Six Dots Make 31, Not 32: When Patterns Break
1, 2, 4, 8, 16... then 31. The pattern breaks exactly when it seems most trustworthy.
Chord Diagram — Points on a Circle
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1
8
Points:
1
Chords:
0
Crossings:
0
Regions:
1
Regions in the Circle
1
n = 1 point
Formula
R(n) = 1 +
C(n,2)
+
C(n,4)
Chords = C(n,2) | Interior crossings = C(n,4)
(Assumes no 3 chords concurrent — generic placement)
Pattern Table
n
C(n,2)
C(n,4)
R = 1+C₂+C₄
2^(n−1)?
Pascal's Triangle — Highlighted Entries Used in R(n)
R(n) uses row n entries at columns 0, 2, 4 (i.e. C(n,0)+C(n,2)+C(n,4) = 1+C(n,2)+C(n,4))