Mathematical Games

Some of these games have worked well with 12 year olds. First the teacher takes on a pupil. Then pupils play against each other in pairs. Then pupils who think they've "got it" try against the teacher.


Players take turns to remove any number of objects from one of n rows. The player who takes the last object wins.

Ans:If a game is played with a finite set of numbers and each move changes only one set and the game terminates, it is equivalent to NIM.

To determine who should win, write the sizes of the piles in binary. Add the columns (no carrying). If the sum of each column is even, the person to play loses. (from Mathematical Carnival)

In the reverse game the player who takes the last counter loses
Ans: The strategy in the reverse game follows the normal game until only one row has more than one counter. Then you take away either all the biggest row or all but one of this row so as to leave an odd number of rows.

Variations include

Noughts and Crosses

Grid Games

  1. Draw a 3x3 grid. Players can color in any amount of squares on one row or column. The winner is the one who fills in the last square.
    Ans: The 2nd player can always win. If the first player fills a single cell in, then the 2nd player should fill 2 cells in to make an 'L'. Otherwise the 2nd player should complete an 'L' or 'T' consisting of 5 cells.
  2. TacTix - Draw a 4x4 grid. Players can color in any amount of adjoining squares on one row or column. The winner is the one who fills in the last square. Ans: Gardner in "Mathematical puzzles & diversions" says that the second player can always win, but there's no simple strategy. The reverse game is much harder. A variation, where the colored in squares needn't be adjoining, is harder still.


  1. 10x10
  2. 2 players take turns to add 1-10 to a common total. First to 100 wins.
    Ans: (First to 89 wins)
  3. Tug of war
        | | | |X| | | |
    Each player has 50 muscle points to gamble. At each turn the one who gambles the most pulls the knot one square their way. The player who pulls the knot to their end wins.
  4. 13 petals (Sam Lloyd) Remove 1 or 2 connected petals. The one who takes the last wins.
    Ans: (preserve symmetry)


  1. On Games and Numbers, J.H. Conway
  2. Mathematical Carnival, Martin Gardner, Penguin 1976
  3. Mathematical puzzles & diversions, M. Gardner, London, 1961.
  4. Mathematical Games, C. Lukacs and E. Tarjan, Souvenir, 1969
See also Mathematical puzzles
Updated 6th April, 1998