Problem 12 · Pigeonhole principle · 4 points
Made by AI and not reviewed by a real person.
Mila draws cards numbered 1 to 12, without looking. What is the fewest cards she must draw to be certain that two of their numbers differ by exactly 3?
- 8
- 7
- 9
- 6
- 5
Show a hint
- Can you split the cards into pairs whose numbers differ by 3?
- Find six cards that avoid such a pair.
Show the answer and solution
Answer: B — 7
Pair the cards as (1,4), (2,5), (3,6), (7,10), (8,11), and (9,12). With 7 cards, Mila must have both cards from at least one pair. But 6 is not enough: she could draw 1, 2, 3, 7, 8, and 9, with no two numbers differing by 3. So the answer is 7; the trap in choosing 6 is thinking that one card from each pair is enough.