Mathematics Problem Of the Week

Spring 2007

POW 5: Odd Checkerboards

Each space in a square checkerboard is labeled with either a "0" or a "1".  A neighborhood of a space is defined to be the space itself and all spaces that touch it (vertical, horizontal, or diagonal).  We'll say a space has odd parity if the sum of all the spaces in its neighborhood is odd, otherwise its parity is even.  For example, in the 3x3 checkerboard shown, each space in the top row has odd parity, while the two bottom corners have even parity.

We'll call an entire checkerboard odd if every one of its spaces has odd parity.

• Find all 3x3 odd checkerboards - or prove that none exist.
• Repeat for 4x4 and 5x5 odd checkerboards. 

Note: You do not need to separately list versions that are obvious rotations of each other. 

For maximum credit you should justify (as efficiently as possible) that you have found all possible solutions.

Due Friday, March 2^nd , at Noon.

Solutions should be submitted to Dr. Maegan Bos’ mailbox in the Math office or sent via e-mail to mbos@stlawu.edu

Presentation counts! The prize-winning entry will be selected from all correct submissions, based on the clarity, creativity and elegance of the solution.

Look for the SLU POW on the Web at http://it.stlawu.edu/~math/activities/