A Latin square is an n × n matrix of n distinct symbols, where each row and column contains exactly one of each symbol.[1]

The number of size-n Latin squares is a rapidly growing function, although not terribly impressive compared to other googological functions:

\begin{eqnarray*} L(1) &=& 1 \\ L(2) &=& 2 \\ L(3) &=& 12 \\ L(4) &=& 576 \\ L(5) &=& 161,280 \\ L(6) &=& 812,851,200 \\ L(7) &=& 61,479,419,904,000 \end{eqnarray*}

No simple formula is yet known for the function \(L(n)\). It is upper-bounded by the function \(n \mapsto (n!)^n\), since each of the \(n\) rows has an arrangement of \(n\) distinct symbols.


  1. Wolfram MathWorld, Latin Square
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