The pigeonhole principle says something almost too obvious: if more objects go into fewer boxes, at least one box gets more than one object.
The power is not the statement. The power is noticing what the boxes are.
First Example
If 13 people are in a room, two people must have birthdays in the same month. There are 13 people and only 12 months.
Remainder Example
Choose 6 integers. When divided by 5, each integer has one of 5 possible remainders. So at least two of the integers have the same remainder.
That means their difference is divisible by 5.
COSMO Teaching Move
Ask students to identify the pigeons and holes before solving. This makes the method feel less like a trick.
Challenge
Prove that among any 8 people, two were born on the same day of the week.
Then make your own problem where the hard part is choosing the boxes.