Worst Case and the Pigeonhole Principle

To be sure means it has to work even on the unluckiest possible run, so luck plays no part in the answer. You might get a matching pair in two draws, but you are not sure of one in two draws. The method is to imagine someone handing you the most awkward socks they can, deliberately, for as long as they can manage — then count how far they can string you along, and add one more draw. That one extra draw is the answer, and everything before it is just counting the unlucky run.

Here is the core idea. The unluckiest first three draws give you one sock of each colour, so no pair. But there is no fourth colour, so the fourth sock must match something you already hold. Four socks guarantee a pair. The rule generalises: with however many categories there are, one more than that number forces a repeat. Notice that the thirty socks in the drawer played no part at all — only the three colours mattered. Big totals are put into these questions precisely to make you think the answer must be big.

Asking for a pair of a specific colour changes the count enormously. To be sure of a red pair, assume the unlucky run hands you every black sock and every blue sock first — that is twenty socks with no red at all. Only then does red start appearing, and you need two of them. So twenty-two. Notice how different that is from four: wanting any matching pair took four socks, wanting a red pair takes twenty-two. The word red is doing all the work.

To be sure of a pair of every colour, the unlucky run empties two whole colours first. Ten black, then ten red — that is twenty socks, and you already have a black pair and a red pair, but not a single blue. Blue is now all that is left, so two more draws finish it. Twenty-two. It is worth noticing that this is the same number as the red-pair question but reached by a completely different unlucky run, so do not assume that matching numbers mean matching reasoning. Build the run each time.

A few phrases always signal this question type: to be sure of, guaranteed, must, certain to, and the smallest number that will always work. By contrast, might, could and at best are asking about a lucky run instead, which is a different and much easier question. The method is three steps every time: work out the unluckiest run that still fails, count how long it is, and add one more draw. Just be careful at the end — if success means a pair you add two, not one. Quiz and exercises now.