Here is a 1-player game played on an n-by-n grid, where n is odd. (I suggest an n of about 13 to 21 for beginners.)
The player places the numbers 1,2,3,...n^2 IN ORDER into the grid.
1 goes in the center square.
Each number (2k+1) must go either below, above, left of, or right of the square with (2k-1) in it.
Each number (2k) can go in any square that is immediately adjacent to a square with an integer already in it.
Numbers can only be placed in empty squares.
The player's score is the number of times the path of odd integers goes completely around the center square clockwise before the player is unable to place any more numbers in the grid.