The number nobody knows
Information theory · word games
The number nobody knows
Notes from the bottom of a word game I built.
Sometimes the medium and the message have a more complicated relationship than<br>expected. “B as in Bravo.” You’ve done it: spelling a word down a bad phone line,<br>reaching past the letters for whole words, because B and D and<br>P all blur into the same mush the moment the line frays.
That is not a quirk of bad reception. It is one of the deepest ideas in modern<br>communication, performed by a human in real time, the same trick that keeps a<br>spacecraft’s signal legible across millions of kilometres of noise. Marshall McLuhan<br>gets the credit for the medium is the message, but Claude Shannon had<br>beaten him to a colder version of it years earlier: to a machine moving your words,<br>the meaning doesn’t matter at all; only the medium does, and which of its signals<br>can be told apart. Bravo and Delta survive a bad line;<br>B and D don’t. You sorted that out by ear, without a flicker of<br>thought. Push the same instinct to its limit (how much can you force through a noisy<br>channel and still be understood with perfect certainty?) and you walk<br>straight into a question Shannon posed in 1956.
I didn’t arrive there as a mathematician; I’m not one. I came from the other<br>direction entirely: I was trying to build a word game that uses deduction instead of<br>clues. Something I expected to be simple.
Let me tell you how the seemingly simplest thing I’ve ever shipped turned out to be<br>balanced on one of the oldest open problems there is.
The “simple” thing
Here is the whole game. A half-finished crossword grid. A queue of letters waiting<br>to be placed. You’re served a letter, you tap the cell it belongs in, and you keep<br>going until the grid is full.
One rule
The thing I wanted out of it seemed almost too small to bother stating: the puzzle<br>should never make you guess. No moment where you’ve done everything right, the logic<br>runs dry, and the game just wants you to pick a cell and hope. Every letter<br>placeable by reasoning alone.
That struck me as the easy part: it’s a crossword, so I figured the hardest part<br>would be building a good dictionary and curating the crosswords. I would like to go<br>back and warn that version of me...
The wall
The dictionary took months. When it was finally good, I did the obvious thing: I<br>pointed the generator at an empty folder and asked it for a few thousand boards to<br>sift through. Fill a grid with real words, hide some, hand the rest back one at a<br>time, keep only the boards a person could finish by pure logic. I’d spend my<br>evenings curating the haul.
It made about forty. Then it stalled, spinning up board after board and discarding<br>every one, because almost none came out deducible. After a few late nights,<br>rewriting the generator, rethinking the problem, it never got past a hundred.
That stopped me cold. A five-by-five grid that has to spell real words across and<br>down is a combinatorial ocean.<br>The space of possible boards is beyond human understanding, but my generator<br>couldn’t find a hundred a person could solve.
This wasn’t a speed problem I could optimise away. It was a wall, and it asked a<br>question I couldn’t answer: could deducible boards really be that rare,<br>only a thin scatter of them playable? Had I misjudged the configuration so badly<br>that the game I’d dreamed up simply couldn’t exist? For a few bad days I believed<br>the project was dead, not hard, not slow, but impossible, my own generator<br>insisting that the open field I had imagined was nothing but a maze of dead ends.
What gave me a glimmer of hope, just enough to keep pushing forward, was how little<br>separated the boards it threw away from the ones it kept. The same grid, the same<br>served letter, and a single uncovered tile is the whole distance between a guess and<br>a certainty. Maybe my view of the problem was the problem.
serve “T” → 2 legal cells. you’d guess.
serve “T” → 1 legal cell. forced.
The same board, a single square apart. First: the served “T” fits two<br>open cells: it spells TOWER going down, or TRAM across the bottom. You’d have to<br>guess. Second: uncover one square, the K of ASK, and the top option<br>spells ATK across. Not a word, so the T is forced to its one legal home.
It wasn’t a bug
For weeks I hunted it like a bug: a flaw in the generator, a dictionary too thin, a<br>stripping pass too greedy. It was none of those.
It was the thing from the bad phone line at the top of this page (a<br>communication problem, old and famous), and I’d been too far down<br>among grids and dictionaries to recognise the shape of it. A puzzle is a channel:<br>I’m the sender, the player is the decoder, I transmit a letter and they have to<br>recover which cell I meant.
Ordinary crosswords are noisy channels. The decoder is usually right, and<br>when they aren’t they shrug, backtrack, and fix it. A little confusion is fine<br>there; in an everyday crossword it is half the fun. But I had promised something<br>stricter than usually right. I had...