2021 AIME II Problems/Problem 8
Problem
An ant makes a sequence of moves on a cube where a move consists of walking from one vertex to an adjacent vertex along an edge of the cube. Initially the ant is at a vertex of the bottom face of the cube and chooses one of the three adjacent vertices to move to as its first move. For all moves after the first move, the ant does not return to its previous vertex, but chooses to move to one of the other two adjacent vertices. All choices are selected at random so that each of the possible moves is equally likely. The probability that after exactly 8 moves that ant is at a vertex of the top face on the cube is 
, where 
and 
are relatively prime positive integers. Find 
Solution 1
On the first step, the ant has 3 choices. One choice is to go on top and there are two choices to stay on the bottom. Let BG be (Base but just got their), BO (Base but originally there), TO (Top but originally there) and TG (Top but just got their). We can list out the possibilities. (Note we do this because when you just got to the top, you cannot go back to the base.)
1 2 3 4 5 6 7 8
BG 0 0 2 6 10 14 26 62
BO 2 2 2 6 18 38 66 118
TO 1 2 2 2 6 18 38 66
TG 0 2 6 10 14 26 62 138
Therefore, the answer is 204/384 = 17/32. We want M+N, so 17+32 = 
See also
| 2021 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
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