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 Extending tic-tac-toe (Posted on 2014-10-23)
In a standard 3x3 tic-ta-toe there are 8 distinct possibilities to create “three in a row”- 3 horizontal, 3 vertical and 2 diagonal.

How many ways are there in a three-dimensional 5x5x5 tic-tac-toe?

 No Solution Yet Submitted by Ady TZIDON No Rating

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 5x5x5 solution | Comment 2 of 6 |
Solution: 603

Method:  Look at the top layer and determine how many "three in a row" have one end at each cube.
` 7  7 11  7  7 7  7 11  7  711 11 17 11 11 7  7 11  7  7 7  7 11  7  7`

The sum of these is 217 (I'll take care of the over-count at the end.)

The 2nd, 4th and 5th layers would be the same.
The 3rd layer has
`11 11 17 11 1111 11 17 11 1117 17 26 17 1711 11 17 11 1111 11 17 11 11`

The sum of these is 338.

The total for the 5 layers is 4*217+338=1206

Because each "three in a row" is counted exactly twice - once at each end - just divide by two to get the solution: 1206/2 = 603

 Posted by Jer on 2014-10-23 10:00:22

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