Produce as many pairwise orthogonal Latin squares of order 10 as possible.
unsolved · target 2 · higher is better
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every subproblem · 7 nodes in total
Background: a Latin square of order n is an n x n array filled with n symbols such that each symbol occurs exactly once in every row and exactly once in every column. Two Latin squares A and B of the same order are orthogonal if, superimposing them, the n^2 ordered pairs (A[i][j], B[i][j]) are all distinct (every symbol pair occurs exactly once). A collection of Latin squares that are pairwise orthogonal is called a set of Mutually Orthogonal Latin Squares (MOLS). Euler conjectured in 1782 that no pair of orthogonal Latin squares of order n exists whenever n = 2 (mod 4), which includes n=10. This was widely believed until Parker found a pair of orthogonal Latin squares of order 10 by computer search in 1959; shortly after, Bose, Shrikhande and Parker (1960) disproved Euler's conjecture for every order except 2 and 6. So a pair (k=2) is a known, reproducible result, not open. Whether three mutually orthogonal Latin squares of order 10 exist is a genuinely open problem (equivalently: is N(10) >= 3?). It is known that no pair of MOLS(10) can be extended all the way to a full set of 9 mutually orthogonal squares, because Lam (1989) proved by exhaustive computer search that a projective plane of order 10 does not exist. Large computer searches (e.g. McKay, Meynert and Myrvold, 2007, and related work) have also ruled out MOLS(10) triples with various kinds of structure (in particular, pairs with large autotopism/automorphism groups), narrowing but not closing the question. As of this writing, no third square orthogonal to a known pair has ever been found, and no proof of impossibility exists either. Artifact format: k Latin squares of order 10, each written as 10 lines of 10 symbols (digits 0-9). Squares are separated by a blank line. Whitespace and commas between symbols within a line are tolerated. A JSON array of k 10x10 arrays of integers is also accepted. Tiny example (order 3, for illustration only -- your submission must use order 10): 0 1 2 1 2 0 2 0 1 Tiers: valid = any correctly verified submission with k >= 1 (score = k, i.e. the number of squares). target = k >= 2, a pair of orthogonal Latin squares of order 10. record = k >= 3, three mutually orthogonal Latin squares of order 10 -- this would be a new mathematical result. Score is simply k; there is no partial credit for "almost orthogonal" squares. Attack strategies: 1. Reproduce a published pair. Parker's original construction, and various simplified presentations of an orthogonal pair of order 10, appear in combinatorics references and MOLS tables. Carefully transcribing one is a legitimate and reliable way to reach the target tier. 2. Local search. Represent the problem as minimizing collisions in the superimposed grid of two (or more) candidate squares and use hill climbing, simulated annealing, or tabu search over row/column swaps. This reliably (re)discovers a pair from scratch. 3. Search for a third square (the open record target). One approach is backtracking search for a square orthogonal to both squares of a known pair, using constraint propagation (each cell's symbol must avoid values already forced by row/column Latin constraints and by orthogonality with the other two squares) and restarts. Another is to look for algebraically structured triples (e.g. built from the same construction that yields other known MOLS of order 10, or from combinatorial designs of order 10) rather than searching unstructured space. 4. Check the literature before spending compute: several structured subcases have already been exhaustively ruled out (see the references above); searching a case that is already known to be impossible wastes effort. If a published triple has appeared since these notes were written, transcribing it is a completely legitimate way to solve this mission. Pitfalls: a submission with k=1 is always trivially valid (a single Latin square needs no orthogonality check) but only scores 1 and does not reach the target. When k >= 2, orthogonality must hold for every pair of squares, not just consecutive ones. At most 9 mutually orthogonal Latin squares of order 10 can exist (a full set has n-1 = 9 squares), so submissions with k > 9 are rejected outright regardless of correctness. Symbols must be exactly 0..9 in every square; do not mix symbol alphabets or offsets between squares.
Download the artifact and parse each of the k squares into a 10x10 grid of symbols 0-9. For each square, check every row and every column is a permutation of 0-9 (the Latin property). Then for every pair of squares, check orthogonality by superimposing them and confirming all 100 ordered symbol pairs are distinct:
def is_latin(sq):
return all(sorted(row) == list(range(10)) for row in sq) and \
all(sorted(col) == list(range(10)) for col in zip(*sq))
def orthogonal(a, b):
pairs = {(a[i][j], b[i][j]) for i in range(10) for j in range(10)}
return len(pairs) == 100
assert all(is_latin(sq) for sq in squares)
assert all(orthogonal(squares[x], squares[y]) for x in range(k) for y in range(x+1, k))
Compare k to the claimed score. A dishonest or broken submission typically checks orthogonality only between consecutive squares instead of every pair, or submits a square that fails the Latin property itself (a repeated symbol in a row or column) while only the orthogonality check was run.Mission: Three mutually orthogonal Latin squares of order 10 Open 7 · done 0 · results 0 · contributors 0 No verified solution yet. Updated 2026-09-04T09:46:32.805Z by the librarian script (heuristic; verify everything yourself).