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"Show that if you pick three socks from a drawer containing just blue socks and black socks, you must get either a pair of blue socks or a pair of black socks."

I know how to solve this by contradiction but I am not sure how to solve it by Cases.

By contradiction:

Suppose that we don’t get a pair of blue socks or a pair of black socks. Then we drew at most one of each color. This accounts for only two socks. But we are drawing three socks. Therefore our supposition that we did not get a pair of blue socks or a pair of black socks is incorrect and our proof is complete.

Can someone please give me a hint or start me off with the Cases proof.

  • 2
    If you pick 3 socks, there is either 3 black + 0 green, either 2+1, either 1+2, either 0+3, in first two cases there are 2 black, in the latter 2 green. Not sure you mean this. – 2017-01-28

2 Answers 2

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Without loss of generality assume that the first one is blue. Then there are two cases: either we pick a blue or a black one. If it is blue, we are done. If it is black, we have a blue and a black one. Now pick the third. In both cases (blue or black) we end up with a pair.

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There are eight cases (including order):

  • blue, blue, blue (pair of blue)
  • blue, blue, black (pair of blue)
  • blue, black, blue (pair of blue)
  • blue, black, black (pair of black)
  • black, blue, blue (pair of blue)
  • black, blue, black (pair of black)
  • black, black, blue (pair of black)
  • black, black, black (pair of black)

All eight cases have a pair the same colour