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Migrating ants

Problem 393 Published on Saturday, 8th September 2012, 05:00 pm; Solved by 601;
Difficulty rating: 50%

An n×n grid of squares contains n2 ants, one ant per square.
All ants decide to move simultaneously to an adjacent square (usually 4 possibilities, except for ants on the edge of the grid or at the corners).
We define f(n) to be the number of ways this can happen without any ants ending on the same square and without any two ants crossing the same edge between two squares.

You are given that f(4) = 88.
Find f(10).