I'm looking for references on this game (name, strategies analysis, ...) :

It's a two player game with two players (Black and White)

A position of the game is a single line (sequence) of black and white stones. If there are an odd numbers of stones, Black plays (White otherwise)

Each turn, a player remove a stone using one of this 3 rules

  1. You can remove the first or last stone of the line.
  2. You can remove a black stone.
  3. You can replace two consecutive white stones by a black stone.

As each move removes a stone, players play alternatively on each turn. When there is only one stone left, if it's a white stone, White wins, it it's a black stone, Black wins.

Example of a game :

  1. $\blacksquare\square\blacksquare\square\blacksquare$ Black removes the middle black stone
  2. $\blacksquare\square\square\blacksquare$ White removes the left black stone.
  3. $\square\square\blacksquare$ Black replaces the two consecutive white stones by a black stone
  4. $\blacksquare\blacksquare$ White resigns as he has no winning moves or just removes a black stone...
  5. $\blacksquare$ Black wins
  • $\begingroup$ I've been working here and there on this since you posted it; there's definitely some interesting patterns, but I don't yet have a complete collection of conjectures. I'll probably give up and post my partial results by the end of another week. Where did you come across this game? Or is it your own invention? $\endgroup$
    – Mark S.
    Feb 9, 2016 at 2:11
  • $\begingroup$ @MarkS. In fact I finally found references a few hours ago. $\endgroup$
    – Xoff
    Feb 9, 2016 at 4:50
  • $\begingroup$ That paper helps a lot! One thing I want to check is if you're still interested in the game where the starting position has an even number of symbols, because I think the paper doesn't discuss that at all (with Black going first). Either way, there's still more interesting stuff to say about this game, though. $\endgroup$
    – Mark S.
    Feb 10, 2016 at 12:26

1 Answer 1


Transposition game by Elise Janvresse, Steve Kalikow, Thierry De La Rue available on https://hal.archives-ouvertes.fr/hal-00372006v2

There is a potential function that you can easily compute for each configuration. White needs to minimize the absolute value of this potential while black needs to maximize it. And each turn, it can only be changed by plus or minus one.

The function is given on top of page 5 (page 6 of the pdf version), example in page 11 and figure page 14.


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