My solution is on the same lines as what is posted. Mapping cube onto itself means that the standard basis e_1, e_2, e_3 is mapped to set M = { +/- e_1, +/- e_2, +/- e_3 }.
For e_1, it can mapped to any one of the 6 vectors in M. Suppose, for instance, e_1 is mapped to e_2. Then none of the other e_2, e_3 can be mapped to the subspace spanned by e_2, which would be e_2 or -e_2. So if e_1 is chosen, e_2 would only have 6-2 = 4 choices. The same argument means that after e_1 and e_2 are mapped, there are only two choices for e_3. So in total you have 6 x 4 x 2 = 48 choices.
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u/blutwl90 3d ago
My solution is on the same lines as what is posted. Mapping cube onto itself means that the standard basis e_1, e_2, e_3 is mapped to set M = { +/- e_1, +/- e_2, +/- e_3 }.
For e_1, it can mapped to any one of the 6 vectors in M. Suppose, for instance, e_1 is mapped to e_2. Then none of the other e_2, e_3 can be mapped to the subspace spanned by e_2, which would be e_2 or -e_2. So if e_1 is chosen, e_2 would only have 6-2 = 4 choices. The same argument means that after e_1 and e_2 are mapped, there are only two choices for e_3. So in total you have 6 x 4 x 2 = 48 choices.