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You can construct orthogonal Latin squares S^{i,j} and T^{i,j} of prime order m where the S^{i,j} = si + j \pmod m and T^{i,j} = ti + j \pmod m and s not equal to t.
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?