Here are two games using the same idea.
Both games are for two people and use an n-by-n grid
drawn on paper. (I suggest an n of 8 to 10 for game 1,
and an n of 12 or so for game 2.)
For both games either player puts a 1 in the upper left
square of the grid.
Players then, on their move, place a number into any
empty square that is next to (in the direction of either
left, right, above, or below) at least one square
with a number already in it. The number the player
puts in the square must be 1 greater than a number
in any of the squares adjacent to the square the player
is writing the number in.
The grid is completely filled in in this way.
Game 1) Players take turns filling in the numbers.
Every time a player writes a prime into a square,
he/she gets a point. Highest score wins.
Game 2) Players make two grids of the same size,
one grid made by each player. When both grids are
complete each player gives their grid to their opponent.
Players then race to see which player can first find
a path (moving up, down, left, right) from the upper
left square of the grid each player is trying to solve
to the grid's lower right square,
moving from 1 to 2 to 3 to...(so that each square
the paths move onto is one number higher than each
path's previous square's number). (Any path that
goes from upper left square to lower right square
while following the rules is a valid path, whether
or not the found path is the path intended by the
A variation: For game 2, when players
are constructing their mazes for their opponents,
they each get a point for each prime in the grid
they construct. So, in essence, these are two
games, a solitaire version of game 1, and game 2.