Knight covering the whole grid?

Question

• A knight (the chess piece)
• $m$ x $n$ grid
• Start from any square

Question:

For which, general case, $m$ x $n$ (neglecting the obvious failes $m,n=1,2$) is this possible/impossible without visiting the same square more than once? Can you mathematically prove this?

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My approach has been "guessing" so far. So don't know how to prove it.

Here are some simple examples:

Obviously you can't start at the mid square and no matter where you start on the periferi the mid square will be left empty.

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On this example we can see my approach (1-12) which means it's possible only if we start at 1,3,5,8,10,12. So this is at least the minimum grid required, $3$x$4$ (least squares involved).

Answer 1

From the wikipedia page of Knight's tour:

Schwenk proved that for any $m \times n$ board with $m \le n$, a closed knight's tour is always possible unless one or more of these three conditions are met:

• $m$ and $n$ are both odd
• $m = 1, 2$, or $4$
• $m = 3$ and $n = 4, 6$, or $8$.

Cull et al. and Conrad et al. proved that on any rectangular board whose smaller dimension is at least 5, there is a (possibly open) knight's tour.