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I think this might be a stupid doubt but I still would want this to be clarified. Here is the following question from the AOPS website for pigeonhole principle.

If a Martian has an infinite number of red, blue, yellow, and black socks in a drawer, how many socks must the Martian pull out of the drawer to guarantee he has a pair?

I am denoting the pairs as follows:- $$(Red, Left) - RL\\(Red, Right) -RR\\(Blue,Left)-BL\\(Blue, Right)-BR\\(Yellow, Left)-YL\\(Yellow, Right)-YR\\(Black, Left)-BlL\\(Black, Right)-BlR\\$$

The answer to the above question is 5 and here is the following answer:-

The Martian must pull 5 socks out of the drawer to guarantee he has a pair. In this case the pigeons are the socks he pulls out and the holes are the colors. Thus, if he pulls out 5 socks, the Pigeonhole Principle states that some two of them have the same color. Also, note that it is possible to pull out 4 socks without obtaining a pair.

However, I am not sure about that answer. This is because we could have $5RL$ socks too, or there might be many possibilities such that one could guarantee you cannot have a pair. For example, $5RL$, $4RL 1BL$ and so on and so forth.

Ideally if we have $n$ socks for each of the 8 combinations mentioned above, then the Martian should pick out $4n+1$ socks inorder to guarantee that he will have a pair. I hope I am right about what I am thinking here.

Palash Ahuja
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  • A pair of socks in this sense means a pair of socks, no further questions. –  Dec 28 '17 at 13:43
  • I think your confusion is about whether you consider the right socks to be different from the left socks or not. If it was gloves, it would be obvious, but I think socks are usually treated as interchangeable. (At least in word problems; in real life YMMV...) –  Dec 28 '17 at 13:45
  • yes, you are right. But, it also says that the person is a Martian. You can't assume that the socks may be different or the same. The assumption may vary. Seems kinda stupid. :) – Palash Ahuja Dec 28 '17 at 13:52

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I imagine the given solution is assuming left and right socks are indistinguishable. If left and right socks are distinguishable, then your math is correct.

BallBoy
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