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Walking on a lattice

Question

A random walk on the three dimensional integer lattice is defined as follows. The walker starts at (0, 0, 0). A standard six sided die is rolled six times. After each roll the walker moves to one of its six nearest neighbors, according to the following protocol: if the die rolls 1, 2, 3, 4, 5, or 6 dots the walker jumps one unit in the +x, −x, +y, −y, z, −z direction respectively. Find the probability that after the sixth roll the walker is back at its starting point (0, 0, 0).

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I got around 4%

But I wonder if there is a better way.

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Each step has a reverse step.

Arriving at (0, 0) means taking any three steps, along with their particular reverse steps.

Consider any of the steps taken. Its reverse must appear among the remaining five.

Consider any of the other four steps. Its reverse must appear among the remaining three.

Consider one of the remaining steps. The other must be its reverse.

All directions have 1/6 probability.

The probability of having a particular outcome in five chances, three chances and one chance,

multiplied together, is the desired result.

[1-(5/6)5] [1-(5/6)3] [1-(5/6)] = (4651/7776) (91/216) (1/6) = 423421/10077696 = 0.042.

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A simple exhaustive search shows that of 46656 possible outcomes, there are 1860 that end at the starting point.

1860/46656 = 0.03986625514403292181069958847737

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