Suppose you want to pack two suitcases with objects of weight a, b, . . . , z. You want to pack them as close to evenly as you can. It can be shown that this is a virtually impossible problem. Despite fifty years of effort, we don’t know how to find the best method of packing, save for trying all of the exponentially many possibilities.
Wow, that is indeed a nice example. It also reminds me of a problem I had to solve at work. I failed to Google up an existing solution, but I figured that was just due to my ignorance of proper terms to describe it. Now I'm not so sure.
This was linked to twice recently, once in a Rationality Quotes thread and once in the article about mindfulness meditation, and I thought it deserved its own article.
It's a transcript of a talk by Persi Diaconis, called "The problem of thinking too much". The general theme is more or less what you'd expect from the title: often our explicit models of things are wrong enough that trying to think them through rationally gives worse results than (e.g.) just guessing. There are some nice examples in it.