Euler's Painting Duality

Brilli the ant is standing on a face of a regular icosahedron. He wants to walk on every other face of the icosahedron exactly once, and then, in the next move, walk back to the face that he started from. How many ways are there for Brilli to do this?

Details and Assumptions: Brilli starts on a fixed face on the icosahedron. Brilli will not walk along edges; he will only walk on an edge to cross to another face. You may find it useful to use the code environment below, which contains a dictionary that encodes the three faces adjacent to each face.

adjacencies = {
1 : [7, 13, 19], 2 : [12, 18, 20], 3 : [16, 17, 19], 4 : [11, 14, 18],
5 : [13, 15, 18], 6 : [9, 14, 16], 7 : [1, 15, 17], 8 : [10, 16, 20],
9 : [6, 11, 19], 10 : [8, 12, 17], 11 : [4, 9, 13], 12 : [2, 10, 15],
13 : [1, 5, 11], 14 : [4, 6, 20], 15 : [5, 7, 12], 16 : [3, 6, 8],
17 : [3, 7, 10], 18 : [2, 4, 5], 19 : [1, 3, 9], 20 : [2, 8, 14]
}
Python 3

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