Puzzle 1 If you have three socks, you're sure to have a match. Why? If the first two match, you're done. But if they don't match, then the third sock must match one of them.
Puzzle 2 If you pick seven socks, it's just possible that you got one of every color, and no match. But picking 8 socks guarantees you that you have a match. This is actually an application of the Pigeonhole principle!
Puzzle 3 Perhaps the easiest way to handle this problem is to work out the probability that you don't get a match. If we suppose there are R red socks, and 25-R indigo socks, then a mismatch happens because you drew, in order, the pair (R,I) or (I,R). The probability of the first event is
R I R ( 25 - R )
-- * -- = -- * ----------
25 24 25 24
and it turns out the probability of (I,R) is the same, so
the probability of a mismatch is
R ( 25 - R ) 1
2 * -- * ----------- = --
25 24 2
yielding
R2 - 25 * R + 150 = 0
with solutions R = 15 or R = 10. Since we know we have more red
socks than indigo ones, this leave us with R = 15 and I = 10.
Puzzle 4: left to your imagination.
Back to The Socks Puzzle.