Experimental Mathematics

Determinants of Latin squares of order {$8$}

David Ford and Kenneth W. Johnson


A latin square is an $n\times n$ array of $n$ symbols in which each symbol appears exactly once in each row and column. Regarding each symbol as a variable and taking the determinant, we get a degree-$n$ polynomial in $n$ variables. Can two latin squares $L,M$ have the same determinant, up to a renaming of the variables, apart from the obvious cases when $L$ is obtained from $M$ by a sequence of row interchanges, column interchanges, renaming of variables, and transposition? The answer was known to be no if $n\le7$; we show that it is yes for $n=8$. The latin squares for which this situation occurs have interesting special characteristics.

Article information

Experiment. Math., Volume 5, Issue 4 (1996), 317-325.

First available in Project Euclid: 13 March 2003

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05B15: Orthogonal arrays, Latin squares, Room squares
Secondary: 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14]


Ford, David; Johnson, Kenneth W. Determinants of Latin squares of order {$8$}. Experiment. Math. 5 (1996), no. 4, 317--325. https://projecteuclid.org/euclid.em/1047565449

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