Math Problem Solver

Problem 1

You are trying to make a checkerboard, but all you have is an infinite supply of sheets of

rectangular paper of all sizes, which are red on one side and blue on the other. You plan to take

some sheets of paper, fold them, and arrange all the folded sheets together (not necessarily side-

by-side) so that the result, when viewed from the top, looks like a checkerboard of alternating

red and blue squares.

Because you are new to origami, you only know how to do one type of fold: you can crease

the paper along a line parallel to the paper’s edges, and then fold the two sides of the crease

together.

For example, Figure 1 shows how to fold a 1×2 checkerboard from a single 1×3 sheet of paper.

(a) Is it possible to fold a 1 × 8 checkerboard (i.e. 8 squares alternating between red and

blue), starting from a single 1 × n strip of paper, for some value of n?

(b) What is the minimum number of sheets (of any size) you need to make a 1 × 8 checker-

board? For example, you can use 4 sheets to make 4 copies of the 1 × 2 checkerboard

below and line them up into a 1 × 8 checkerboard.

(c) What is the minimum number of sheets you need to make an 8 × 8 checkerboard?

Feel free to explore variants, such as allowing other kinds of folds.