Cut a cake with straight flat cuts, moving nothing between cuts, and count the most pieces you can make. One cut gives 2, two give 4, three give 8 — so four give 16, surely? Four give 15. It is the space-shaped cousin of the circle-regions trap, and it breaks one cut sooner.
The answer key prints on a separate page.
The counts 1, 2, 4, 8 are so plainly the powers of two that almost everyone writes 16 for the fourth cut, and being wrong is the lesson. The most pieces n flat cuts make is 1 + (the cuts) + (the pairs of cuts) + (the triples of cuts): one whole cake, one more piece for each cut, one more for each PAIR of cuts (a pair of planes meets in a line), and one more for each TRIPLE (three planes meet at a point). That is 1 + C(n,1) + C(n,2) + C(n,3), which only shadows 2ⁿ for the first four values.
The reason four cuts fall short is that the fourth plane cannot meet the other three in enough new places to double the pieces — in three dimensions there simply are not that many pairs and triples yet. Counting the pairs and triples instead of trusting the doubling is the whole method, and it is the same inclusion-exclusion that governs the circle, one dimension up.