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In the limit, a random walk on the lattice is a Wiener process whose probability distribution after time t is normal and rotationally symmetric. So for the discrete case if each block moves for n steps picking a direction uniformly at random at each step then you would end up with an approximately circular "pile"
wence
Well doctor, it started when I dreamt I was in a snowstorm in a Minecraft-like world, and wondered what algorithm would be used to control the settling of cubic snowflakes on the landscape...
If a column of snowcubes start at cell X on a grid, and must each move to a randomly chosen neighbour cell including diagonals, how so you weight the choice so that you end up with roundish, not square snowdrifts?
When I said 'ratio in terms of pi' I'd assumed there was a simple-ish ratio.
Thanks for the exact answer @wence !
I now think a practical way of approximating it would be to draw a bigger version of the original diagram on a grid, and just count squares in the blue and red areas.