The best square to start a Knight's Tour on each board size

Published: Aug 27, 2026 Est. read time: 5 minutes

Two squares on the 5x5 board are both perfectly legal places to begin. From one of them the standard solving method cannot fail. From the other it fails about half the time. Nothing on the board tells you which is which.

A companion article works out which squares can never start a Knight's Tour at all. That settles a question about legality, and on the odd boards it rules out a great many squares. It leaves the more practical question untouched: among the squares that do work, are some easier than others?

They are, and on the small boards they are not remotely equal.

How this was measured

Warnsdorff's rule is the standard approach to the puzzle: always move to the square with the fewest onward moves, and choose at random when several are tied. It is simple enough to follow by hand and good enough to finish most boards.

Every legal starting square on every board size was played out that way tens of thousands of times, counting how often the rule reached a complete tour and how often it left the knight stranded. The figures below are those success rates.

It is worth being clear about what they are. They measure how much room for error a square leaves a good simple strategy. They are not a measurement of human play, and you are not Warnsdorff's rule: you will be worse than it at counting onward moves and better than it at noticing that a corner of the board is about to be cut off. What carries over is the ranking, and the ranking is not subtle.

The 5x5 splits into three tiers

That top row is a stronger claim than the rest, and for a reason worth spelling out. From the four corners and the centre square you can enumerate every line the rule permits, every tie broken every possible way, and count them. From a1 there are thirty-two such lines and all thirty-two finish. Not one dead-ends. There is no wrong answer available to find.

From c1 there are dead ends, and breaking ties at random walks into one about half the time.

So on this board the distance between the best legal opening and the worst legal opening is the distance between a puzzle you cannot lose with the standard method and a coin toss. That is worth knowing before you spend twenty minutes on it.

The 6x6 hides its only difficulty in the middle

Every square on the outer ring finishes every time. The ring inside it is close behind, between about 98 and 100 per cent. The four squares in the middle, c3, c4, d3 and d4, drop to around 76 per cent.

Four awkward squares out of thirty-six, and they are the four a beginner is most likely to pick, because the middle of the board looks like the safe place to be. It is the safe place to be on move twenty. It is a poor place to be on move one.

On the 8x8 it barely matters where you start

Across all sixty-four squares the success rate runs from about 97 to about 99 per cent. The entire board sits inside roughly two percentage points.

This is worth saying plainly, because "start in a corner" is advice repeated all over the internet, including in the strategy tips on our own rules page. On the small boards it is exactly right. On the classic board it is close to irrelevant. Start wherever you like, and accept that an 8x8 tour is decided by what you do in the middle of it rather than by where you began.

The 10x10 behaves the same way, between about 97.5 and 99.5 per cent. So does the 9x9, with a single exception.

The centre is the best square on one board and the worst on another

On the 7x7, d4 is the dead centre of the board, and it is the easiest square on it by a wide margin: about 99 per cent, against 71 to 81 per cent for every other legal square. Nothing else on that board comes close to it.

On the 9x9, e5 is the dead centre of the board, and it is the hardest square on it: about 94 per cent, against 97 to 99 per cent everywhere else. It is the only square on the 9x9 that is meaningfully worse than its neighbours.

The same piece, the same shape of board, two sizes apart, and the opposite answer. We do not have a clean explanation for the reversal, and would rather say so than invent one. It is a good illustration of why the folk rules about knights and the centre are worth less than they sound. The board size changes the answer, and it does not change it in a direction anyone could have guessed.

What happens if you just play legal moves

For a baseline, here is the same experiment with no strategy at all: choose at random from whatever moves are available and keep going until nothing is left.

Random play gets about halfway across every board and then runs out of room. That consistency is the interesting part. It is not that the larger boards are harder to get started on, it is that a knight left to its own devices paints itself into a corner at roughly the same point no matter how much board there is to work with.

It is also the gap the strategy closes. On the 5x5, picking a corner and following the rule turns a one in fifteen hundred shot into something that cannot fail.

Where to start, in one table

On the three odd boards there is a prior question to clear first. A square has to be the same colour as the corners before any of this applies to it. Legal first, then easy.

The quickest way to feel the difference is to play the 5x5 twice, once from a corner and once from c1, and notice how much earlier the second one closes in around you. Both are on the play page.