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.)

In this way, maybe some kind of interesting design will emerge.
(Tongue in cheek)
:)

Thanks,
Leroy Quet

1.  It's played on two side-by-side grids, each 9 cells wide (although
it can be played on different grids, 9-cells wide seemed ideal.)
2.  You own one grid, and your opponent owns the other.
3.  Players take turns.  On each turn you write down a word in the
upmost available horizontal row on your grid.  Your score for turn is
the number of letters that your word shares with your opponent's most
recently played word, times the number of letters in the longest word
that you formed *vertically* when you added the new word
horizontally.  For example, if you wrote "tan" on your first turn, and
then on your second turn, you wrote "cat" such that the "t" from cat
is directly below the "a" from "tan", then you would have formed the
word "at" vertically.  Since "at" is 2 letters long, your score for
that turn is 2 times however many letters "cat" shares with your
opponent's most recently played word.
4.  the first letter of your word may be in any cell, as long as the
word fits completely in your grid.
5.  No word may be played twice, and you may not play a word that
contains in its entirety a word previously played by any player.
6.  There's a bonus if your word is an anagram of your opponent's most
recently played word.
7.  Game ends after say, 8 turns.  Highest score wins.
8.  If you think your opponent played a fake word, you can challenge
it, and if it is a fake word, you're opponent gets zero points for his
turn.  If your challenge fails, you must skip your next turn.  Fake
words *are not* removed from the board after they've been played.

9.  Since the score is undefined for the each player's first turn, the
first turn of the game works differently:

-To begin, a word is picked randomly from the dictionary, and then
player 1 writes that word into his topmost gridline.  He also adds any
number of points he wants to either player's score column.  Then
player 2 decides whether to play as player 1 or player 2.  This acts
as a pie rule, and the randomly chosen initial word ensures that each
game will proceed very differently.
-The score for player 2's first turn is the just the number of letters
that his first word shares with the randomly chosen word that player 1
wrote down on his first turn.