Scoring Misère

Misère games are played by the same rules as the normal ones with one notable exception: while in the normal game the player unable to move loses, in the misère games, the player unable to move wins.

In the Scoring misère, like in Scoring, the players are presented with one or more piles (or heaps) of objects (chips, counters, pebbles.) A move consists in removing a number of objects from a single pile. In Scoring (normal or misère), a player, on a single move, is allowed to remove one or more objects up to a prescribed maximum.

Strangely, the misère games are by far more difficult than their normal counterparts. A winning strategy is known for the straight Nim and its various incarnations (Nimble, Plainim, Date Game, or, say, Silver Dollar Game With No Silver Dollar.) For Nim, the winning strategy is to play as in normal Nim until all non-empty heaps with one exception, contain a single counter. Then make a move so as to leave an odd number of single counter heaps.

Scoring, especially with several heaps, is often (and mistakenly) identified with Nim. In particular, it is very easy to give an example when the above Nim misère strategy does not work for the Scoring misère. I've no doubt you would run into such a situation if you play with the applet below. (In addition to the above, it implements one other strategy and makes a random selection between the two.)

 

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

A two heaps misère admits a perfect strategy which has been implemented in a separate applet.

References

  1. E. R. Berlekamp, J. H. Conway, R. K. Guy, Winning Ways for Your Mathematical Plays, Volume 2, A K Peters, 2003
  2. J. H. Conway, On Numbers And Games, A K Peters, 2001
  3. R. Guy, fair game, Comap's Explorations in Mathematics, 1989

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