Moser's Circle Problem

How many regions does a circle divide into when n points on its boundary are connected by all chords? The answer follows a pattern — until it doesn't.

or click the circle

Click on the circle boundary (or use the button) to place points. Max 10.

n = 0 points
Predicted 2^(n−1)
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Actual Regions
1
R(n) = 1 + C(n,2) + C(n,4)
Pascal's Triangle — C(n,2) and C(n,4) highlighted
Method: Euler's formula V − E + F = 2. Count chord intersections (V), chord segments between them (E), apply formula. Inner regions = F − 1.
P1: n=5 → 16 regions (matches 24 = 16). Formula: 1+10+5 = 16 ✓
P2: n=6 → 31 regions (NOT 32). Formula: 1+C(6,2)+C(6,4) = 1+15+15 = 31. First failure of the power-of-2 pattern.