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Difference between revisions of "2018 AIME II Problems/Problem 8"

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==Solution==
 
==Solution==
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We solve this problem by working backwards. Notice, the only points the frog can be on to jump to <math>(4,4)</math> in one move are <math>(2,4),(3,4),(4,2),</math> and <math>(4,3)</math>. This applies to any other point, thus we can work our way from <math>(0,0)</math> to <math>(4,4)</math>, recording down the number of ways to get to each point.  
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We solve this problem by working backwards. Notice, the only points the frog can be on to jump to <math>(4,4)</math> in one move are <math>(2,4),(3,4),(4,2),</math> and <math>(4,3)</math>. This applies to any other point, thus we can work our way from <math>(0,0)</math> to <math>(4,4)</math>, recording down the number of ways to get to each point recursively.  
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<math>\begin{tikzpicture}
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<math>(0,0): 1</math>
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\draw[step=0.5cm, color=gray] (0,0) grid(4,4);
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<math>(1,0)=(0,1)=1</math>
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\end{tikzpicture}</math>
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<math>(2,0)=(0, 2)=2</math>
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<math>(3,0)=(0, 3)=3</math>
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<math>(4,0)=(0, 4)=5</math>
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<math>(1,1)=2</math>, <math>(1,2)=(2,1)=5</math>, <math>(1,3)=(3,1)=10</math>, <math>(1,4)=(4,1)= 20</math>
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<math>(2,2)=14, (2,3)=(3,2)=32, (2,4)=(4,2)=71</math>
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<math>(3,3)=84, (3,4)=(4,3)=207</math>
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<math>(4,4)=2\cdot \left( (4,2)+(4,3)\right) = 2\cdot \left( 207+71\right)=2\cdot 278=\boxed{556}</math>
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{{AIME box|year=2018|n=II|num-b=7|num-a=9}}
 
{{AIME box|year=2018|n=II|num-b=7|num-a=9}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 10:22, 24 March 2018

Problem

A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$, the frog can jump to any of the points $(x + 1, y)$, $(x + 2, y)$, $(x, y + 1)$, or $(x, y + 2)$. Find the number of distinct sequences of jumps in which the frog begins at $(0, 0)$ and ends at $(4, 4)$.

Solution

We solve this problem by working backwards. Notice, the only points the frog can be on to jump to $(4,4)$ in one move are $(2,4),(3,4),(4,2),$ and $(4,3)$. This applies to any other point, thus we can work our way from $(0,0)$ to $(4,4)$, recording down the number of ways to get to each point recursively. $(0,0): 1$ $(1,0)=(0,1)=1$ $(2,0)=(0, 2)=2$ $(3,0)=(0, 3)=3$ $(4,0)=(0, 4)=5$ $(1,1)=2$, $(1,2)=(2,1)=5$, $(1,3)=(3,1)=10$, $(1,4)=(4,1)= 20$ $(2,2)=14, (2,3)=(3,2)=32, (2,4)=(4,2)=71$ $(3,3)=84, (3,4)=(4,3)=207$ $(4,4)=2\cdot \left( (4,2)+(4,3)\right) = 2\cdot \left( 207+71\right)=2\cdot 278=\boxed{556}$


2018 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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All AIME Problems and Solutions

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