Four questions, one idea: the answers begin like the powers of two and then break, one after another. A pancake stops doubling at the third cut (7, not 8), a cake at the fourth (15, not 16), a circle of chords at the sixth point (31, not 32), and a strip of dominoes at the very third column (3, not 4).
The answer key prints on a separate page.
Section 1 cuts a pancake with straight lines: 1, 2, 4, and then the third cut makes 7, not 8. Section 2 cuts a cake with flat planes and breaks one step later, at 15 instead of 16. Section 3 joins points on a circle with every chord and holds out longest — 2, 4, 8, 16 — before the sixth point gives 31, not 32. Section 4 covers a two-wide strip with dominoes and breaks earliest of all: 1, 2, and then 3, not 4. Each looks like the powers of two for just long enough to be completely convincing, which is the whole reason they are worth setting together.
The lesson is not one formula but one habit — work out the real rule instead of trusting the run of numbers. Three of the four break by the SAME rule: the count is a sum of pieces you can count, one dimension at a time (the pancake is 1 + the cuts + the pairs of cuts that cross; the cake adds the triples; the circle is 1 + the chords + the crossings). The dominoes break by a DIFFERENT rule, and that is the point of including them: each length is the two lengths before it added together, a Fibonacci recurrence, because the last column is either one upright domino or two flat ones. Same trap, different rule underneath.
