Algebraic and Geometric Combinatorics by Eric Mendelsohn

By Eric Mendelsohn

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H f . Each entry in S ( m , p ) has second coordinate 1 or 2. The statement of the corollary is obviously true for a subsquare satisfying Proposition 4(b). Let L be a subsquare satisfying Proposition 4(c). Since p I u, gcd@, d ) = 1 and i, = i l -k (x - l)d (mod p ) , 1S x S u, i, takes the value of each congruence class m o d p exactly u/p times. Considering the intersection of L and an A-column of P(m,p) it is easy t o see that L certainly contains an entry with second coordinate 1 or 2 (actually both).

But then the (bl + a 2 + a2- a l , 1) in the other column is also in G, belonging to S(m. This again leads to a new cell of S ( m , p ) in G, and repeating this argument until we reach a cell containing (al+ b l , 1) in the second column we deduce that G contains entries from t h e two columns corresponding to two columns of a subsquare of A ( m ) . (G may contain further entries from the same two columns, but we only need this first ‘cycle’ for the argument). Because the two columns correspond to the column of a subsquare, G also contains cells from S ( m , p ) in a mixed column, and here it both contains an entry, which was originally in the C-area and an entry which was originally in the B-area.

And (:(u + 3), u ) is~a C(p>cell. To prove this we must prove that But this follows from the fact that j S m / q and W e now consider two cases. Case 1. Then, since the entry of ($(u + l), u ) has ~ the same first coordinate as the entry of (t(u - l ) , l ) L (namely a + b + $(u- 3)j - I. mod m),we get from the second coordinates (using (2)) giving (applying (3)) ju = 5 - 2 i , - jl+ 2d (mod p ) A direct constructionfor latin squares withouf proper subsquares 37 Comparing this to (4)we get: If j,, = 1 - i l (mod p), 1- i l = 5 - i l - ( d + 1 ) + 2d (mod p), implying d=-3 (modp).

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