Conjecture about parallel and series circuits

[bcrowell]

If we have n (possibly unequal) resistors, we can combine them in various ways to produce a device with two terminals. In many cases, the equivalent resistance of this device can be found by repeatedly breaking the circuit down into parallel and series parts. Conjecture: n=5 is the lowest for which a circuit exists such that this strategy cannot be applied.

Can anyone prove this, or provide a counterexample for n<5?

 

[Curl]

What do you mean "have n resistors". Are you trying to find equivalent resistance between two points? The simplest one I can think of right now is the resistor cube but that is n=12. Can you show me one for n=5?

 

[Meir Achuz]

n=5 is the Wheatstone bridge circuit.

 

[Curl]

Shouldn't be that hard to prove. With n=4 you have 4 elements. Connect two together, and you get 2 resistors in series. Now with the 3rd resistor, you can either put it in series again, or connect it in between the first two to make a T.

If you do the first, then you have 3 resistors in series. The forth one must either go at the ends or somewhere in between, making either 4 or 3 or 2 resistors in series (effectively shorting out other resistors). This can be solved using equivalent combination.

If you use the 3rd element to make a T, then the fourth must either be in parallel with two other resistors (three possibilities) or it can go in series and short one resistor out. This can also be solved using equivalent combination.

So yeah, this is a sketch of proof for n=4. With n=3 it's easier (the proof is actually inside the proof for n=4) and n=2 and n=1 are trivial.
For n=6+ you can find a counterexample, such as a Wheatstone bridge with resistors attached in series.

 

[PJC01]

I cannot provide a definitive answer to this conjecture without further information or experimentation. However, I can offer some thoughts and considerations.

Firstly, the concept of parallel and series circuits is well-established in electrical engineering and has been extensively studied and tested. The idea of combining resistors in different configurations to achieve a desired resistance is a common practice.

Secondly, the statement "n=5 is the lowest for which a circuit exists such that this strategy cannot be applied" is a bold claim and would require significant evidence to support it. It would be helpful to know the specific criteria or conditions under which this conjecture was made, as well as any assumptions that were made in the process.

Furthermore, it is important to note that the behavior of electrical circuits can be complex and unpredictable, especially when dealing with non-ideal components. As such, it is possible that there could be a circuit with n<5 resistors that cannot be simplified using the parallel and series strategy. However, this would likely be a rare and specific case and not a general rule.

In conclusion, while the conjecture presented is interesting, it is important to approach it with caution and continue to explore and test it further before making any definitive statements or conclusions. As with any scientific theory, it is always open to revision and refinement as new evidence and insights are gained.