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Published online by Cambridge University Press: 04 November 2009
Abstract
The counting of the number of Intercalates, 2 × 2 subsquares, possible in a latin square of side n is in general a hard problem. N2–Free latin squares, those for which there are no intercalates, are known to exist for n ≠ 1,2,4. N2–complete latin squares, those which have the property that they have the maximum number of N2's possible, must be isotopic to and thus of side 2k. The maximum for n ≠ 2k is in general unknown. We propose an intermediate possibility, that of N2–ubiquitous. A latin square is N2 ubiquitous if and only if every cell aij is contained in some 2 × 2 subsquare. We show these exist for n ≠ 1,3,5,7. It is also determined for which n, C–ubiquitous latin squares exist for every partial latin square, C with four cells. We also enumerate the number of times each 4-cell configuration can appear in a latin square and show that this number depends only on n and the number of intercalates.
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