Codeword

It is a filled crossword with the clues taken away and the letters hidden. Every letter square carries a number instead, one number per letter of the alphabet, and the same number means the same letter everywhere in the grid. Nothing tells you what the words are — the grid tells you what the words can look like, and that turns out to be enough.

• Every number stands for one letter, and every letter has one number.
• The same number is always the same letter, wherever it appears.
• Two different numbers can never be the same letter. That rule does most of the solving.
• Every across and down entry of three or more squares is a real word.
• The letters printed at the start are the way in, and they cannot be changed.

There is no theme, no general knowledge and no setter's wordplay — only spelling. That makes it the most translatable puzzle on this site: a solver who would never attempt a cryptic crossword can finish a 13 × 13 codeword, because the only thing being tested is whether a string of letters looks like English.

Two, usually — and never fewer, because a grid with nothing filled in is a discouraging thing to open even when it is solvable. The real number is not chosen for looks. Letters are revealed one at a time until the rest of the key is forced, so the count is whatever that particular grid needs. Big grids need less help: more entries means more crossings, and crossings are the whole argument.

The 9 × 9 is the hard one, which surprises people. It has a third of the crossings of the big grid and no entry longer than five letters, so a wrong idea survives much longer before something contradicts it — and it routinely needs five or six letters given away before the rest can be deduced at all. If you want a long think, take the small grid.

A badly built one can, and it is the worst failure a codeword has: two solvers hand in different grids, both spelled entirely with real words, and both are right. Nothing in the puzzle can settle it. So every grid here is checked before it is dealt, and checked against the generous list rather than the friendly one — all 172,823 words of the shipped dictionary, not the 25,322 familiar ones the grid was built from. A second answer spelled in obscure words is still a second answer.

The check is a propagation argument, which is why it is worth more than a solve. Each entry keeps the list of every word that could still fit it; when every survivor agrees on the letter in one position, that letter is true in any answer, so writing it in cannot lose one. If that argument alone reaches a complete key, there is no other key — and, because nothing along the way was a choice, no step of the solve needs a guess. Scale of the thing being ruled out: a grid using 21 different letters has 26 × 25 × … × 6, about 3.4 × 10²⁴, possible keys.

The one in the longest entry that already has a given letter in it. Length is information: a three-letter entry could be any of the 677 common three-letter words this site is willing to deal, and knowing one of its letters barely dents that. A seven-letter entry — the longest in any grid here — has 4,240 candidates, but it also has three or four crossings, and two known letters in the right places usually leave a handful.

The other early lever is repetition. A number that appears fifteen times is a vowel or one of E, S, T, R, N; a number that appears twice in the same short entry is doing something specific — English has plenty of words with a doubled letter and very few with a doubled J. And watch the last squares of entries: a shared final number across three or four entries is almost always S, and a shared pair at the end is usually ED, ER or LY.

More than which letters you have used. Because the key is one-to-one, a letter you have placed is a letter that is gone — no other number can be it. So the tracker is an elimination list, and the remaining letters are the candidate set for every unsolved number in the grid. Late in a solve that is the fastest tool on the board: eleven numbers left and eleven letters left, and one of those letters is Q, which has to be followed by U.

It is also the thing that catches the classic mistake. Write a letter against a second number and the board says so immediately, both numbers flagged, rather than accepting it and telling you twenty minutes later that the finished grid is wrong. Most implementations let that contradiction sit there, because they only check the answer at the end — and by then you cannot tell which of the two placements was the bad one.