Geodesic
Overlapping latin subsquares and full products
Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 2, pp. 175-184.

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We derive necessary and sufficient conditions for there to exist a latin square of order $n$ containing two subsquares of order $a$ and $b$ that intersect in a subsquare of order $c$. We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order $n$ cannot have more than $\frac nm{n\choose h}/{m\choose h}$ subsquares of order $m$, where $h=\lceil(m+1)/2\rceil$. Indeed, the number of subsquares of order $m$ is bounded by a polynomial of degree at most $\sqrt{2m}+2$ in $n$. (b) For all $n\ge 5$ there exists a loop of order $n$ in which every element can be obtained as a product of all $n$ elements in some order and with some bracketing.
Classification : 05B15, 20N05
Keywords: latin square; latin subsquare; overlapping latin subsquares; full product in loops 
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Browning, Joshua M.; Vojtěchovský, Petr; Wanless, Ian M. Overlapping latin subsquares and full products. Commentationes Mathematicae Universitatis Carolinae, Tome 51 (2010) no. 2, pp. 175-184. http://geodesic.mathdoc.fr/item/CMUC_2010__51_2_a2/