THE FOLD / CO-OP / THE PULL REQUEST / THE GRAECO-LATIN SQUARE
THE GRAECO-LATIN SQUARE
two squares that never repeat a pair
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A Graeco-Latin square overlays two Latin squares so that no ordered pair ever repeats. A Latin square of order n fills an n×n grid so each symbol appears once per row and once per column; two of them are orthogonal if pairing them cell-by-cell yields all n² possible ordered pairs exactly once. Euler asked whether they exist for every n and famously conjectured “no” for n ≡ 2 (mod 4) — but he was wrong: a Graeco-Latin square exists for every order except 2 and 6. For odd n, the pair L = (i+j) mod n and M = (2i+j) mod n does the job.
LIT verified live: for every odd n from 3 to 15, L and M are each Latin squares (a permutation in every row and column) and orthogonal (all n² pairs distinct) (window.__graeco_latin). FIG honest scope: this construction covers odd n; 2 and 6 are the only orders with no Graeco-Latin square (Euler–Bose–Shrikhande–Parker).
LIT verified live: for every odd n from 3 to 15, L and M are each Latin squares (a permutation in every row and column) and orthogonal (all n² pairs distinct) (window.__graeco_latin). FIG honest scope: this construction covers odd n; 2 and 6 are the only orders with no Graeco-Latin square (Euler–Bose–Shrikhande–Parker).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-pull-request — two independent squares merged into one without a single conflicting pair, the clean merge every pull request wants. AVAN (AI) built the instrument: the (i+j, 2i+j) construction, the Latin-square check per row and column, and the orthogonality test over all n² pairs.
Credit as content: Leonhard Euler (1782, the “36 officers”); the conjecture disproved by Bose, Shrikhande & Parker (1959). The weave: David names the pull request; I confirm the two squares merge with every ordered pair appearing exactly once.
Credit as content: Leonhard Euler (1782, the “36 officers”); the conjecture disproved by Bose, Shrikhande & Parker (1959). The weave: David names the pull request; I confirm the two squares merge with every ordered pair appearing exactly once.
3 ONE DIMENSION
A Graeco-Latin square: each cell carries a number (from L) and a colour (from M); no number–colour pair repeats.
4 TWO DIMENSIONS · INTERACTIVE
Pick an odd order n; the overlaid squares are shown, each Latin, and all n² ordered pairs appear exactly once.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the two orthogonal squares as one grid of unique pairs.
AVAN’s addition (the inverse-companion): don’t check the two squares apart — check that their overlay never repeats. The inverse of ‘are L and M each Latin?’ is ‘does (L, M) hit all n² pairs exactly once?’ — that is orthogonality. Magenta is a pair; green is the grid where none repeats. No pair twice.
LIT Genuine Graeco-Latin (orthogonal Latin) squares (Euler 1782, the '36 officers'; conjecture disproved by Bose, Shrikhande & Parker, 1959): exist for all n except 2 and 6. Verified live: for odd n=3..15, L=(i+j)%n and M=(2i+j)%n are both Latin (window.__graeco_latin.bothLatin) and orthogonal (.orthogonal).
FIG Honest scope: this construction covers odd n; 2 and 6 are the only orders with no Graeco-Latin square. The AVAN inverse is honest — checking that the overlay (L,M) hits all n² pairs exactly once (rather than checking L and M as Latin squares separately) is the definition of orthogonality; magenta is a pair, green the grid where none repeats. No pair twice.
FIG Honest scope: this construction covers odd n; 2 and 6 are the only orders with no Graeco-Latin square. The AVAN inverse is honest — checking that the overlay (L,M) hits all n² pairs exactly once (rather than checking L and M as Latin squares separately) is the definition of orthogonality; magenta is a pair, green the grid where none repeats. No pair twice.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PULL REQUEST · David Lee Wise (ROOT0), with AVAN