**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.

In the beginning, I found the number of patterns in three cases where single-color thread was involved. Now, let us find the number of patterns

**p**in case of different-color threads.

__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 are located on a straight line.

A3. Different-color threads might be used for different segments of the line. Let us use the adjective

*painted*for a segment that is covered by a thread and the verb

*paint*for the action itself.

A4. Only professional connections are used (click

*In the beginning*link above for definition of

*professional*).

__Calculation:__

1. Let us take a button with

**n**holes (

**n**is a natural number equal to or greater than 2). As all holes are located on a straight line, there are

**n-1**segments between the outermost holes. Let us call the number of segments

**m**, where

**m=n-1**.

2. In the simplest case where

**n=2**,

**m=1**and

**c=1**(e.g. black), how many patterns are there? Just

**1**(a black thread is keeping the button attached). When

**n=3**,

**m=2**and

**c=1**, there are

**3**possibilities: black+black, black+blank (

*blank*signifying a missing thread) and blank+black. Should we continue thinking like this we will find that

**p = [(1+1)^m]-1 = (2^m)-1**(which we have

__already__found). The

**-1**represents the all-blank pattern, where the button is not attached to the cloth.

3. What if

**n=2**,

**m=1**and

**c=2**(e.g. black and red)? There are

**2**possibilities: black or red. When

**n=3**,

**m=2**and

**c=2**, there are

**8**possibilities: black+black, red+red, black+red, red+black, black+blank, blank+black, red+blank and blank+red. Should we continue thinking like this we will find that

**p = [(2+1)^m]-1 = (3^m)-1**.

4. After spending some time and effort, we find that the number of patterns depends on

**c**(number of colors available) and

**m**(number of segments to color) in the following way:

**p = [(c+1)^m]-1.**

5. So far, we have not asked the question how many colors there are. According to Wikipedia, in the frequency range of the perceivable light (430-750 THz) people could see 10 million different colors.

6. Therefore, in the case of Kolmogorov’s button (

**n=4, m=3, c=10,000,000**)

**p = [(10,000,001)^3] - 1**

**≈**

**1*10^21**