Tic tac toe
In the game of Noughts and Crosses there are 8 distinct winning
lines. How many distinct winning lines are there in a game played
on a 3 by 3 by 3 board, with 27 cells?
Age
11 to 14
Challenge level
Problem
The ideas in this problem have been expanded and made into another, more detailed investigation. See Marbles in a Box.
| In the game of Tic Tac Toe (i.e. noughts and crosses) there are 8 distinct winning lines.
Investigate how many distinct winning lines there are in a game played on a 3 by 3 by 3 board, with 27 cells. A winning line connects 3 cells. |
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Student Solutions
We received only one correct solution to this problem.
There are only 49 winning lines, which no doubt many of you will find hard to believe. Duplication of lines obviously helped muddy the approaches that were used with this problem.
Laura of West Flegg GM Middle School used an approach which classified the type of winning line possible with accompanying sketches :
9 vertical lines, 18 horizontal and 22 diagonal lines.
(Of the diagonal lines 6 go from front to back, 6 go from left to right, 6 go from top to bottom while four lines connect the 8 corner cells.)
a total of 49 lines
(27 of length 1, 18 of length Root 2 and 4 of length Root 3)