17.6.1. Identifying Pigeons and Holes

Try these first

1. 1.

   Explain the pigeonhole principle in your own words.

   Hint

   Think about socks in drawers.

2. 2.

   What is the midpoint of (3,5) and (7,9)?

   Hint

   Add x-coordinates and y-coordinates separately, then divide by 2.

3. 3.

   What does the pigeonhole principle state?

   • If items equal holes
   • all containers are full.
   • More items than containers results in at least one full container.
   • There are empty containers.

   Hint

   Think about sock drawers!

4. 4.

   True or False: Two distinct integers selected from 1 to 8 cannot sum to 9.

   • True
   • False

   Hint

   Check the pairs methodically.

5. 5.

   Demonstrate that selecting 5 coordinates (x,y) within a certain range guarantees common parity.

   Hint

   Remember your mapping from earlier!

6. 6.

   Given any integer k, what is the least number of 1's you can arrange to find a multiple of k consisting only of 0s and 1s?

   Hint

   Consider the remainders calculated.