20210113, 10:10  #1 
Romulan Interpreter
Jun 2011
Thailand
3×3,049 Posts 
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. Last fiddled with by LaurV on 20210113 at 10:10 
20210113, 10:21  #2 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
176A_{16} Posts 
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. Or, as a mathematician might say: the possibilities get interesting very fast. 
20210113, 10:38  #3 
Apr 2012
2^{2}·7·13 Posts 
https://en.wikipedia.org/wiki/Gabriel_Kron
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. 
20210113, 17:38  #4 
Feb 2017
Nowhere
23×181 Posts 
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^{n*(n1)/2}. So I think that would be an upper bound for your sequence.

20210113, 19:04  #5 
Apr 2010
Over the rainbow
3·839 Posts 
nerd sniping, xkcd?
https://xkcd.com/356/ 
20210113, 19:17  #6  
Bamboozled!
"ð’‰ºð’ŒŒð’‡·ð’†·ð’€"
May 2003
Down not across
10,501 Posts 
Quote:


20210113, 20:00  #7 
"Viliam FurÃk"
Jul 2018
Martin, Slovakia
2^{3}×41 Posts 
Does each resistor need to participate in the resistance? I.e. whether a 3star is a valid connection, or is equivalent to either one of R1+R2, R1+R3, R2+R3.

20210114, 00:09  #8 
Jan 2017
2×43 Posts 
I think the most reasonable interpretation is that you have two special nodes, and then ask what resistances you can construct between them.

20210114, 00:56  #9 
"Viliam FurÃk"
Jul 2018
Martin, Slovakia
2^{3}·41 Posts 

20210114, 02:30  #10  
Romulan Interpreter
Jun 2011
Thailand
3·3,049 Posts 
Quote:
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 analogtodigital line (ADC) of the MCU and connect all the keys there, with different resistor dividers, or some R2R 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 metaldome switches (which give a zeroohm resistor when you push them), which we used in our tests, but cheap rubber keypads, which have a carbon pill contact. The carbon pills give a contact resistance which could be anything between few ohms and few kiloohms, dependent on the materials used, contact surface, and pressure. That is how a carbon microphone 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 , 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. Of course, we had to redesign, and make place to connect an 8 keys matrix.... Last fiddled with by LaurV on 20210114 at 06:16 

20210114, 04:49  #11 
Undefined
"The unspeakable one"
Jun 2006
My evil lair
2×3^{4}×37 Posts 
