Socks in the Dark
The problem
A drawer holds red, blue and black socks, all loose and indistinguishable by touch. The room is completely dark. How many socks must you take to be certain of holding **two complete pairs**, where socks are paired up within a colour?
- Socks of the same colour feel identical; you cannot identify a colour in the dark.
- Socks of one colour are grouped two at a time, so three red socks count as one pair and four red socks count as two.
Take a minute
Reveal hint 1
Failure is cheapest for an adversary who spreads the colours as unevenly as possible: three of one colour plus one of each of the others uses five socks and yields only one pair.
Reveal hint 2
Ask what the sixth sock is forced to do once five have been drawn in that worst-case pattern.
Solution
Answer socks.
Derivation
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Count pairs within a colour.
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Find a failing draw of five.
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Describe every failing draw.
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Check all shapes of six.
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Conclude the minimum.
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Note the underlying principle.