Your sock drawer is quite a mess. You have eight black socks, seven white socks, two blue socks, and one weird striped sock. They are all indistinguishable in the dark. It is dark, and you want to select the least number of socks in each scenario.
1) The socks have been put into pairs though not necessarily matching pairs. How many pairs (no rematching allowed) do you need to select to be sure of getting at least two matching socks (which could be from different pairs)?
2) The socks are all loose. How many socks do you need to select to be sure of getting at least one matching pair?
3) Like #2, but now, you want two matching pairs.
Submit your answer to Gene Wirchenko <email@example.com>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, March 12, 2014 at noon Pacific Time. I will post the answer shortly after.