Combinatorics Project
Methods tried
L & R
I calculate the squares that would be produced by the methods described in Lewis and Russell's paper and count \(a\), \(b\), \(c\), \(d\), and \(e\). This method can easily reach order \(20\).
Naive orthogonal search
I do a brute force search for orthogonal pairs of squares and count \(b\), \(c\), \(d\), and \(e\). Order \(6\) with this method is too slow to be quickly done with my laptop.
Row complete with orthogonal check
I search for all row complete squares at an order and then compare each pair looking for the most orthogonal one. This only works at composite orders and can go as far as order \(8\) on my laptop (or \(9\) if given enough time).
Row complete with orthogonal check and shuffle symmetry
As the previous except that it accounts for symmetry of the form of swapping a row to the top and then re-normalizing. This one runs a small but measurable amount faster than the previous.
Row complete with orthogonal check and full symmetry
As the previous except that it accounts for symmetry of the form of flipping the square across the Y-axis and then re-normalizing. This one runs more significantly faster than the previous but it still takes some time to do order \(9\).
