Difference between revisions of "2018 AIME I Problems/Problem 14"

(Solution)
Line 4: Line 4:
  
 
==Solution==
 
==Solution==
(incomplete, someone help format this)
+
(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. <math>P_2, P_1, S, P_5</math> all equal the sum of their adjacent elements in the previous jump. <math>P_3</math> equals the previous <math>P_2</math>. <math>P_4</math> equals the previous <math>P_5</math>. Each <math>E</math> equals the previous adjacent element in the previous jump.
+
Make a table showing how many ways there are to get to each vertex in a certain amount of jumps. <math>P_2, P_1, S, P_5</math> all equal the sum of their adjacent elements in the previous jump. <math>P_3</math> equals the previous <math>P_2</math>. <math>P_4</math> equals the previous <math>P_5</math>. Each <math>E</math> equals the previous adjacent element in the previous jump. \
  
 
Jump & E & P_3 & P_2 & P_1 & S & P_5 & P_4 & E  
 
Jump & E & P_3 & P_2 & P_1 & S & P_5 & P_4 & E  
  
0 0 0 0 0 1 0 0 0
+
0 0 0 0 0 1 0 0 0 \
1 0 0 0 1 0 1 0 0
+
1 0 0 0 1 0 1 0 0 \
2 0 0 1 0 2 0 1 0
+
2 0 0 1 0 2 0 1 0 \
3 0 1 0 3 0 3 0 1
+
3 0 1 0 3 0 3 0 1 \
4 1 0 4 0 6 0 3 1
+
4 1 0 4 0 6 0 3 1 \
5 1 4 0 10 0 9 0 4
+
5 1 4 0 10 0 9 0 4 \
6 5 0 14 0 19 0 9 4
+
6 5 0 14 0 19 0 9 4 \
7 5 14 0 33 0 28 0 13
+
7 5 14 0 33 0 28 0 13 \
8 19 0 47 0 61 0 28 13
+
8 19 0 47 0 61 0 28 13 \
9 19 47 0 108 0 89 0 41
+
9 19 47 0 108 0 89 0 41 \
10 66 0 155 0 197 0 89 41
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10 66 0 155 0 197 0 89 41 \
11 66 155 0 352 0 286 0 130
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11 66 155 0 352 0 286 0 130 \
12 221 - - - - - - 130
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12 221 - - - - - - 130 \ \
  
  
 
The number of ways to jump to the ends within 12 jumps is <math>221 + 130 = \boxed{351}</math>.
 
The number of ways to jump to the ends within 12 jumps is <math>221 + 130 = \boxed{351}</math>.

Revision as of 11:50, 9 March 2018


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 $221 + 130 = \boxed{351}$.