Here is a game played on a n-by-n grid drawn on paper. The game is for 2 to 4 players. The size of the grid (n) should be relatively small as far as my games go, n = 4 to 6 for a 2-player game.
In the first phase of the game players take turns, each placing ANY integer from 1 to n^2 into any empty square of the grid on a turn.
(The same integer may be used more than once in the grid.)
In the second phase of the game, each player has a marker (each marker is small enough to fit inside a single square of the grid). Each player places his/her marker in a different corner square at the start of the second phase.
Players move in a predetermined order (such as clockwise by which corners the players start the second phase in).
The first player to move moves to any (above, below, right of, or left of, or DIAGONAL TO) adjacent square.
Players thereafter take turns moving their markers from square to adjacent square (in the directions of orthogonally or diagonally).
Say, the last player to move (whom we will call "player A") moved from a square numbered j to a square numbered k. Then on the next player's (Player B's) move, Player B MUST move, IF it is possible, to a square such that GCD(j,k) equals GCD(p,q), OR such that |j-k| = |p-q| -- where p = the number of the square Player B was on, and q = the number of the square Player B is moving to.
In other words, if it is possible, player B must move so that the numbers of his/her consecutively-moved-on squares have either the same greatest common divisor or same absolute difference as the last two consecutively-moved-on squares of Player A -- where player A is the player who moves just before player B (and where who exactly are players A and B changes each move -- player B always being the player currently moving).
If there are no such adjacent squares that have the same GCD or absolute difference, then player B may move to ANY adjacent square.
If a player is forced to move onto a square already occupied by another player's marker, then the SECOND player to occupy the square is removed from competition, and his/her marker is removed from the grid. (A player landing on an already occupied square is a 'drat', as in "Drat!".)
Play continues until there is one remaining player, who is then the winner.
Note: In the small number of trials where I played myself, it seemed that all games are either long or short, but never middle-lengthed.
Maybe some other math rules, besides GCD or absolute difference, would perhaps make this a more fun game.
PS: This game has been edited to allow for diagonal moves. Otherwise, with only orthogonal moves, one player may be unable to win no matter what.