Because resistors are expensive

Electricity and Magnetism Level 3

What is the minimum number of 15 Ω 15 \Omega resistors that are required to build a composite resistor of 10 Ω 10 \Omega ?

Challenge: Generalize the problem.

Assumption: You are only allowed connecting resistors in series and parallel obeying standard laws of combination.


The answer is 3.

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Agnishom Chattopadhyay by Agnishom Chattopadhyay , 22, India

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4 solutions

Aditya Kumar
Nov 24, 2015

It is quite evident that U can't have a direct series connection. Now in order to get a 10 Ω 10\Omega resistance U need to have a series combination of 2 15 Ω 15\Omega resistances and to that combination we have to add a parallel resistance of 15 Ω 15\Omega . This combination overall produces a net resistance of 10 Ω 10\Omega . So minimum resistances used is 3 .

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To generalize this lets consider x ohms which needs to be converted to y.

Case 1: if y<x The circuit needs to be connected in parallel.

This means that 1/x +1/nx=1/y where n is a rational number =>1/15+ 1/15*n=1/10 gives n=1/2 and the answer is 3 in parallel

Case 2: if y>x the circuit is connected in series and then in parallel => x + xn=y Iterate this process till y<x and solve it as usual.

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Arkodipto Dutta
Apr 17, 2016

Connect two 15 ohm resistances in series and add one 15 ohm resistance parallel to the equivalent resistance of the series combination. So minimum number of resistances used is 3 (2 in series and 1 parallel to them).

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Ashish Menon
Nov 29, 2015

15 and 15 resistors in series. And 15 resistance to their combination

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Try generalising the problem!

Agnishom Chattopadhyay - 5 years, 6 months ago

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What is the minimum number of x-ohm resistors that are required to build a composite resistor of y ohms?

we need to look at the ratio x/y in lowest terms.

I have an observation that n resistors can produce a maximum denominator y equal to F(n+1) where F is the fibonacci sequence.

for example, 6 resistors can have a maximum denominator of y = F(7) = 13

also, by the reciprocal theorem, the numerator x has the same upper bound.

using this observation we may find a lower bound for n given x/y.

Antonio Valente Macarilay - 5 years, 6 months ago

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