Northcott's game

In every row of a rectangular board, there are two checkers: one white and one black. A move consists in sliding a single checker in its original row without jumping over another checker. You play white, computer plays black. As usual, the player to make the last move wins.


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Northcott's is obviously a partizan game. After every move it's possible to detect what color checker has been moved and thus determine which player performed the move. On the other hand, I may argue (and I believe you would agree with me, especially after trying the game several times) it's still an impartial game in disguise. Moreover, it's another Nim in disguise.

Indeed, every row may be looked at as a heap of beans (or row of checkmarks) with the number of beans defined by the distance between two checkers. A notable difference though is that in this variant it's also possible to add beans to a heap. In the regular Nim (Turning Turtles, Nimble) we only could remove beans from a heap. Interestingly, the moment one player increases the size of a heap, the other player may simply reduce the heap to its original size. In which case the first player faces precisely the same Nim-position as before the two last moves. So the Northcott's Nim introduces the notion of reversible moves. It's easy to see that the game still must end. For with every pair of reversible moves the number of beans that might be added to a heap dicreases and eventually becomes 0.

Variants of Nim that allow reversible moves are known as bogus nim. In the Northcott's game it's possible to bend rules a little by permitting each player to move checkers of either color. This will violate the ending condition: a game will never end. A trifle that removes the purpose in a game: to win. In some weird sense we still have a bogus nim in which it matters not whether a player makes right or wrong moves.

Reference

  1. E.R. Berlekamp, J.H. Conway, R.K. Guy, Winning Ways for Your Mathematical Plays, v1, A K Peters, 2001.
  2. J.H. Conway, On Numbers And Games, A K Peters, 2001
  3. R. K. Guy, Fair Game, Comap's Explorations in Mathematics, 1989

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