# A knight chased by three knights

Two players are playing a variant of chess on a 8x8 grid. The first player controls a white knight that starts in the top-left corner. The second player controls three black knights that start in the other three corners. The first player aims to avoid being captured for as long as possible, while the second player aims to capture the white knight. Players alternate in making moves: the first player moves his white knight, then the second player moves all his black knights and so on. Passes are not allowed. The white knight is allowed to capture black knights.

If both players play optimally what would be the outcome of this game? Can white escape indefinitely? Can black guarantee to capture white no matter what he does? Bonus question: what would happen if the second player had just two black knights instead of three?

• Can White Knights capture Black Knights? Commented May 30, 2020 at 1:41
• Yes! I will add this in. Commented May 30, 2020 at 1:42

Observation:

The white knight starts on a white square. Two of the black knights start on black squares, with the third black knight on a white square.

Implication:

After each move by white, the white knight will be on a square that is of the same color as that of two black knights. These two black knights will never have an opportunity to capture the white knight.

Conclusion:

The white knight need only avoid being captured by the third black knight, which can be done indefinitely.

• Ninja'd again! How unlucky am I! Commented May 30, 2020 at 2:22

First,

The starting cell of White is white, the starting cell of the right top and left bottom corner Black knights are black, and the starting cell of the bottom right Black knight is White. Let the Black knight starting in the right bottom corner be called the Capturer.

Also,

We know that a Black knight could capture a White knight means that the colour of their cells are different before the capture. As after every White move, the only Black Knight whose colour of its cell is different from White Knight is the capturer. So the White knight has to avoid the Capturer. I will edit after trying is it possible to avoid the capturer indefinitely.

Bonus,

If the spawning position of the Black knights are the right top and left bottom, then by the above arguments, the White knight can escape indefinitely. Or else, this is same for the three knight position.

Observation:

To show that the white knight can escape indefinitely from one black knight remark that

...

at the first move the white knight lefts the corner square. It can be easily checked that a knight placed on any square of a chessboard with all corner squares cut, always have at least three moves. Also it can be easily checked that whenever distinct squares are the white and the black knight are placed, the white knight has at most two moves leading to a possible capture at the next move of the black knight.