Solution to November’s Challenge

The Challenge: A chessboard of size n x n, colored with black and white squares in the usual alternating pattern, is displayed on a computer screen. You are allowed to select any rectangular sub-array of the squares and by pressing a mouse button reverse the black and white colors in that sub-array. Find the minimum number of mouse clicks necessary to change all the squares of an n x n chessboard to the same color; that is, to all black or to all white.

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Solution by Chase Schuler, Grand Rapids Forest Hills Northern High School

To make the chessboard one color, you start by clicking on every other row. That will yield a chessboard of alternating black and white columns. Then you click on the alternating columns. This process will require 8 clicks for the 8 x 8 chessboard. This process applies to any n x n chessboard; the number of clicks required is equal to n, provided n is even. If n is odd, the chessboard will require n-1 clicks. These two formulas can be summed up by using the floor function as follows:

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Also solved by: David Zhang (Ann Arbor Huron H.S.); Sarah Bier (Coldwater H.S.); Thomas Fields, Mitchell Kosowski, Katelyn Van Slyke, Elise Miller, Greg Calhoun, Drew Monks, Emily West (Grand Rapids Forest Hills Northern H.S.); Andrew Kraus (Grand Rapids Catholic Central); Joe Pawlowski, Stefan Progovac, Roger Klein (Grosse Pointe South H.S.); Nikhil Sebastian (Saginaw Heritage H.S.); Myron Chang, Lisa Tian, Lee Victor, Emily Wang(Plymouth-Canton Educational Park); Rachelle Grandia (Rockford H.S.); David Kye (Troy H.S.); Dustin Welch (Vestaburg H.S.); Albert Holzgang (Marquette H.S.).

Partial solutions were submitted by: Michael Finn, Amanda Pitts, Courtney Plouff, Amanda Winn (Grand Rapids Forest Hills Northern H.S.); Cody Van Tuyl (Kalamazoo Loy Norrix H.S.); Amy Pavlov (Marysville H.S.); Eric Zech (Plymouth-Canton Educational Park); Joshia Tan (Andover H.S.); David Scott (Rockford H.S.); Hannah Zhou (Troy H.S.); Marie Tominna (Troy Athens H.S.); Margaret Mueller (Ladywood H.S.); Jordan Beard (Big Rapids H.S.).