What is Numerical Tic-Tac-Toe?
Numerical Tic-Tac-Toe is played on the usual 3×3 board, but with the numbers 1 to 9 instead of X and O. The first player owns the odd numbers 1, 3, 5, 7 and 9, and the second player owns the even numbers 2, 4, 6 and 8. On each turn you write one of your numbers that has not been used yet in an empty square.
You win by completing a line of three numbers, across, down or diagonally, that adds up to exactly 15. The line can mix both players' numbers: what counts is who fills its last square. A full line with any other total wins nothing, and if the board fills without a line of 15 the game is a draw.
Both players have to keep track of which numbers are still free, for themselves and for the opponent. A line holding two numbers with a total between 6 and 14 is a threat for whoever still holds the missing number. A computer search of every position shows that the first player wins with perfect play. The centre is not the winning opening here: the first player wins by putting 1, 3, 7 or 9 on the middle square of an edge.
Rule changes
| Rule | Standard Tic-Tac-Toe | Numerical Tic-Tac-Toe |
|---|---|---|
| What you place | Standard Tic-Tac-ToeX for the first player, O for the second. | Numerical Tic-Tac-ToeA number. The first player places the odd numbers 1, 3, 5, 7 and 9, the second the even numbers 2, 4, 6 and 8, each once. |
| How to win | Standard Tic-Tac-ToeComplete a line of three of your own marks. | Numerical Tic-Tac-ToeComplete a line of three numbers that adds up to 15, whoever placed the other two. |
| Draws | Standard Tic-Tac-ToeA full board without a line is a draw. | Numerical Tic-Tac-ToeA full board without a line that adds up to 15 is a draw. |
| Perfect play | Standard Tic-Tac-ToeA draw. | Numerical Tic-Tac-ToeA win for the first player, who must open with 1, 3, 7 or 9 on the middle square of an edge. |
| Board | Standard Tic-Tac-Toe3×3. The table setup also offers 4×4 and 5×5. | Numerical Tic-Tac-ToeAlways 3×3. |
History
Numerical Tic-Tac-Toe was invented by the American mathematician Ronald Graham, known for his work in combinatorics and for Graham's number. The game can be generalised to larger square boards with more numbers.
The target of 15 is the sum of every row, column and diagonal of the Lo Shu, the 3×3 magic square of the numbers 1 to 9, and there are exactly eight ways to choose three different numbers from 1 to 9 that add up to 15, one for each line of the board.