1999 Gauss 8 Problem 20

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?

\textbf{(A)} P \textbf{(B)} Q \textbf{(C)} R \textbf{(D)} T \textbf{(E)} W

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Gauss contests here.