Fox and Geese

The fox steps one square, or jumps a neighbouring goose into the empty square beyond and eats it. The geese step one square and never capture. The geese win by leaving the fox with no move at all; the fox wins by surviving 200 plies, or by eating the flock down below 4 — the point at which, as the table below shows, it can no longer be cornered from any arrangement. Everything runs in your browser; nothing is uploaded.

The minimum is four, and four is useless

Solved exhaustively on this board by working backwards from every position where the fox has no move — 2,897,559 positions in all. For each flock size, how many of the arrangements with the geese to play are wins for the geese:

Three geese can never trap the fox — not from any placement, with either side to move. Four can, from precisely 84 arrangements out of 101,745. So the fewest geese that still win is four, and the answer is close to useless: you would have to be handed one position in twelve hundred. The minimum is a curiosity, not a way to play.

The game turns over between six and seven

The number worth knowing is not the minimum but the point where this stops being the fox's game. Six geese win from 9.36% of positions; seven win from 91.02%. One extra bird carries the flock from hopeless to overwhelming, and that single step is larger than every earlier step added together.

The share rises at every size, as it must — an extra goose can always be parked somewhere harmless — but it rises nothing like evenly. Going from four geese to six buys nine points. Going from six to seven buys eighty-two.

All of it belongs to this variant. Sets disagree about whether geese may move backwards and whether the fox may chain several jumps in a turn, and either choice would move these numbers. Here the geese move freely in any direction and the fox takes one goose per turn.

Why the game above has a stopping rule

Fox and Geese has no natural end for the fox — its win condition is simply not to be caught, so a game can run for ever. The first version of the game here did exactly that, sitting at four geese for four hundred moves with the flock unable to corner anything and the fox with no reason to go anywhere. The board now stops when the flock drops below 4, which is not a chosen number but the one the table above computes, and otherwise after 200 plies, which is the fox surviving. The two figures are pinned to agree, and the small rows of the table are re-solved on every build.