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Peter has many socks, all the same except that they are in three different colours.
He is leaving to catch an early train to go on a business trip, and he does not want to wake his wife,
so he packs in the dark. He needs eleven pairs of socks, each sock in each pair the same colour.
How many socks must he take from his drawer to be sure of achieving this?
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In this problem, the words "has many socks" mean, by default,
"has many enough to make all necessary manipulations to get the guaranteed outcome".
It is a nice entertainment problem, and if you solve it for the first time in your life,
it is better do it step by step.
1) Getting 1st match.
How many socks should he take to be sure that there is at least one matching pair of the same color ?
In the worst case, he will take 3 socks of different color (unmatched),
but any 4th sock will match with one of the 3, just taken.
So, in the worst case, after taking 4 socks, Peter has 1 match and 2 socks remaining
in his hands of different color.
2) Getting 2nd match.
How many socks should he take to be sure that there are at least two matching pairs of the same color ?
In the first step worst case, Peter takes 4 socks, has 1 matching pair and 2 non-matching socks.
At the second steps, Peter takes any 5th sock from the great collection.
In the worst case, it does not match the two different color socks that Peter has after step 1.
So, Peter takes any 6th sock from the great collection.
And then he inevitably has his second match plus two unmatching socks in his hands.
So, in the worst case, after 2nd step he takes 6 socks, has 2 matches and 2 socks remaining
in his hands of different color.
3) After that, the pattern is clear: to provide next match, in the worst case Peter should take
2 socks from the drawer after every previous match.
4) Following this pattern, I created the table below (for worst cases)
# of matching pairs 1 2 3 4 5 6 7 8 9 10 11
# of socks to take 4 6 8 10 12 14 16 18 20 22 24
ANSWER. Under given conditions, it is enough to take 24 socks randomly
to achieve the goal (to have 11 matching pairs of socks).
Solved.