Difference between revisions of "2018 AIME I Problems/Problem 14"
(→Solution) |
|||
| Line 1: | Line 1: | ||
| − | Let <math>SP_1P_2P_3EP_4P_5</math> 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. | + | |
| + | |||
| + | Let <math>SP_1P_2P_3EP_4P_5</math> be a heptagon. A frog starts jumping at vertex <math>S</math>. From any vertex of the heptagon except <math>E</math>, the frog may jump to either of the two adjacent vertices. When it reaches vertex <math>E</math>, the frog stops and stays there. Find the number of distinct sequences of jumps of no more than <math>12</math> jumps that end at <math>E</math>. | ||
==Solution== | ==Solution== | ||
| − | (incomplete) | + | (incomplete, someone help format this) |
| − | Make a table | + | 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 | ||
| + | |||
| + | 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 <math>221 + 130 = \boxed{351}</math>. |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
Revision as of 11:49, 9 March 2018
Let 
be a heptagon. A frog starts jumping at vertex 
. From any vertex of the heptagon except 
, the frog may jump to either of the two adjacent vertices. When it reaches vertex , the frog stops and stays there. Find the number of distinct sequences of jumps of no more than 
jumps that end at .
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. 
all equal the sum of their adjacent elements in the previous jump. 
equals the previous 
. 
equals the previous 
. Each 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 
.
