Moser's circle problem

Problem in geometry

Moser's circle problem asks how many regions a circle can be divided into by choosing n {\displaystyle n} points along the circumference of the circle and joining each pair of points by a straight line. The greatest possible number of regions with n {\displaystyle n} points is given by r G = ( n 4 ) + ( n 2 ) + 1 = 1 24 ( n 4 − 6 n 3 + 23 n 2 − 18 n + 24 ) , {\displaystyle r_{G}={n \choose 4}+{n \choose 2}+1={\frac {1}{24}}(n^{4}-6n^{3}+23n^{2}-18n+24),} resulting in the sequence 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ...

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Moser's circle problem

Problem in geometry

Moser's circle problem asks how many regions a circle can be divided into by choosing n {\displaystyle n} points along the circumference of the circle and joining each pair of points by a straight line. The greatest possible number of regions with n {\displaystyle n} points is given by r G = ( n 4 ) + ( n 2 ) + 1 = 1 24 ( n 4 − 6 n 3 + 23 n 2 − 18 n + 24 ) , {\displaystyle r_{G}={n \choose 4}+{n \choose 2}+1={\frac {1}{24}}(n^{4}-6n^{3}+23n^{2}-18n+24),} resulting in the sequence 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ...

From Wikipedia

Moser's circle problem asks how many regions a circle can be divided into by choosing n {\displaystyle n} points along the circumference of the circle and joining each pair of points by a straight line. The greatest possible number of regions with n {\displaystyle n} points is given by r G = ( n 4 ) + ( n 2 ) + 1 = 1 24 ( n 4 − 6 n 3 + 23 n 2 − 18 n + 24 ) , {\displaystyle r_{G}={n \choose 4}+{n \choose 2}+1={\frac {1}{24}}(n^{4}-6n^{3}+23n^{2}-18n+24),} resulting in the sequence 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... (sequence A000127 in the OEIS). Though the first five terms match the geometric progression 2 n − 1 {\displaystyle 2^{n-1}} , the two sequences differ for n ≥ 6 {\displaystyle n\geq 6} . As Leo Moser noted in 1949, this sequence demonstrates the risk of generalising from only a few observations.

Text: Wikipédia, CC BY-SA 4.0. · Image: Cmglee (CC BY-SA 4.0) ·

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