The one tally this game can never show you
4 min read
Here is a small thing you can use. On the five-letter board there are twenty-one shapes a tally could take — every pair of numbers that adds to five or less. Twenty of them can actually happen. One of them cannot, ever, and knowing which one is occasionally worth a guess.
You will never be told four hits and one near.
Why it cannot happen
Four hits means four of your five letters are sitting in the right places. That leaves exactly one position unaccounted for on each side: one letter of your guess, and one letter of the answer.
For the reply to include a near, that leftover guess letter would have to appear in the answer somewhere it has not already been counted. But every other position in the answer is spoken for — each one was claimed by a hit. The only unclaimed letter in the answer is the one at that last position. So either your leftover letter equals it, in which case it is a fifth hit and the reply is five and zero, or it does not, in which case there is nothing left for it to match and the reply is four and zero.
The same argument runs on any board, so every board has exactly one forbidden tally:
| Board | Shapes that fit | Shapes that occur | The impossible one |
|---|---|---|---|
| Easy (4 letters) | 15 | 14 | 3 hits, 1 near |
| Regular (5 letters) | 21 | 20 | 4 hits, 1 near |
| Hard (6 letters) | 28 | 27 | 5 hits, 1 near |
It survives repeated letters, which is the surprising part
The six-letter board allows the answer to repeat a letter, and repeats are where tidy arguments about word games usually fall apart. This one holds, because of the order the counting happens in.
Hits are claimed first, and a claimed letter is spent. Take the answer LETTER and guess LETTEE: five positions match, and the leftover guess letter is an E. The answer does contain other E’s — but both of them were already claimed by hits, so there is no unspent E for the sixth letter to match. The reply is five hits, zero near.
That is the same rule that decides every duplicate-letter case in the game: exact matches are taken out of the pool before anything else is counted, and no letter is ever counted twice. If you are keeping score by hand in a two-player game, that ordering is the entire thing you have to get right.
What to do with it
- It is a check on your own arithmetic. If you have convinced yourself the answer must have four of your letters placed and one more floating, you have made an error somewhere further back. Not an unlikely conclusion — an unreachable one.
- One hit short of the word is closer than it feels. The board can only say 4 and zero, which means one substitution away. There is no “nearly, but also something is out of place” state to worry about.
- It shrinks the branching by less than you would hope. Twenty reachable shapes rather than twenty-one is a two per cent saving, which is a good illustration of why this feedback rule is so much weaker than a grid of coloured tiles: the whole vocabulary is small enough that removing one word from it barely registers.
Checked, not asserted
Every claim above is pinned by a property test that runs the real scoring function over every answer in every pool against a slice of each dictionary, and fails if a forbidden tally ever appears or if any of the others stops appearing. A proof in prose is worth something; a proof that breaks the build if it stops being true is worth rather more.