# If repetition loses, who wins KvK?

Suppose we're playing a version of chess where you lose if you repeat once an earlier position (position = arrangement of pieces + who's move it is). We're down to the wire, and we've just arrived at the position after the 30th capture, with only the two kings left on the board. With best play, who wins?

I think the question originated with the British go player Matthew Macfadyen 6-dan, but was solved by John Rickard. Avoiding loops is a thing in go however the puzzle here requires zero knowledge of that game.

As far as I know the solution has never been published. So here we chess.stackexchange: good luck!

EDIT: With just the two kings, stalemate cannot happen. Assume there is no 50-move Rule, no Draw by Triple Repetition & no Dead Position Rule. This problem does not concern these rules. And in any case, in compositions these rules are applicable by default only to retro problems, which this is not.

• Deeming stalemate a loss entails identifying which player stalemated the other. So perhaps you'd consider it reasonable, when the position is dead, to identify the player who played from an alive position to a dead one. What the result would be depends on how you define your variant. Or perhaps your variant doesn't have the same notion of "dead position" as chess, and the position is dead only at KvK? Commented Jun 28, 2021 at 7:23
• In KRvKN, in chess, capture of the R, or a R check which forces black to take the R, kills the position because mate is now impossible. But perhaps, in your variant, play must go on because the result is not yet settled? Depends on how you score it. Commented Jun 28, 2021 at 7:24
• @Steve Bennett. The solution should specify what makes a position winning, and how to win from there Commented Jun 29, 2021 at 3:11
• You're looking for the complete strategy to win this game no matter the starting position of the two kings? That seems broad. Commented Jun 29, 2021 at 3:27
• I tried it on 3x3 board, the first mover wins in all possible lines. In 4x4 board, I haven't tried all possible lines but the first mover seems to have the advantage. If we were able to generalize this to 8x8 board, I would say the color who has the first move wins. But of course this is neither a complete nor a proved correct answer. This problem might be brute-forced by a computer easily if you implement the rules. Commented Aug 12, 2021 at 8:35

## 2 Answers

I am gussing in the conditions you explained, the winner would be the king, who make an opposition (direct, diagonal, distant) first. Because this would lead to a chasing where at the end the other king is limited in movement, and can only move in a rank or file, hence the repetition of the position which has happend before.

• I think this is not correct. For example, if the white king is on e3 and the black king is on e6, White can play 1. Ke4 to form opposition, but then after 1... Kd6 2. Kd4 Ke6, it is White that would have to repeat a position to maintain the opposition.
– m90
Commented Nov 25, 2021 at 5:42
• @m90 This position is exactly what I meant. I assumed from the question that the lost is happening when the arragment of pieces is already repeated and the loser will be the one who has to move now. In your described position, whites repeat the arrangement of pieces by 3. Ke6 and now it's balck's turn to move. So the black loses. But If I misunderstood the question, then I guess maybe the answer could be changed to: " The loser will be the King, which makes the opposition first." Commented Nov 25, 2021 at 7:44

It's been long enough. Here's John's beautiful answer.

Suppose the Black king has just captured the 30th unit, to leave the two kings alone. White now chooses a 2x2 square currently containing the king, and moves clockwise or anti-clockwise within this 2x2 zone for the rest of the game.

At any point, bK can obstruct at most two of the squares in the zone, so wK can continue to move in the remaining squares.

Now here's the point. Suppose wK moves to repeat a position. Then wK just moved from the same square that it did the first time the position occurred. So this prior position has also been repeated - by the last Black move. So Black loses.

Why can't Black execute this strategy? Because the position just after Black captured the last unit has no ancestor with just two pieces. If this repeats, Black can't point to an earlier position that must also have repeated. So the strategy can only work for White.

Note that this is not "draw by triple repetition". A single recurrence is enough.

• Except if the last move came from the square where the white king captured the last piece? Commented Jun 20, 2023 at 18:52
• @RemcoGerlich Yes - I've fixed the solution thanks. Hope it's clearer why it works, and why it only works for White Commented Jun 21, 2023 at 5:40
• Note that this is not draw by triple repetition. A single recurrence is enough Commented Jun 21, 2023 at 12:56