mathematics - Haselbauer-Dickheiser Test no. 3: Circle divided by lines between a blue dots

This is the test no. 3 from Haselbauer-Dickheiser Test.

3.

These three circles below all have blue dots on their circumference which are connected by straight lines. These lines divide the circles into smaller regions. The first circle, with two blue dots, is separated into two regions. The second circle, with three blue dots, is separated into four regions. The third circle, with four blue dots, is separated into 8 regions.

Given a circle where seven blue dots are placed anywhere on its circumference, what would be the maximum number of regions into which it could be divided?

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Answer

The answer is

57.

This is a well-known problem called

Moser's circle problem. The sequence given by "maximal number of regions with $n$ blue dots" for increasing values of $n$ is $1,2,4,8,16,31,57,\dots$ (OEIS A000127). It's famously deceptive because the first few terms make it look like it's going to be simply the powers of 2, as another answer guessed, but it isn't.