A game is played on the board shown. In this game, a player can move three places in any direction (up, down, right or left) and then can move two places in a direction perpendicular to the first move. If a player starts at $S$, which position on the board ( $P, Q, R, T$, or $W$ ) cannot be reached through any sequence of moves?
(A) $P$
(B) $Q$
(C) $R$
(D) $T$
(E) $W$

![](https://cdn.mathpix.com/cropped/2024_04_20_46ead6524a8d61e21c51g-102.jpg?height=217&width=279&top_left_y=325&top_left_x=1408)
A game is played on the board shown. In this game, a player can move three places in any direction (up, down, right or left) and then can move two places in a direction perpendicular to the first move. If a player starts at $S$, which position on the board $(P, Q, R, T$, or $W$ ) cannot be reached through any sequence of moves?
(A) $P$
(B) $Q$
(C) $R$
(D) $T$
(E) $W$

|  |  | $P$ |  |  |
| :--- | :--- | :--- | :--- | :--- |
|  | $Q$ |  | $R$ |  |
|  |  | $T$ |  |  |
| $S$ |  |  |  | $W$ |

## Solution

If $S$ is the starting position we can reach position $R$ immediately. From $S$ we can also reach $P$ and then $W$ and $Q$ in sequence. To reach position $T$, it would have to be reached from the upper right or upper left square. There is no way for us to reach these two squares unless we are allowed to move outside the large square which is not permitted.

ANSWER:(D)