Discrete Mathematics Level 5

A single snowflake rests at the origin, \((0, 0)\), on an infinite cartesian plane. At time \(t = 0\), it divides and replicates itself, moving exactly \(1\) unit in each of the \(8\) major compass directions (N, NE, E, SE, S, SW, W, and NW), leaving \((0, 0)\) vacant.

At \(t = 1\), 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:

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So, to iterate the first few steps:

  • There is \(1\) snowflake at time \(t = 0\)
  • At time \(t = 1\), there are \(8\) snowflakes pre-merge.
  • No snowflakes coincide, so there are \(8\) snowflakes post-merge as well.
  • At time \(t = 2\), there are \(64\) snowflakes pre-merge.
  • Post-merge, only \(33\) snowflakes remain.

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

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