Why some squares can never start a Knight's Tour

Published: Aug 15, 2026 Est. read time: 4 minutes

If you have played the 5x5 board and lost with the knight stranded after twenty-odd moves, you may have assumed you misplayed it. Often you did not. On the 5x5, twelve of the twenty-five squares cannot begin a complete tour at all. Choose one of those and the game is already lost on move one, however well you play afterwards.

This is not a flaw in the puzzle. It is a genuine and rather elegant property of how knights move, and once you can see it you can also see which squares are safe.

A knight always changes colour

Colour the board like a chessboard. A knight move is two squares one way and one the other, so it changes your position by an odd amount overall. That means every knight move lands on the opposite colour. There are no exceptions and there is no way to avoid it.

So a tour, which is a single unbroken sequence of knight moves, alternates colours the whole way through: light, dark, light, dark, and so on to the end.

On odd boards the two colours are not equal

A 5x5 board has twenty-five squares. They do not split evenly. Thirteen share a colour with the four corners, and twelve are the other colour.

Now count what a complete tour needs. Visiting all twenty-five squares means a sequence of twenty-five squares that alternates colour. Write it out and the pattern is forced: it uses thirteen of whichever colour it starts on, and twelve of the other.

The board only has thirteen of one colour. So the tour must start on that colour, the corners' colour. Start on one of the twelve minority squares and you would need fourteen of them, and only twelve exist. No amount of skill fixes that.

Every odd board behaves the same way. Every even board does not, because there the two colours are equal and the argument has nothing to bite on.

Which squares work

On the three odd boards the rule is the same: a square can start a tour only if it is the same colour as the four corners. On the even boards every square works.

For the 5x5 specifically, these thirteen work:

a1, c1, e1, b2, d2, a3, c3, e3, b4, d4, a5, c5, e5

and these twelve cannot, no matter how you play:

b1, d1, a2, c2, e2, b3, d3, a4, c4, e4, b5, d5

The two halves of that answer are not proved the same way

Worth saying plainly, because it is the interesting part.

The impossible squares are settled by the colouring argument above. It is a proof. It does not need a computer and it does not need checking case by case.

The possible squares are not proved by that argument at all. Knowing a square is the right colour only tells you nothing rules it out. To know a tour genuinely exists from a square, somebody has to find one. Every count in the table above was confirmed that way, by searching for an actual tour from every candidate square and verifying it move by move.

That asymmetry is not a curiosity. Searching is very good at showing a tour exists and hopeless at showing one does not: on the 7x7 a search for a tour from an impossible square was still running after two hundred million positions, with nothing to show, because "no" means exhausting every arrangement rather than finding one. The colouring argument answers in a sentence what a computer cannot answer at all.

A related fact about round trips

A closed tour is one that finishes a knight's move away from where it started, so it loops. On an odd board a closed tour is impossible from every square, not just most of them. The same counting argument does it: a closed tour of all the squares must use exactly as many of one colour as the other, and an odd board does not have equal numbers to give.

That is why you will see 8x8 closed tours everywhere and never a 5x5 one.

What this means when you play

1. On 5x5, 7x7 and 9x9, look at the corners before your first move. Pick a square sharing their colour and a tour exists. Whether you find it is then genuinely down to you.
2. On 6x6, 8x8 and 10x10, start anywhere. Every square works, so a loss is a loss on merit.
3. 8x8 is a kinder first board than 5x5, which is why the game recommends it. The small board looks easier and is not: fewer squares means fewer escape routes, and half the openings are dead before you begin.

The game will tell you when you have opened on a square that cannot work. It does not stop you playing it, because exploring a doomed board is a perfectly good way to develop a feel for how quickly a knight runs out of room. Try it on the play page.