> Start by writing a row of n 0's on the piece of paper. (n is a
> positive integer decided beforehand by the players. I suggest an n
> between 5 and 10 for a 2 person game. Slightly more for more players.)

> After writing the row of n 0's, write the value of n to the right of
> this row.

> Next, the players take turns. On a player's move, he/she copies the
> row (which will be of 0's and 1's) immediately above, but with either
> one 1 changed to a 0, or one 0 changed to a 1. (The player can change
> any one digit she/he chooses, under restrictions -- see below.)

> Next, that same player writes down (to the right of the row) the
> lengths of the runs of both 0's and 1's in the row he just wrote down.
> Each "run" is made up completely of 0's or completely of 1's, and is
> bounded by runs of the other digit or by the edge of the row. (No two
> consecutive runs are of the same digit.)
> It doesn't matter if a run is of 0's or 1's. All that matters in this
> game is where each boundary is between each run of 0's and the
> adjacent run of 1's.

> * A player, though, cannot change a digit on his move such that the
> multiset of run-lengths (of the row of 0's and 1's just created) has
> already occurred in the game.
> (A "multiset" is a list of numbers where the order of the numbers in
> the list is unimportant, but the number of occurrences of each number
> is indeed important. For example, {1,2,1,3) and (2,1,1,3) would be
> considered to be the same multiset, but (1,2,1,3) and (1,2,3,3) would
> not be the same.)

> The last player able to move is the winner.

> Sample game. Simple example:
> (n=5)

> 00000     5
> 00010     3,1,1
> 10010     1,2,1,1
> 10011     1,2,2
> (Can't do 10111 here, for example, because the run-length multiset
> 3,1,1 already occurred.)
> 00011     3,2
> 00001     4,1

> The player who wrote 00001 wins, because 10001 (run-lengths 1,3,1),
> 01001 (1,1,2,1), 00101 (2,1,1,1), 00011 (3,2), and 00000 (5) each have
> a multiset of run-lengths that already occurred.

> FYI: The total number of moves in a game is no more than the number of
> (unrestricted) partitions of n. (So, there is a maximum of 7 moves in
> an n=5 game.)

> Thanks,
> Leroy Quet
> PS: More of my games here:
> http://gamesconceived.blogspot.com/