Skew room squares exist for all odd values greater than 5. If n is prime it is a simple matter to generate a skew starter. But 667 is not prime. 667=23*29 which means a computer search has to be done in order to generate one. I would be satisfied with the skew room square of side 667, even though we can show it exists we can't seem to construct it. Any suggestions on this particular problem are appreciated.
The reference provided 10 minutes ago may not be appropriate.
But, one may try some direct product construction as follows.
Start from skew starter S_1 in Z_{23} and skew starter S_2 in Z_{29}.
Find a permutation P=(p_0,p_1,...,p_{28}) such that
P-I = (p_0-0, p_1-1, ..., p_{28}-28)
And
P+I = (p_0+0, p_1+1, ..., p_{28}+28)
Are both permutations in Z_{29}. For example, take p_s=2s.
For each pair {x, y} in S_1, construct 29 pairs {(x,p_s), (y,s)}, s=0,1,…,28.
This generates 11x29 pairs.
Add 14 more pairs {(0,u),(0v)} such that {u,v} is from S_2.
It may be easy to verify if the 11x29+14 pairs form a skew starter in Z_{23}xZ_{29}.
If it still does not work, please let me know.
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