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…you can't tell whose turn it is.
It's a bit of a trick question. The idea is that by just looking at the board's current contents, you can't tell whether it's White's turn or Black's turn. So there must be two game histories, one of x ply, where x is odd and it's White's turn, and one of y ply, where y is even and it's Black's turn, that produce the same board contents. From among the x and y values of all such pairs of game histories, we pick the smallest, and that's the answer. But if the answer is odd, we know it's White's turn and if the answer is even, we know it's Black's turn, so we really do know whose turn it is after all.
But seriously folks, my guess is that the answer is the shortest game in which at least two chips have been removed due to no longer being connected to a Dvonn. It is then ambiguous as to whether one of those chips was moved on top of the other before they were removed.
Sounds about right.
To be specific, I guess it's four (jumping) ply after the board is filled.
It seems to me that the state of the board after the first black chip has been placed, assuming that that chip is placed next to a dvonn chip, cannot be differentiated from an endgame position, where it may be either black's turn to play, or white's turn to pass. Or, if passing doesn't count as a turn, just add a white chip.
So depending on the definition I'd rather say the answer is 4 or 5 moves after the start of the game :)
New puzzle! Sweet.
I really like antony's answer. Motly because it's the right one, but also because I missed it. My defence is, one could probably tell if an unfinished game is in setup phase or moving phase, because at setup phase there would be two neat stacks for unplaced stones (one white one black, possibly one white at the bottom of the black pile), while at endgame there would be just jumbled pieces in the discard box.
If we go to the moving phase, the answer is no moves at all, if a pass counts, as black can have all the edges. Otherwise the answer is two. At that point situation can be such, that black may or may not have had all the edges and thus you cannot tell, which two high pile was played first.
Yes, great insight, antony!
However, isn't the completely empty, apart from 3 dvonns, position legal late in the game after either player's move? In which case 3 would be the answer, no? I'm presuming a real board where dvonn-on-dvonn is distinguishable from a solo dvonn. Otherwise, 1 seems to be the don't-know-what-phase answer.
If we're constrained to the moving phase, then one might say that if it's player X's move, and X must pass, then it's actually Y's move. In OtB tournament play, do you actually need to wait for your opponent to acknowledge that he has no moves, or can the one with moves left simply hammer them out back to back?
Phil> …completely empty, apart from 3 dvonns, position legal late in the game
It's a legal position late in the game, and the last move may have been made by either White or Black, but the game is over, so it's neither White's turn nor Black's turn. However, you can't tell by looking at the board whether it's Black's turn (early) or no one's (late).