Zeus starts at the origin $(0,0)$ and can make repeated moves of one unit either up, down, left or right, but cannot make a move in the same direction twice in a row. For example, he cannot move from $(0,0)$ to $(1,0)$ to $(2,0)$. What is the smallest number of moves that he can make to get to the point $(1056,1007)$ ?
(A) 2112
(B) 2161
(C) 2063
(D) 2111
(E) 2113
For Zeus to arrive at the point $(1056,1007)$, he must travel up and to the right from his starting point $(0,0)$.

More specifically, Zeus will need to make at least 1056 moves to the right $(R)$.

Since Zeus cannot move in the same direction twice in a row, then no two moves $R$ can be next to each other. In other words, there must be at least one move between each pair of $R$ s.

Since there are 1056 moves $R$ and there must be at least one move between each pair, then there are at least 1055 more moves (at least one between the $1^{\text {st }}$ and $2^{\text {nd }} R$, between the $2^{\text {nd }}$ and $3^{\text {rd }} R$, and so on).

Therefore, at least $1056+1055=2111$ moves are needed.

We will show that there is actually a sequence of 2111 moves that obey the given rules and that take Zeus to $(1056,1007)$.

For Zeus to end at a point with $y$-coordinate 1007, he will need to make at least 1007 moves up $(U)$.

So we put one $U$ between the $1^{\text {st }}$ and $2^{\text {nd }} R$, between the $2^{\text {nd }}$ and $3^{r d} R$, and so on up to between the $1007^{\text {th }}$ and the $1008^{\text {th }} R$.

This leaves us with $1055-1007=48$ spaces between $R$ s to fill.

We want to fill these spaces with moves that do not affect Zeus' position (since we already have him moving 1056 moves $R$ and 1007 moves $U$ ) and so that none of the moves are $R$ (since the spaces to be filled are between adjacent moves $R$ ).

We can do this by making the first 24 of these moves $U$ and the remaining 24 of these moves down $(D)$. This results in Zeus moving an additional 24 units up but then an additional 24 moves down, which is no net change in position.

We have shown that Zeus needs at least 2111 moves to get to $(1056,1007)$ and that he can actually get to $(1056,1007)$ in 2111 moves following the given rules, so the smallest number of moves that he needs is 2111 .

ANSWER: (D)