Orienting a Die
A standard 6-sided die has the property that the numbers on any two opposite sides add up to seven.
How many distinct ways are there of orienting the numbers on a standard 6-sided die?
Details and Assumptions:
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"Distinct" ways implies that you can't pick it up and reorient it to have the numbers appear on the same faces as in the original orientation.
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It has nothing to do with the orientation of the pips on a single side.
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1 solution
WLOG, you can pick the vertex where sides 1 , 2 , and 3 will meet. (They have to meet at some vertex, since no two of them add up to 7 so they can't be opposite each other.) You can also WLOG label one of these sides as 1 . That leaves only 2 possible orientations for the other two faces (of these three). Once you have labeled these three faces, the die is defined, since you only now have one possible choice for labeling sides 4 , 5 , and 6 .