1 1 1
1 1 1 1 0 1 1 1 1 0 0 1 0 0 0 0 1 1 0 4 4
1 1 1 2 0 1 1 1 0 1 0 1 0 0 0 1 0 1 0 3 4
1 1 1 3 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 3 3
1 1 2 2 0 1 1 0 1 1 0 1 0 0 1 0 0 1 0 2 4
1 1 2 3 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 2 3
1 1 3 3 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 2 2
1 2 2 2 0 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 4
1 2 2 3 0 1 0 1 1 0 1 1 0 1 0 0 1 0 0 1 3
1 2 3 3 0 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 2
1 3 3 3 0 1 0 0 1 1 1 1 0 1 1 0 0 0 0 1 1
2 2 2 2 0 0 1 1 1 1 0 1 1 0 0 0 0 1 0 0 4
2 2 2 3 0 0 1 1 1 0 1 1 1 0 0 0 1 0 0 0 3
2 2 3 3 0 0 1 1 0 1 1 1 1 0 0 1 0 0 0 0 2
2 3 3 3 0 0 1 0 1 1 1 1 1 0 1 0 0 0 0 0 1
3 3 3 3 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0

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. ]


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