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?
How about 6x6x6 ?
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 7
11 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 11
11 11 17 11 11
17 17 26 17 17
11 11 17 11 11
11 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
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Posted by Jer
on 2014-10-23 10:00:22 |
5x5x5 solution
