Reuben has a 4x4 checkerboard. He uses scissors to cut out each square from the board. He then randomly arranges the 16 pieces into four rows and four columns.
Disregarding rotations, determine the probability that this layout is in a precise checkerboard pattern.
How will the answer change if rotations are taken into consideration?
It doesn't matter which color the first piece (say the upper left) is, as, if necessary, the completed board can be rotated 90°.
The piece next to it has probability 8/15 of being the required opposite color. The next has probability 7/14 of being the same color as the first, and the next has probability 7/13 of continuing the alternation.
8 * 7 * 7 * 6 * 6 * 5 * 5 * 4 * 4 * 3 * 3 * 2 * 2 * 1
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15*14*13*12*11*10* 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2
8! * 7! / 15! = 1/6435 ~= 0.000155400155400155
If it's required that the top left be a specific one of the two colors, the probability is 1/2 this, or 1/12870 ~= 7.77000777000777 x 10^-5
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Posted by Charlie
on 2022-05-16 10:42:35 |
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