Coloring and counting rectangles on the board

Summary We describe the Rectangles on the Board project, an adaptation of an activity for elementary and middle school students that appears in [1]. Students are challenged to determine the coloring of the instructor's 10 × 10 board, given the restrictions that (1) all 100 squares are colored in one of four colors and (2) the colors form four rectangular regions, one in each color. Our extensions of this project involve counting, symmetry, geometry, and logical reasoning. For example, given the color of some squares, students infer the color of other squares based on geometry. In turn, they use logic and their understanding of this geometry to count the minimal number of squares needed to be revealed so that they can determine all of the squares' colors. Similar reasoning leads them to a best next “guess,” when playing the game. Students use combinatorics and symmetry to count the number of ways to color the board. Notes for the instructor This project is suitable for mathematics courses at all levels. We have implemented versions of the activity in elementary, middle, and high school classes, as well as in undergraduate mathematics and graduate mathematics education courses. The game-like aspect of the project is engaging and appealing to students. The project can be used to spur all-class discussions, or can be used to promote cooperative learning. Depending on how deeply your class desires to delve into the activity and its extensions, it can take from one to three 50-minute class periods to complete.