Products Remember, Sums Forget
Four tiny Latin squares, one shared cross, four bent cages. Replay the deduction one move at a time, then take the clues apart to see why a product can never bluff and a sum sometimes must.
The Deduction, Step by Step
Each step fills what that one observation forces, and highlights the cells it used. Cells in rust are new this step.
Clue Decoder
A bent three-cell cage: the two ends (rust) may match; the middle may match neither. Pick an operation and a target, and see every fill that produces it.
Seven fills of a bent cage. Two pairs of them share a sum (5 and 7), so a sum clue can bluff. No two share a product, because a product of primes has exactly one factorization. The one product that keeps a secret is 6: it fixes the digits but not their order.
Only 48 Boards
Every 3×3 Latin square on 1, 2, 3 is a set of diagonal stripes, and the cross lets a neighbour either continue them or reflect them. Deal through the possibilities.
The short version: 12 can only be 2·2·3 because every whole number has exactly one prime factorization, so a product cage is a complete record of its digits. 7 can be 2+3+2 or 3+1+3; the sum has thrown that information away. Products remember, sums forget — and in this board, the shared cross is what remembers on the sums’ behalf.
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