2018 AIME II Problems/Problem 8
Problem
A frog is positioned at the origin of the coordinate plane. From the point , the frog can jump to any of the points
,
,
, or
. Find the number of distinct sequences of jumps in which the frog begins at
and ends at
.
Solution 1
We solve this problem by working backwards. Notice, the only points the frog can be on to jump to in one move are
and
. This applies to any other point, thus we can work our way from
to
, recording down the number of ways to get to each point recursively.
(2,0)=(0, 2)=2$$ (Error compiling LaTeX. Unknown error_msg)(3,0)=(0, 3)=3$$ (Error compiling LaTeX. Unknown error_msg)(4,0)=(0, 4)=5$$ (Error compiling LaTeX. Unknown error_msg)(1,1)=2
(1,2)=(2,1)=5
(1,3)=(3,1)=10
(1,4)=(4,1)= 20$$ (Error compiling LaTeX. Unknown error_msg)(2,2)=14, (2,3)=(3,2)=32, (2,4)=(4,2)=71$$ (Error compiling LaTeX. Unknown error_msg)(3,3)=84, (3,4)=(4,3)=207$$ (Error compiling LaTeX. Unknown error_msg)(4,4)=2\cdot \left( (4,2)+(4,3)\right) = 2\cdot \left( 207+71\right)=2\cdot 278=\boxed{556}$
Solution 2
We'll refer to the moves ,
,
, and
as
,
,
, and
, respectively. Then the possible sequences of moves that will take the frog from
to
are all the permutations of
,
,
,
,
,
,
,
, and
. We can reduce the number of cases using symmetry.
Case 1:
There are possibilities for this case.
Case 2: or
There are possibilities for this case.
Case 3:
There are possibilities for this case.
Case 4: or
There are possibilities for this case.
Case 5: or
There are possibilities for this case.
Case 6:
There are possibilities for this case.
Adding up all these cases gives us ways.
2018 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.