Problem:
A bee is moving in three-dimensional space. A fair six-sided die with faces labeled , , and is rolled. Suppose the bee occupies the point . If the die shows , then the bee moves to the point , and if the die shows , then the bee moves to the point . Analogous moves are made with the other four outcomes. Suppose the bee starts at the point and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Without loss of generality, assume that the first roll is . In order for the bee to traverse four distinct edges of a cube, the second roll cannot be or , so there are 4 rolls ( , , and ) that are allowed at this stage. Each new roll must represent a perpendicular direction for the bee, and there are 3 choices that remain in compliance for the third roll-2 of which extend into three dimensions, and 1 of which creates a " C " shape. In the former case, there are 2 choices for the fourth roll, while in the latter case, there are 3 choices (including the one where the bee traverses four edges forming a square). In total, the number of compliant paths is . The total number of paths is , and the probability that the path represents exactly four edges of a unit cube is
The problems on this page are the property of the MAA's American Mathematics Competitions