The Sock Drawer

She must pull out exactly 3 socks. With only 2 colours, the worst case after 2 pulls is one of each. The third pull must match one of them. The total number of socks in the drawer does not matter.

This is a direct application of the Pigeonhole Principle: if you distribute more items than there are categories, at least one category must contain more than one item. Here the categories are colours (just 2) and the items are socks. Step 1: There are only 2 possible sock colours: red and blue. Think of these as 2 pigeonholes. Step 2: Worst-case scenario on pull 1: she gets a red sock. Worst case on pull 2: she gets a blue sock. After 2 pulls, she has one of each colour — no pair yet. Step 3: Pull 3: no matter what colour comes out, it must be either red or blue. Either way, it matches one of the two socks already in her hand. Step 4: So after exactly 3 pulls, she is guaranteed at least one matching pair — regardless of what order the socks come out. Step 5: The number of socks in the drawer (10 red + 10 blue = 20) is completely irrelevant to the answer. Only the number of distinct colours matters. Step 6: General rule: to guarantee a matching pair from k colours, you need k + 1 draws. Key Insight: The Pigeonhole Principle turns a question about probability into a question about logic. Once you reframe it as 'how many categories exist?' rather than 'what are the odds?', the answer becomes immediate. The total number of socks is a red herring.