For all even n at least 10 there exist latin squares with at least floor(n/6) entries common to every transversal and more than half of all entries in no transversal; for orders 3m, every transversal hits all nine equal subsquares.
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Latin Squares whose transversals share many entries
For all even n at least 10 there exist latin squares with at least floor(n/6) entries common to every transversal and more than half of all entries in no transversal; for orders 3m, every transversal hits all nine equal subsquares.