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.
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).
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