Here is another game that is (guess what) played on an
n-by-n grid drawn on paper. (n is finite.)
(I suggest an n of about 5 to 10 for beginners.)
I do not know if I have stolen this idea from anywhere,
as is the case with all of my games I post here.
But this game does not seem that familiar.
This game can be played either as a two-person game or solitaire.
Players take turns placing 1 through n^2 into the empty squares
of the grid, the integers placed in order and a single integer
is placed into an empty grid-square every move.
(Player 1 places the odd numbers into the grid, player 2 places
the even integers into the grid.)
Players could place any marks they choose into the grid's squares,
actually; but I suggest placing numbers so as to keep track of
who moved where when.
An integer can be placed by a player any number of squares
(as long as the integer is placed within the grid) --
and in the direction of either up, down, left, right, or diagonally
-- from the last integer the player's opponent placed into the grid.
Player 1 can place the 1 in any square. Players can "jump over" any
integers already in the grid when going from the last integer by their
opponent to their current integer.
A player gets a point every time the number of squares from the
square he just filled-in to the last square filled in by his
opponent is greater than (not equal to or less than) the number
of squares from the last square filled-in by his opponent to the
square filled-in before by the player in his/her previous move.
In other words, if s(m,m-1) is the number of squares from the
square with the (m-1) in it to the square with the m in it,
then a player (placing the integer m on his current move) gets
a point if s(m,m-1) > s(m-1,m-2).
(And so players cannot score until the second move by player 1
at the earliest.)
Play continues until every square (in the directions of
up/down/left/right/diagonally from the last square either player
has put an integer in) each already have integers in them.
If there is a natural bias towards one player getting more points
than the other, then do what I suggest for all my games with a bias:
Just play two rounds, switching who is player 1 and who is player 2,
and then each player adds up their scores from both rounds to get
their grand score.
Simply play both the even integers and the odd integers as if you
are both players, but getting only one score, trying to get the
highest possible score you can for an n-by-n grid.
The strategies for the solitaire version vary, however, from the
strategies for the competitive version of this game.
(I bet there are some interesting strategies either way.)