Latin squares with a unique intercalate
Suppose that n=1 or 5 mod 6 and n≥7. We construct a Latin square
Ln of order n with the following properties:
- Ln has no proper subsquares of order 3 or more.
- Ln has exactly one intercalate (subsquare of order 2).
- When the intercalate is replaced by the other possible subsquare on the
same symbols, the resulting Latin square is in the same species as Ln.
Hence Ln generalises the square that Sade famously found to complete
Norton's enumeration of Latin squares of order 7. In particular,
Ln is what is known as a self-switching Latin square and
possesses a near-autoparatopism.