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 $(4, 4)$.
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. 
$(0,0): 1$
$(1,0)=(0,1)=1$
$(2,0)=(0, 2)=2$
$(3,0)=(0, 3)=3$
$(4,0)=(0, 4)=5$
$(1,1)=2$, $(1,2)=(2,1)=5$, $(1,3)=(3,1)=10$, $(1,4)=(4,1)= 20$
$(2,2)=14, (2,3)=(3,2)=32, (2,4)=(4,2)=71$
$(3,3)=84, (3,4)=(4,3)=207$
$(4,4)=2\cdot \left( (4,2)+(4,3)\right) = 2\cdot \left( 207+71\right)=2\cdot 278=\boxed{556}$
A diagram of the numbers:

~First