Choco Banana

The same predicate, affirmed once and denied once

Both halves of the rule are about the word rectangle, and the two directions could not be less alike as inference. “Is a rectangle” is closed under bounding boxes: any two shaded cells known to be in one group drag the entire box between them in with them, so a single pair can fill a block. It pushes. “Is not a rectangle” is closed under nothing. A banana that happens to be a rectangle right now is not yet wrong; it only becomes wrong when it is sealed shut and can no longer grow. So that rule cannot advance a position — it can only reject one, and only at the last moment.

So you would expect the veto to be the cheap half. It is not. Across the 68 shipped boards the denied half writes 4.2–4.5% of the cells and the affirmed half writes 3.5–5.5% — both small next to the group-size arithmetic, which writes about two thirds. But take each one out of the solver and the loss is lopsided the other way: without rect the search needs 17,452 assumptions to get through the set, without anti it needs 103,958. The rule that can only say no is six times more expensive to do without than the rule that can say yes.

The reason is that the affirmed half has already moved house. Its real work happens inside size, where a clued bar of area v is matched against the rectangles of that area that still fit — that is “is a rectangle”, with a number attached. rect standing on its own has almost nothing left to do. The denied half has no such second home.

Ablate them as rules of the puzzle instead of as inferences and both turn out to be load-bearing. Without “a banana is not a rectangle”, 26 of the 38 6×6 boards and 18 of the 30 8×8 boards stop having a unique answer, the median board picking up 2 and 4.5 answers respectively.

A number is a factorisation, not a quantity

The clue gives a size and withholds the colour, so reading it is an arithmetic question about which shapes that area admits. A connected set of 1 or 2 cells is a rectangle however you draw it, so 1 and 2 are always chocolate. Going the other way, a chocolate bar of area v exists on an H×W board only when v = a·b for some a ≤ H and b ≤ W — so on a 6×6 board the numbers 7, 11, 13, 14, 17, 19, 21, 22, 23… are all banana, because they are simply not in the 6×6 multiplication table. The count of areas a bar may have on an n×n board is OEIS A027424: 1, 3, 6, 9, 14, 18, 25, 30, 36, 42.

Four answers no clue set can tell apart

Because the clue language is exactly “the size of your group”, the strongest clue set possible is a number in every single cell. So an answer can be pinned down if and only if it is the only legal grid with its size map — and that is not always true. On a 3×4 board, a 2×3 bar in any of the four corners leaves six banana cells in an L, and every cell of all four grids reads 6. Writing all twelve numbers still leaves four answers standing.

They are rare but they are real: exhaustively, 0.0% of the answers on boards up to 3×3 are stuck like this, rising to 0.7% on 3×4 and settling near 0.1% on 4×5. The generator has to detect them — it walks away when the adversarial clue-buying loop finds no cell where the two answers disagree.

Five rungs

value reads the number and nothing else. rect fills the bounding box of every chocolate bar. size is the group-size arithmetic — a group can be no smaller than what it already holds and no larger than what it can still reach, both ends saturate, and a clued bar has to be one of the finitely many rectangles of that area that fit around the cell without swallowing a banana. anti is the denied half. probe is singleton consistency on top. Cumulatively they settle 19.8%, 34.4%, 69.7%, 79.8% and 100% of a 6×6 grid.

probe is sound but not complete here. Over 581 clue prefixes it never once claimed to finish a board that had more than one answer — but 56 times a board had exactly one answer and probe still needed a guess to find it. The gap only ever points one way.