Overview
Nothing punishes pattern-matching quite like this sequence. Five terms double perfectly, and the sixth quietly refuses.
How to solve 1, 2, 4, 8, 16, …?
- Stop extrapolating and count what the picture actually contains.
- Start with one region (the empty disc). Every chord you add splits regions, and so does every interior crossing.
- A chord needs 2 of the points and a crossing needs exactly 4, so r(n) = 1 + C(n,2) + C(n,4).
The key insight
Powers of two and 1 + C(n,2) + C(n,4) agree for n ≤ 5 by pure coincidence, because C(n,4) is still tiny there. From n = 6 the polynomial grows far slower than the exponential, and the illusion collapses.
Variations & echoes
- The same trap appears in the 'next number' puzzles of IQ tests: finitely many terms never determine a sequence.
- Try counting the regions for n = 7 (57) and n = 8 (99) to feel the polynomial take over.
Frequently asked questions
Why is 32 wrong when the pattern is so clear?
Because the doubling was never the rule — it was a numerical accident of the first five cases. The rule is geometric, and it produces 31.