20210114, 05:30  #12 
Romulan Interpreter
Jun 2011
Thailand
3^{3}×347 Posts 
re Gabriel Kron, and graph theory (presented above by other posters): we have this puzzle turning in our head for a very long time (years), being forgotten and coming back periodically. We tackle with it from time to time, when no other things to do. This time it was brought back to our attention few days ago after somebody here in the forum was discussing the partitions and sets in a parallel thread, and we followed the links provided there, and links from those links, etc., and reached the wiki's Bell numbers page. We recognized the beginning of the sequence, and "pop!" there it was, and that's why "now". But as said, our sequence grows faster, so it is not Bell numbers. It may be some variation of it, and it may also split in two (like, odd number of resistors, versus even number of resistors, due to some symmetry).
With graphs, the issue is that you have in fact a multigraph, there can be a lot of edges between the same two nodes (think 3 resistors connected in parallel, that's a graph with 2 nodes and 3 edges between the two nodes), and also, there can be graphs which are not connected, are different, and yet they give the same resistance (for example, take 3 resistors, put one between our initia/final nodes, and then make another subgraph with the other two, not connected to both nodes, there are many ways to do that, with the remaining 2 resistors connected to each other or not, connected to one of the initial/final node or not, but not to the other, etc, in all cases, you can only have one resistance). That's not easy, unless someone comes with a better coding from resistor nets to more palatable graphs (connected, nonmulti, etc), which coding eludes me for now. So, as said, we play with it from time to time, when we have free time, or the job requests it, but we never allocate it the proper time and work to solve it, and we do not have a general solution. It may be something very simple, it may be not. Last fiddled with by LaurV on 20210114 at 06:20 
20210114, 06:25  #13  
Romulan Interpreter
Jun 2011
Thailand
3^{3}×347 Posts 
Quote:
(WATCH FOR IT! at about 3 minutes) Last fiddled with by LaurV on 20210114 at 06:34 

20210115, 00:05  #14  
Apr 2012
Brady
101111101_{2} Posts 
Quote:
I "cheaped" out in answering the question because there seems to be an infinity of choices. Way back when I was learning circuit theory we used "Spice" to develop circuits. I experimented with Buckminster Fullers' Tensegrity designs as well as polytopes (where Karmarkar's algorithm could possibly be modeled). For different topologies I could obtain different numeric values (depending upon the components) which is why Diakoptics caught my interest. This was in the late 70's. Arithmetically, partition theory would provide one type of answer. Aside from some circuit designs from the '50's that I came across in archived journals, this Dover reprint "Ingenious Mathematical Problems and Methods by L.A. Graham has Problem #91 Resistance 'Cross the Cube which applies to your question as one approach.. I think. Polya's " Patterns of Plausible Reasoning" I believe has a bit on Ramsey theory (combinatorial bead counting) which also may apply to your question and provide an algebraic solution. Minesweeper (R.W.Kaye and the NP problem, cellular automata and switches (open/closed)) are other aspects which could be put into circuit form...which does not seem to contradict any of your conditions. It seems to me that many different configurations are possible depending on your initial conditions. Scientific American printed a textbook for Amateur Scientists in the 60's where there was a design for a circuit (Pircuit I think it was called) which illustrated chaos and chaotic orbits..but uses active components. The ACL2 site is always worth burrowing into regarding testing/engineering. As a final edit, here are two links that may be of interest to approach LaurV's question. The first link provides a bit of background to the second. https://www.newyorker.com/culture/an...elostin2020 https://en.wikipedia.org/wiki/Graham%27s_number Last fiddled with by jwaltos on 20210115 at 00:46 Reason: final edit 
