Probability Level 5

A single snowflake rests at the origin, (0,0)(0, 0), on an infinite cartesian plane. At time t=0t = 0, it divides and replicates itself, moving exactly 11 unit in each of the 88 major compass directions (N, NE, E, SE, S, SW, W, and NW), leaving (0,0)(0, 0) vacant.

At t=1t = 1, all snowflakes which occupy the exact same points are merged into one, and then the process repeats for every snowflake, resulting in the following pattern:

So, to iterate the first few steps:

  • There is 11 snowflake at time t=0t = 0
  • At time t=1t = 1, there are 88 snowflakes pre-merge.
  • No snowflakes coincide, so there are 88 snowflakes post-merge as well.
  • At time t=2t = 2, there are 6464 snowflakes pre-merge.
  • Post-merge, only 3333 snowflakes remain.

How many snowflakes are there at time t=1000t = 1000, post-merge?

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