2006 AMC 12A Problems/Problem 20

The following problem is from both the 2006 AMC 12A #20 and 2006 AMC 10A #25, so both problems redirect to this page.

Problem

A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. At each vertex the bug will choose to travel along one of the three edges emanating from that vertex. Each edge has equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?

$\mathrm{(A) \ } \frac{1}{2187}\qquad \mathrm{(B) \ } \frac{1}{729}\qquad \mathrm{(C) \ } \frac{2}{243}\qquad \mathrm{(D) \ } \frac{1}{81} \qquad \mathrm{(E) \ } \frac{5}{243}$

Solution

Solution 1 Let us count the good paths. The bug starts at an arbitrary vertex, moves to a neighboring vertex (3 ways), and then to a new neighbor (2 more ways). So, without loss of generality, let the cube have vertex $ABCDEFGH$ such that $ABCD$ and $EFGH$ are two opposite faces with $A$ above $E$ and $B$ above $F$ . The bug starts at $A$ and moves first to $B$ , then to $C$ .

From this point, there are two cases.

Case 1: the bug moves to $D$ . From $D$ , there is only one good move available, to $H$ . From $H$ , there are two ways to finish the trip, either by going $H \to G \to F \to E$ or $H \to E \to F \to G$ . So there are 2 good paths in this case.

Case 2: the bug moves to $G$ . Case 2a: the bug moves $G \to H$ . In this case, there are 0 good paths because it will not be possible to visit both $D$ and $E$ without double-visiting some vertex. Case 2b: the bug moves $G \to F$ . There is a unique good path in this case, $F \to E \to G \to H$ .

Thus, all told we have 3 good paths after the first two moves, for a total of $3\cdot 3 \cdot 2 = 18$ good paths. There were $3^7 = 2187$ possible paths the bug could have taken, so the probability a random path is good is the ratio of good paths to total paths, $\frac{18}{2187} = \frac2{243}\Rightarrow \boxed{\mathrm (C)}$ .

Solution 2 As in Solution 1, the bug can move from its arbitrary starting vertex to a neighboring vertex in 3 ways. After this, the bug can move to a new neighbor in 2 ways (it cannot return to the first vertex). The total number of paths (as stated above) is $3^7$ or $2187$ . Therefore, the probability of the bug following a good path is equal to $\frac{6x}{2187}$ for some positive integer $x$ . The only answer choice which can be expressed in this form is $\frac2{243}\Rightarrow\boxed{\mathrm (C)}$ .

2006 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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All AMC 12 Problems and Solutions

2006 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Last Question
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All AMC 10 Problems and Solutions

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2006 AMC 12A Problems/Problem 20

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