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, author Masha Gessen admits that two professional mathematicians, both students of Kolmogorov, have given different answers.
I have some answers, including the case where threads of different color are used. Based on my findings, I have even submitted a new sequence to OEIS. Now it occurred to me that there could be parallel threads of different color connecting any 2 buttonholes and the problem becomes more complicated. Once again I will pose it in terms of networks.
Social networks (like those created using social networking websites) are interesting things. Once 2 users are connected, there is only one type of connection possible, so the user relationship can be coded 1 (there is a connection) and 0 (there is no connection). Buttons are objects of a similar type. Everybody sews their buttons using a single-color thread, so the segment between 2 buttonholes is either coded 1 (there is a connecting thread) or 0 (no connection). Even if 2 buttonholes were connected by many differently colored threads, the only visible color is the color of the topmost thread.
Wider reality is much different. There, the human relationship coding is more complex. 0 still means no connection, but 1 could be personal love relation (1.1 could mean weak, 1.2 – normal and 1.3 – obsessive), 2 could be personal hate relation (there could exist 2.1, 2.2, 2.3, 2.4 etc.), 3 could be business cooperation relation, 4 could be business competition relation etc. Coding becomes much more complex, especially where there exist relations that are thought of as mutually exclusive, e.g. when one has an obsessive-love-hate-person-business-cooperation-competition relations with her boss (you know what I mean).
There are other similar examples. When connecting 2 cities (corporations, states) by different types of communication channels, the situation is complex too. The connections could be coded like this: 0 (no connection), 1 (telephone), 2 (telegraph), 3 (low-speed), 4 (hi-speed), 5 (landline), 6 (satellite) and XY (combinations between some of the first six/more types already listed). Having said this, Kolmogorov’s button problem looks very differently than the one 5-year-old Andrey Kolmogorov envisaged. Now it looks like this:
There are n cities located on the vertices of a convex n-gon and c types of communication lines available. Any city can be connected to any other by o communication lines that are numbered and can be of any type (it is realistic to assume that o≥c, meaning that technological achievement is over-implemented rather than under-implemented). A network exists if at least 2 cities are connected by at least 1 communication line. How many different networks can be built?
There can be built p networks, where
p = [(c+1)^(o*m)]-1
In the case of Kolmogorov’s network where: a) the number of cities is 4 (n=4, therefore m=6), b) there are 2 types of communication lines (c=2) and c) there are opportunities for 2 communication lines between any 2 cities (o=2), the number of different networks (p) will be p = [(2+1)^(2*6)]-1 = 531,440