This activity is based on the children's book The King's Chessboard by David Birch. The story provides a rich problem for children to explore patterns that are not linear. While much of the pattern is within the reach of intermediate students, I have added an extension for middle school students who are studying exponential equations. The King's Chessboard is one activity out of a unit called Solving Algebra and Pattern Problems.
Included is the lesson, student explorations sheets, answers and a middle school extension.