On the Bechtel terrace one of the chessboards had the following grid:
Imagine the chess game being played on this board. Since "X" marks the spot, pay special attention to the moves where a piece is captured (denoted with an 'x' in standard chess notation). Each capture happens on a unique square and gives you a single letter.
This puzzle has two answers. The squares where White captures a (black) piece give the answer COTTON BALL, suggesting a white circle. The squares where Black captures a (white) piece give the answer HARD SKI SLOPE, suggesting a black diamond.