2018 AIME I Problems/Problem 14

Let $SP_1P_2P_3EP_4P_5$ be a heptagon. A frog starts jumping at vertex $S$ . From any vertex of the heptagon except $E$ , the frog may jump to either of the two adjacent vertices. When it reaches vertex $E$ , the frog stops and stays there. Find the number of distinct sequences of jumps of no more than $12$ jumps that end at $E$ .

Solution

(incomplete, someone help format this)

Make a table showing how many ways there are to get to each vertex in a certain amount of jumps. $P_2, P_1, S, P_5$ all equal the sum of their adjacent elements in the previous jump. $P_3$ equals the previous $P_2$ . $P_4$ equals the previous $P_5$ . Each $E$ equals the previous adjacent element in the previous jump.

Jump & E & P_3 & P_2 & P_1 & S & P_5 & P_4 & E

0 0 0 0 0 1 0 0 0 \\ 1 0 0 0 1 0 1 0 0 \\ 2 0 0 1 0 2 0 1 0 \\ 3 0 1 0 3 0 3 0 1 \\ 4 1 0 4 0 6 0 3 1 \\ 5 1 4 0 10 0 9 0 4 \\ 6 5 0 14 0 19 0 9 4 \\ 7 5 14 0 33 0 28 0 13 \\ 8 19 0 47 0 61 0 28 13 \\ 9 19 47 0 108 0 89 0 41 \\ 10 66 0 155 0 197 0 89 41 \\ 11 66 155 0 352 0 286 0 130 \\ 12 221 - - - - - - 130

The number of ways to jump to the ends within 12 jumps is $221 + 130 = \boxed{351}$ .

2018 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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2018 AIME I Problems/Problem 14

Solution