Nim - Variants

There are many variants of Nim that are interesting to play and some that are just Nim in disguise!

Tac Tix

In Mathematical Puzzles and Diversion, Gardner calls this the most exciting Nim variant.  It was invented by Piet Hein who also invented Hex.  It is played using a square (or rectangular) grid of counters, commonly 4x4.  Players take turns to remove a contiguous group of counters from any row or column (i.e. spanning no gaps).  The player who takes the last counter loses.

In the diagram, the first player took 1 counter from the second row (or perhaps the third column!).  If the second player also wants to use the second row, he can take the two on the left, or the one on the right, but not all three.

Gardner said that on a 4x4 board, there was no simple strategy, but it was know that the second player had a win.  He said the 6x6 game was unsolved and interesting to play.  He also gave a couple of problems on a 4x4 board for readers to solve.

If, instead, the person who takes the last counter wins, then the game becomes trivial.  If the board has an even number of counters on each side, then the second player wins using a half-turn symmetry strategy.  If there are an odd number of counters on each side, then the first player has a winning strategy by taking the centre counter first, then using the symmetry strategy.

More to follow . . . .