Cut a pancake with straight lines and count the most pieces you can make. One cut gives 2, two cuts give 4 — so three cuts give 8, surely? Three give 7. It is the flat cousin of the cake-number trap, and it breaks the soonest of all: at the very third cut.
The answer key prints on a separate page.
The counts 1, 2, 4 look exactly like doubling, and the third cut is where that stops: it can cross the first two cuts, but a third line simply cannot make enough new crossings to double the pieces again. The most pieces n straight cuts make is 1 + (the cuts) + (the pairs of cuts): one whole pancake, one more piece per cut, and one more for each PAIR of cuts, because each pair crosses exactly once inside. That is 1 + C(n,1) + C(n,2), which shadows 2ⁿ only for the first three values.
Counting the pairs instead of trusting the doubling is the whole method — and it is the same idea as the cake one dimension down, and the circle one shape over. A pancake's cuts cross in points, a cake's planes cross in lines and points, a circle's chords cross in points; each time the answer is one plus the parts you can count, never a power of two.