Five points on a 2D plane. Each pair has a cost equal to the Manhattan distance — horizontal plus vertical. You need to connect all of them with minimum total wire.
The natural instinct is to be greedy: always wire each point to its nearest unconnected neighbor. It feels optimal. The edges are short. The total looks low. Let me show you what that produces.
I tried this on my first attempt. I was so confident. Then I saw the actual minimum. Five units of unnecessary wire that I could have saved with a different strategy.