AMC 10 · 2010 · #3
Grade 1 countingPick an answer.
AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The word "guarantee" points straight at Tool #14 (Extreme Principle): to be certain, plan for the unluckiest possible order and see how long you can avoid a pair. With only 4 colors you can dodge a match for at most 4 pulls, so the very next pull forces a repeat. Tool #3 (Eliminate Possibilities) then checks the answer against the choices, ruling out the too-small numbers (that only might pair) and the too-big numbers (that count far past the first guaranteed match).
Imagine the unluckiest draw
Assume the worst luck: every sock is a brand-new color. With only four colors, you can pull at most 4 socks and still hold no pair.
You can only stay pair-free while every sock brings a new color, and there are just 4 colors to hand out.
1.OA.A.1Extreme PrinciplePull one more sock
The next sock has no unused color left, so it repeats one you hold and finishes a pair: 5 pulls always work.
Once each color is used up once, there is nowhere new for the next sock to go except onto a color you already have.
Once each colour has been used once, the next sock has nowhere new to go.
▸ Why?
More socks than colours means at least two must land on the same colour.
▸ Why?
With every fresh colour already taken, only a repeat remains as an option.
Check against the choices
4 socks can be all different, so (A) 3 and (B) 4 promise nothing; (D) 8 and (E) 9 overshoot. The smallest sure count is (C) 5.
The right answer is the exact turning point: one less still risks no pair, and anything more is overkill.
1.OA.A.1Eliminate PossibilitiesTo be sure of a matching pair, plan for the worst luck: with 4 colors you could pull one of each, so the 5th sock has no new color left and must make a pair.
- Imagine the unluckiest draw
- Pull one more sock
- Check against the choices