Resistors
 

How many ways you can connect a maximum of N resistors, assuming:
a) all resistors are equal ?
b) all resistors are different, and given ?
c) all resistors are different and you can chose such as:
c1) you have the smallest possible number of resulting values ?
c2) you have the largest possible number of resulting values ?

This number grows faster than sequences found in oeis which start with the same or similar numbers (1, 2, 5, 18, ...). I can come with the calculus for any particular N, but a general formula eludes me. The c2 case is the most interesting.

Example, zero resistors you can connect in only one way, and have infinite resistance. One resistor you can connect in two ways and have either R, or infinite. Two resistors you can connect either in series or in parallel, which makes 5 ways totally, with resistance: either infinite, R1, R2, R1*R2/(R1+R2), or R1+R2. If they are equal, you only get 4 ways: infinite, R/2, R, 2R. When N gets larger, the possibilities get ugly very fast.

I think you can add c3 and c4 maybe.
c3: The most even linear distribution of values.
c4: The most even logarithmic distribution of values.

[QUOTE=LaurV;569151]When N gets larger, the possibilities get ugly very fast.[/QUOTE]Or, as a mathematician might say: the possibilities get interesting very fast. :razz:

[url]https://en.wikipedia.org/wiki/Gabriel_Kron[/url]
Would his approach apply? Could this work for conductors (Mhos) as well?
Would 3D configurations apply? Would any electric/magnetic fields created be considered as well? Could the whole works be immersed in something augmenting/diminishing the value of resistance/conductance? Perhaps these questions don't apply but I'm trying to get a handle on the boundary conditions a little more.

It seems reasonable to model "connecting the resistors" by a graph in which the vertices model the resistors, and the edges model the connections. If you rule out "parallel" edges and "loops" the model is a "simple graph." The number of simple graphs with n vertices is known to be 2[sup]n*(n-1)/2[/sup]. So I think that would be an upper bound for your sequence.

nerd sniping, xkcd?
[url]https://xkcd.com/356/[/url]

[QUOTE=firejuggler;569193]nerd sniping, xkcd?
[url]https://xkcd.com/356/[/url][/QUOTE]Wonderful!

Does each resistor need to participate in the resistance? I.e. whether a 3-star is a valid connection, or is equivalent to either one of R1+R2, R1+R3, R2+R3.

[QUOTE=Viliam Furik;569203]Does each resistor need to participate in the resistance? I.e. whether a 3-star is a valid connection, or is equivalent to either one of R1+R2, R1+R3, R2+R3.[/QUOTE]
I think the most reasonable interpretation is that you have two special nodes, and then ask what resistances you can construct between them.

[QUOTE=uau;569229]I think the most reasonable interpretation is that you have two special nodes, and then ask what resistances you can construct between them.[/QUOTE]

Yes, thank you, I think I understand it properly now.

[QUOTE=retina;569153]Or, as a mathematician might say: the possibilities get interesting very fast. :razz:[/QUOTE]
That was the "ElectronicCrank" inside me, not the "MathCrank". They do eight-hours shifts...
The c3 and c4, however, sounds very interesting, albeit they are subsets of the general case, you do all combinations and keep those which are linear (or log) and ignore the others.

Here I could share an interesting story, not long ago we did a project which involved 8 keys (push buttons) but we only had 2 or 3 inputs available, and somebody came with the idea to use an analog-to-digital line (ADC) of the MCU and connect all the keys there, with different resistor dividers, or some [URL="https://en.wikipedia.org/wiki/Resistor_ladder"]R-2R[/URL] network. This worked nice in theory, and you could clearly read different voltages when pressing the keys one by one, but it was quite difficult to distinguish between key combinations, like for example, pressing two or three keys that made higher individual voltages was generating a lower voltage (as they were kinda "parallel" in that case, making a lower total resistance) and that created (almost) the same effect as pressing a single key which made a lower individual voltage. After a lot of experiments and calculus, we decided to use two ADC lines and connect 4 buttons (with the right resistor nets) to each line. In this case, we could differentiate all keys combinations more accurate, because only 16 cases, and not 256, and the input keypad worked perfectly, no matter what you pressed.

Then we sent samples to the customer, and total fiasco. Unknown to us, and that's happens when you (the customer) do NOT share all data, and when you (designer/manufacturer) do not ask, they were not using metal-dome switches (which give a zero-ohm resistor when you push them), which we used in our tests, but cheap [URL="https://www.alibaba.com/showroom/conductive-carbon-pills.html"]rubber keypads[/URL], which have a carbon pill contact. The carbon pills give a contact resistance which could be anything between few ohms and few kilo-ohms, dependent on the materials used, contact surface, and [U]pressure[/U]. That is how a [URL="https://en.wikipedia.org/wiki/Carbon_microphone"]carbon microphone[/URL] used to work, ages ago, when they were invented. Yep, if you press one key harder the resistance is very low, while if you press it softer, only a little, the resistance is much higher. Therefore, pressing a single key harder may look the same for the system like pressing two or more keys softer (remember, they are all connected to the same ADC line of the MCU, it can read the voltage, and decide what's pressed), where the individual resistances of the keys in combination will be higher, but all in parallel give a smaller total resistor.

Of course, the design was anything but usable. You could not press any key reliable, single or not. Total screw up. In fact, it was a very clever design :razz:, you could use one single key to generate all possible key combinations (and any resistance, in fact), if you could control your amount of force when pressing it. :lol: :rofl:

Of course, we had to redesign, and make place to connect an 8 keys matrix....

[QUOTE=LaurV;569234]... it was a very clever design ...[/QUOTE]It makes the engineer feel good. It makes everyone else suffer. :davar55:

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 [URL="https://en.wikipedia.org/wiki/Bell_number"]Bell numbers[/URL] 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 multi-graph, 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, non-multi, 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.

[QUOTE=firejuggler;569193]nerd sniping, xkcd?
[URL]https://xkcd.com/356/[/URL][/QUOTE]
Well, related to that, see Dave Jones' [B]brilliant[/B] [URL="https://www.youtube.com/watch?v=v1YrANSmOGY"]blog #25 here[/URL]. :lol:
(WATCH FOR IT! at about 3 minutes)

[QUOTE=LaurV;569248]Well, related to that, see Dave Jones' [B]brilliant[/B] [URL="https://www.youtube.com/watch?v=v1YrANSmOGY"]blog #25 here[/URL]. :lol:
(WATCH FOR IT! at about 3 minutes)[/QUOTE]
I like his style! Good video.
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.
[url]https://www.newyorker.com/culture/annals-of-inquiry/three-mathematicians-we-lost-in-2020[/url]
[url]https://en.wikipedia.org/wiki/Graham%27s_number[/url]