# Multiply

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:

Animated Image

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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