2021 Fall AMC 10A Problem 24
Problem 24 of 25HarderGeometryCombinatoricsProblem-Solving Techniques
Each of the edges of a cube is labeled or Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the faces of the cube equal to
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Solution
Label one face Each face must contain two s and two s, so first split by the pattern on the four edges of
Case opposite edges of have the same label. There are such patterns on For either pattern, once the label of one vertical edge, say is chosen, the face-sum conditions force all remaining labels. This gives labelings.
Case opposite edges of have different labels. There are such patterns on For each, the labels of two adjacent vertical edges may be chosen in ways, and then the remaining labels are forced by the face-sum conditions. This gives labelings.
The total is
Thus, E is the correct answer.
Tagged: arrangements with restrictions · cube geometry · casework