Answer to old chestnut: shoes and socks in the dark:

To ensure a pair of socks, you only need to pull out 3 socks. If the first two are the same color, they make a pair. Otherwise, you have one of each color, and the third one must match one of them.

To ensure a pair of shoes, if you don't check for left/right shoes first, you need to pull out 11 shoes. If you only take 10, you could get all left shoes, or 5 white lefts and 5 black rights, and not have a pair. Having 11 ensures you have at least one shoe of 3 of the 4 distinct types (white left, white right, black left, black right), and that means that for one of the colors, you have both types of shoes.

If you can tell left and right shoes apart by feel, then you only need to pull out 7 shoes: one for one foot and six for the other. If you pull out 5 or fewer left shoes and 5 or fewer right shoes, it is possible to get all lefts of one color and all rights of the other. By pulling out six shoes for one foot, you ensure you get both colors for that foot. Then you just need any one shoe for the other foot to make a match.

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