Impossible 2x2
A 2x2 restricted to two letters and two digits has no valid completion (no order-2 Graeco-Latin square)
How to use it
A corner case worth knowing by name. Find a 2×2 of cells spanning two rows and two columns where only two letters and two digits remain available. Those four cells would need to hold the four pairs of two letters by two digits — an order-2 Graeco-Latin square, which does not exist. So any candidate whose placement would squeeze the corner into exactly that shape is impossible: block it.
Spot it as a candidate that, if placed, leaves the four cells of a rectangle holding only two letters and two digits between them. Verifying the collapse is a bounded check — no chains, no deep search.
Why it is sound
A 2×2 submatrix of a Graeco-Latin square would itself be a Graeco-Latin square of order 2, which does not exist: with only two available letters and two available digits there is no way to place four distinct pairs while keeping both layers Latin in the rows and columns. If a candidate would force a 2×2 into that two-letter/two-digit shape, it is impossible. Pure logic, proved by exhaustive 2×2 case check (zero valid completions).
A worked example
The diagram shows the shape on a hand-built candidate position — no fixture grid produces this pattern naturally, so the page shows the deduction in its pure form; the position itself need not be completable, only the local 2×2 shape matters. The three shaded cells of the 2×2 offer only letters {A, B} and digits {1, 2}. The pivot, r1c1, could still take A1 among its options. But place A1 there and the rectangle is cornered: its four cells must host four distinct pairs using only two letters and two digits — and no order-2 Graeco-Latin square exists (a fact provable by checking every arrangement of a 2×2). So A1 at the pivot is impossible: block it. The catch, and the reason this stays a rare, last-resort weapon: the elimination is only valid when the two-letters/two-digits squeeze is really forced by the placement, not merely possible — which is why the collapse must be checked before claiming the block.