**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 third 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.

A3. Angles and distances are not taken into account.

__Calculation:__

**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), the holes are located on the vertices of aconvex

**n**-gon (a polygon with

**n**vertices). My second solution to the problem (click

__here__) is that a professional tailor has

**{2^[(n-1)*n/2]}-1**ways to attach the button. Whom do I call

*professional tailor*? He/she is a person who realizes that the thread must connect a buttonhole with a buttonhole, i.e. one who realizes that cross-border button-to-cloth connections are not allowed.

**1, 2, .., n-1, n**utilized holes out of

**n**total holes, and is therefore

**(2^n)-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).

**[(2^n)-1]*{{2^[(n-1)*n/2]}-1}**

**f(n)**is the sum of all patterns that are “professional”, “unprofessional” and “combinatorial”, i.e.

**f(n) = {{2^[(n-1)*n/2]}-1} +(2^n)-1 + [(2^n)-1]*{{2^[(n-1)*n/2]}-1}**

or

**f(n) = {2^[n*(n+1)/2]}-1**

In the case of Kolmogorov’s button f(n)=1,023.

In the case of Kolmogorov’s button f(n)=1,023.

**n**-hole button? Yes, there are. Checking Assumption 1 will give you ideas. There is a group of infinite number of patterns, which are angle-specific, thread-length-specific and number-of-thread-specific (but are not

**n**-specific and even button-specific).

Are there any other ways to attach an

**n**-hole button? Yes, there are. There is another group of patterns, which are color-specific and I will expand on them soon.