Pigeonhole Principle (Easy)

AMC 10 Q1–8 depth — contest entry, not school drills.

Pigeonhole Principle: core idea

Basic form.

A drawer has $10$ black and $10$ white socks (no other colors). What is the minimum number you must pull out to **guarantee** a matching pair?

Hint: Two colors → worst case $2$ unmatched, then the third matches. Answer: 3.

1. What is the minimum number of people needed to **guarantee** that two share a birth month?
2. From a standard $52$ -card deck, what is the minimum number of cards you must draw to **guarantee** two cards of the same suit?
3. Choose integers from $\{1,2,\ldots,10\}$ . What is the minimum size of a subset that **guarantees** two numbers summing to $10$ ?
4. What is the minimum number of people needed to **guarantee** that two were born on the same day of the week?
5. Balls are placed into $4$ identical boxes (each box can hold many). What is the minimum number of balls to **guarantee** some box has at least $2$ balls?
6. Pick integers from $\{1,2,3,4,5\}$ . What is the minimum size guaranteeing two **consecutive** integers (like $2$ and $3$ )?
7. What is the minimum number of people needed to **guarantee** that two share the same zodiac sign (there are $12$ signs)?
8. Pick integers from $\{1,2,\ldots,20\}$ . What is the minimum size guaranteeing two with the **same remainder** when divided by $5$ ?
9. A bag has $3$ red, $3$ blue, and $3$ green marbles. What is the minimum drawn (without looking) to **guarantee** two of the same color?