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.
,
,
,
A diagram of the numbers:
5 - 20 - 71 - 207 -
3 - 10 - 32 - 84 - 207
2 - 5 - 14 - 32 - 71
1 - 2 - 5 - 10 - 20
1 - 1 - 2 - 3 - 5
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.
Solution 3 (General Case)
Mark the total number of distinct sequences of jumps for the frog to reach the point as
. Consider for each point
in the first quadrant, there are only
possible points in the first quadrant for frog to reach point
, and these
points are
. As a result, the way to count
is
Also, for special cases,
Start with , use this method and draw the figure below, we can finally get
(In order to make the LaTeX thing more beautiful to look at, I put
to make every number a
-digits number)
So the total number of distinct sequences of jumps for the frog to reach is
.
~Solution by (Frank FYC)
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