Question 1127420: Peter has many socks, all the same except that they are in five 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?
Found 3 solutions by math_helper, Additi123, ikleyn:
Answer by math_helper(2461)   (Show Source):
You can put this solution on YOUR website! 
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Clearly 
Say the colors are A,B,C,D, and E.
When Peter first starts picking socks from the drawer, the greatest number of socks he can pick without getting a single pair is 5 (A,B,C,D,E). Upon drawing the 6th sock (A) he is guaranteed to have a pair. At this point, he either draws an A which he holds unmatched OR he draws another color that pairs up (and hence the pair count goes up by 1).
Worst case pairing table:
Draw: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
# pairs: 0, 0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7
Starting with draw #6, if we assume any color on any draw, then the drawn color either replaces a previously unmatched color or it forms a pair, there is no other option.
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Alternative solution:
What is the least number of pairs he can have with 14 socks drawn? That would be 5 pairs.
A,B,C,D,E, A,B,C,D,E, A,B,C,D
He would also hold 4 of the 5 colors, unmatched. Now the next draw (#15) would have to be the one color he is not holding (E) in order to avoid making another pair. At this point though, the next draw (#16) definitely pairs up, bringing the pair count to 6. He'd have to then draw the color just paired on #17 in order to avoid making a pair. Finally, draw #18 is another must-match situation, bringing the pair count to 7.
Answer by Additi123(1)  (Show Source):
You can put this solution on YOUR website! Peter can either have a even number or odd number of socks.
We can think of the worst case scenario that can happen to odd number and even number of socks.
ODD NUMBER: 5 socks are different colors so the 5 socks will not be paired.
Expression for odd number: (socks - 5)1/2 = pairs
We are subtracting 5 because we want to get rid of the socks that will not be paired.
We are multiplying by 1/2 because a sock is 1/2 of a pair.
EVEN NUMBER: 4 socks are different colors so the 4 socks will not be paired.
Expression for even number: (socks-4)1/2 = pairs
Finding number of socks for 11 pairs:
Odd--> (socks-5)1/2 = 7 Even--> (socks-4)1/2 = 7
socks -5 = 14 socks - 4 = 14
socks = 19 socks = 18
Since even number of socks is less than even, it gives us the minimum number of socks Peter needs to take to get 7 pairs.
Thus Peter needs to take at least 18 socks to have 11 pairs of socks.
Answer by ikleyn(52781)  (Show Source):
You can put this solution on YOUR website! .
Peter has many socks, all the same except that they are in five different colors.
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 color.
How many socks must he take from his drawer to be sure of achieving this?
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It is easy for me to disprove the solution of the other person:
to have 11 socks matching, the number of the taken socks must be at least 22. (ha-ha-ha)
And 22 is the happiest case, if you are lucky 11 times in a row
(which practically NEVER may happen (!) )
So, the correct solution MUST BE different (!)
* * * THE CORRECT SOLUTION * * *
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) How many socks should he take to be sure that there is at least one good pair of the same color ?
In the worst case, he will take 5 socks of different color,
but any 6-th sock will match with one of 5, just taken.
So, 6 socks is enough in this case.
2) How many socks should he take to be sure that there are at least two good pairs of the same color ?
In the previous case, worst case was 5 socks of different colors.
After adding any 6-th sock, we have one good pair and match,
but still may have 4 remaining socks of different colors.
So, we should add the sock #7 to create possible worst case, and then add #8 to have a match guaranteed.
So, the answer in this case is 8, which is "add 2 to 6".
3) After that, the pattern is clear: to provide next match, we should add 2 socks from the drawer after every previous match.
4) Following this pattern, I create the table below
# of matching pairs 1 2 3 4 5 6 7 8 9 10 11
# of socks to take 6 8 10 12 14 16 18 20 22 24 26
5) ANSWER. 26 socks.
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The solutions and the answers of two other tutors, @math_helper and @additi123,
are both incorrect. Ignore them and us MY CORRECT solution.
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