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Is there any winning strategy in International Checkers for 3 or more kings vs 1 king?

I've reached a draw because the opponent was moving his single king on the diagonal from one end to another, and I could find no way to stop him doing so and made a trap for the king.

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2 Answers 2

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The answer by @RemcoGerlich is essentially correct. Some extra info below. For further reference see the Dutch book Drie tegen een is gemeen, that contains a mathematical proof that 3 vs. 1 kings is a draw (which predates the age of perfect knowledge endgame databases by almost a decade!).

The answer depends crucially on both the board geometry and the king movement.

For short-ranged kings (American checkers and Italian draughts) 2 kings can force a win against 1 king. The same applies for orthogonally capturing kings (Frisian draughts) and kings that need to end a jump immediately behind the last jumped piece (Thai draughts).

For long-ranged kings, on square boards larger than 8x8 (10x10 International and 12x12 Canadian draughts), 3 kings against 1 king is a draw but 4 kings can always force a win.

On an 8x8 board (e.g. American pool checkers and Russian draughts), it is a win if the majority side occupies the long diagonal, and a draw otherwise. The key position is called Petrov's triangle (in Russian, use a browser translator):

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On rectangular boards of dimensions N x (N ± 1), N x (N ± 2) and N x (2 N - 1), there are two single corner diagonals that allow special tactics to enable a 3 vs. 1 forced win.

In International draughts, the 3 vs 1 drawn engame is seen as the main culprit for the >90% drawing percentage at grandmaster level matchplay. Many game tweaks have been proposed.

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3 kings vs 1 king is usually not enough for force a win, because (as you discovered), you can't catch a king that can safely stay on the main diagonal.

The rules say that 1 king vs 1 king (where neither king is immediately lost) is an immediate draw; 2 kings (or a king and a piece) vs 1 king is a draw if no captures occur within five moves, three kings (or two kings and a piece, or one king and two pieces) vs 1 king is a draw if no captures occur within 15 moves.

A game is also a draw if only king moves are played for 25 consecutive turns.

With four kings against a single king, there are various ways to trap the king on the long diagonal; this page shows a few (the bottom four diagrams). I must admit I don't understand the last one, I don't really play this game...

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For the bottom-right diagram, suppose a single black king is located in the upper-left part of the board. Then white puts his left-most king on the same diagonal as the black king. This gives it a choice of two captures, to one of the single corners on the long diagonal. Whatever black's chosen jump, white then puts the remaining king on that diagonal in the opposing single corner. If black is located in the bottom-right triangle, the same tactic is used with the right-most white king. –  TemplateRex Aug 22 at 12:20
Thanks, that helps! –  RemcoGerlich Aug 22 at 13:13

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