*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. Then I found the number of patterns in case of different-color threads where the buttonholes were located on a straight line. Now, let us explore a more realistic situation where the buttonholes are located on the vertices of a convex

**n**-gon. In order to temporarily avoid some difficulties let me reword the Kolmogorov’s problem:

*There are*

**n**cities located on the vertices of a convex**n**-gon. There are**c**types of communication lines available. Any city can be connected to any other one by only one communication line (that can be of any type). A network exists if at least**2**cities are connected by a communication line. How many different networks (**p**) can be built?From our last exercise with the Kolmogorov’s button we know that the number of networks (

**p**) depends on: a) the number of communication-line types (

**c**) and b) the number of possible communication lines (

**m**). In the case of a convex

**n**-gon (with

**n**edges and

**n*(n-3)/2)**diagonals)

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

Therefore, the number of networks is

**p = [(c+1)^m]-1 = {[(c+1)^[(n-1)*n/2]}-1**

For example, in the case of 10,000,000 types of communication lines and 4 cities (in the language of Kolmogorov’s button problem: a button with 4 buttonholes and threads with 10,000,000 different colors, i.e. the number of colors perceiveded by the human eye), the number of different networks is

**p={(10,000,000+1)^[(4-1)*4/2]}-1 ≈ 1*10^42**

Unlike the communication lines of different type (that are unseen by the user), the differently colored threads can make different patterns when diagonals intersect

(

*red-over-black*looks different than

*black-over-red*). This is a subject I will expand on in the future.