A list of math problems related to parity

1. Parity Problems
1. Begin with five cups, four of which are down and 1 of which is up. Each move consists of turning over
exactly 2 cups. The goal is to turn all of the cups upright. Can you do it? Why or why not? Play
again but start with a different number of cups up or down. Or start with a different total number of
cups. What do you observe?
2. Write down six 0’s and five 1’s on a piece of paper. Then begin crossing out pairs of digits: either two
1’s, two 0’s, or a 1 and a 0. If the digits crossed out are the same, then write a new 0. If the digits
crossed out are different, then write a new 1. Continue in this way. What happens?
3. There are 9 people at a party. Prove that it is impossible for each of them to shake hands with exactly
3 other people.
4. Arrange some number of two-color counters in a line in front of you. Have some of each color to begin
with. You may remove a yellow counter so long as you turn over the adjacent (neighbor) counters after
doing so. You may not remove a red counter. Play this game several times with a partner. and take
note of what you see. Write it down!
5. A spacemouse, floating through space, finds a 3 × 3 × 3 block of cheese cubes (think of a Rubik’s cube
made of cheese). If it begins at one of the corner cubes and gobbles up a small cube at a time moving
to the next cube of cheese (through a face), can our space mouse end at the cube at the center of our
block?
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