In Gödel, Escher, Bach, Douglas Hofstadter lays out how self-reference underpins some staggering philosophical truths. Relatedly, it’s satisfying and funny. Pangrams - sentence containing every letter - are also satisfying, and though perhaps I never actually LOL’d at “the quick brown fox jumps over the lazy dog”, I think they’re sort of funny (maybe they are a weak form of Tom 7’s “improper hierarchy”).

So this, a sentence in Douglas Hofstadter’s later book I am a Strange Loop, is pretty captivating:

This pangram tallies five a’s, one b, one c, two d’s, twenty-eight e’s, eight f’s, six g’s, eight h’s, thirteen i’s, one j, one k, three l’s, two m’s, eighteen n’s, fifteen o’s, two p’s, one q, seven r’s, twenty-five s’s, twenty two t’s, four u’s, four v’s, nine w’s, two x’s, four y’s, and one z.

A self-descriptive sentence like this is called an autogram. Coming up with any old self-referential pangram isn’t hard (although this ersatz pangram’s quotient of glyphs wants six), but a pangrammatic autogram is very tricky. Try inventing such a sentence from scratch naïvely, and this is how far you get:

This sentence contains two a’s, one b, three c’s, one d,

Now how many e’s will you tally? There are six so far, so once you’ve added your tally there will be seven. But “seven” contains two extra e’s! The self-reference in such a pangram is clearly much more tangled than this sentence’s own simple narcissism.

Here’s a widget that renders these sentences with any falsehoods highlighted in red:

Wait a second - Hofstadter’s pangram is wrong! It only contains twenty-one t’s! See if you can fix it by clicking the little ‘+’s and ‘-s’ (are they too little for phone screens? Sorry. Web design is hard. Why aren’t you reading this on a proper computer, anyway? Why are your fingers so fat? It’s not my fault) to adjust the letter counts. I can’t make it right. This may be funny but it is not satisfying.

So, how do we go about generating them? The naïve approach above can get us surprisingly far if we are careful about the order in which we tally the letters. This sentence is correct:

This pangram tallies one b, one c, five a’s, two d’s, one j, one k, two m’s, two p’s, one q, one z, one y, two g’s, three l’s, one u, five w’s, thirteen o’s, three f’s, three v’s, six h’s, and six r’s.

We’ve tallied twenty letters here but we’ll always get stuck at about this point. Suppose we somehow managed to tally twenty five letters by this method. To add the final tally, we would need to use letters that have already been tallied, so the sentence would become false.

Do you hate computers? If you would like to see yours suffer, you can watch it trying to complete this impossible task here:

start stop step

In theory it will admit defeat eventually, but if my calculations are correct, its stupid metal brain will oxidise into dust before that happens.

One way to think about the difficulty of generating pangram-autograms is to compare it to a slightly simpler problem: what if we don’t care if the individual letter counts are correct, just that they add up to the right total? In that case, the only thing that matters about the phrase “one a” is that it tallies one letter, but contains four. It doesn’t matter what those four letters are. Generating these pseudo-autograms turns out to be a variant of the “subset sum” problem. This is an NP-complete problem, which is a fancy way of saying it’s “pretty tricky”, but it can be solved quite speedily for the values we’re interested in here. Before writing this article, I thought I could just write some code to generate these pseudo-autograms, then test loads of them until I found a true autogram among them. But the number of pseudo-autograms is vaster than I thought (I made the rookie mistake of underestimating the factorial function!) and proper autograms are a very slim subset of pseudo-autograms. My little laptop would take far too long.

After realising this, I had a look online to see if anyone has come up with a solution, and it turns out Chris Patuzzo has found rather a good one. He describes it rather well in this podcast so in the interest of variety I’ll try to describe it in the opposite direction from his.

One of the best known problems in computer science is “Boolean satisfiability”, or “SAT” for short. Suppose I tell you “either I am a mongoose, or I am speaking”. This might be true, if I was truly a mongoose, or if I was truly speaking aloud. But if I tell you “I am a mongoose, and mongeese cannot speak, and I am speaking” then you don’t need any evidence to infer that I’m lying. If I was truly a mongoose making the claim aloud, then my assertion of the muteness of mongeese would stand on shaky ground indeed. If mongeese are really speechless and you heard me speaking, you would surely doubt my mongoosehood. If I was indeed a mongoose, which was truly a nonverbal creature, it would not be possible for me to speak. SAT, then, is the problem of detecting unsatisfiable assertions like this, which can never be true. As a bonus, if the claim can be true, a SAT-solver can tell you the situations where it would be. When given “Either I am a mongoose, or I am speaking”, a SAT-solver would spit out two valid situations:

You are indeed a mongoose, but you are not speaking

You are indeed speaking, but you are not a mongoose

Such a simplistic example may not make it clear, but a SAT-solver is quite a spectacular thing, because it turns logical puzzles inside out. Chris Patuzzo figured out a way of feeding the claim “there is a pangram-autogram” into a SAT-solver, and the SAT-solver spat out situations where that claim is true i.e. pangrammatic sentences. I have of course been glossing over the details; SAT-solvers do not really understand English, you feed them formal logical expressions, and they spit out lists of all the combinations of truth and falsehood that can make your overall expression true. The logical expression that I’m glossing over when I write “I am a mongoose” is very simple, it contains a single unit of truth or falsity (i.e. my mongoosehood). But if the formal expression that we imagined for the assertion “there is a pangrammatic autogram” was so simple, then the SAT-solver would simply tell us our assertion is true as long a “there is indeed a pangrammatic autogram”. This is not much use. Patuzzo’s work was to express “there is a pangrammatic autogram” in such specific terms that the SAT-solver had no choice but to spit out everything we need to know to write out the actual sentence.

I would love to harp on about this until I feel I’ve really explained, but I think it would take too long. One more analogy: this technique means that instead of having to answer the question “how do I find pangrammatic autograms?”, you just have to answer “given a sentence, how do I figure out if it’s a pangram-autogram?”. The latter is obviously much easier. Patuzzo clearly thought this was a pretty neat idea (and I agree), because he created an entire programming language, called Sentient, which helps you express many different problems in ways that bully SAT-solvers into solving them for you, as he did for the problem of pangrammatic autograms.

This is pretty satisfying. In a few minutes my little laptop had generated this acmatic beauty using Patuzzo’s tool:

This pangram tallies five a’s, one b, one c, two d’s, twenty nine e’s, nine f’s, four g’s, six h’s, thirteen i’s, one j, one k, three l’s, two m’s, twenty one n’s, fifteen o’s, two p’s, one q, seven r’s, twenty five s’s, eighteen t’s, four u’s, five v’s, eight w’s, two x’s, four y’s, and one z

But, of course it would a cruel joke if I didn’t also extend his tool a little bit so it could work on larger numbers. Otherwise I wouldn’t be able to tell you that this article contains five hundred and sixty-four a’s, eighty-seven b’s, one hundred and seventy-five c’s, two hundred and forty-four d’s, eight hundred and fifty-five e’s, one hundred and sixty-four f’s, one hundred and eighty-six g’s, three hundred and twenty-six h’s, five hundred and thirty-one i’s, eleven j’s, forty-five k’s, two hundred and ninety-four l’s, one hundred and eighty-one m’s, five hundred and five n’s, five hundred and thirty-five o’s, one hundred and thirty-nine p’s, nine q’s, three hundred and ninety-nine r’s, five hundred and sixty-four s’s, seven hundred and thirty-nine t’s, two hundred and seventy-four u’s, eighty-one v’s, one hundred and thirty w’s, thirty-three x’s, one hundred and sixty-nine y’s, and eighteen z’s.