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:
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.
- 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.