17.7.2. Proof using Pigeonhole Principle

Try these first

1. 1.

   Define the Pigeonhole Principle in your own words.

   Hint

   Think about distributing items in boxes.

2. 2.

   What is the midpoint formula?

   Hint

   It involves averaging the corresponding coordinates.

3. 3.

   What does the Pigeonhole Principle state?

   • If ten items are put into nine boxes
   • at least one box has two items.
   • All items will fit into the boxes available.
   • Every box will contain one item.
   • If objects are grouped into boxes
   • each box must hold the same amount.

   Hint

   Remember the connection between items and containers.

4. 4.

   True or False: The midpoint of two points can be a non-integer.

   • True
   • False

   Hint

   Think about averaging even and odd coordinates.

5. 5.

   Consider a group of ten friends, each picking from five distinct fruits. Show how the Pigeonhole Principle guarantees fruit overlaps among their selections.

   Hint

   Group the choices into the types of fruits available.

6. 6.

   Using the Pigeonhole Principle, prove that among any ten consecutive integers, at least two of them will produce the same remainder when divided by five.

   Hint

   Count the number of integers and the possible remainders.