__Andrey Kolmogorov__asked questions like

**How many patterns can you create with a thread while sewing on a four-hole button?**

Unfortunately, there is no information on the Web what the answer is and had Kolmogorov found it himself. In a book about Grigori Perelman, its author Masha Gessen admits that two professional mathematicians, both students of Kolmogorov, have given different answers. I am not a mathematician, but would like to present my second solution to the problem.

__Assumptions:__

A1. It is mandatory to utilize the holes. There are many ways to sew on a button without utilizing the holes but let us forget about them for the time being.

A2. All holes lay on the vertices of a convex polygon.

__Calculation:__

1. Let us take a button with

**n**holes (

**n**is a natural number equal to or greater than 2). Except for the case where

**n=2**(click here for details of my first solution),

**n**holes mean a convex

**n**-gon. The number of diagonals of such a

**n**-gon is

**n*(n-3)/2**. The total number of segments (sides+diagonals) is

**n+(n*(n-3)/2)=(n-1)*n/2**.

2. Let us call a segment

*black*if a thread connects the holes and

*blank*if there is no connecting thread. The total imaginable number of patterns is

**2^[(n-1)*n/2]**, where

**2**is the number of colors (black and blank).

3. The total actual number of patterns

**f(n)**is

**{2^[(n-1)*n/2]}-1**, as we should not count the case when the color of all segments is blank (in which case the button is not attached at all).

**In the case of Kolmogorov’s button f(4)=63.**

For those who disagree with the assumption that there is only one way to utilize button’s holes a separate text is coming soon.