GeekDad Puzzle of the Week Solution – Capacitor Combinatorics

Reading Time: 4 minutes

Last week’s engineering-based puzzle as posted:

capacitors_715x500
Looking back to my Physics coursework, one of the most interesting (and challenging) aspects was calculating the overall capacitance of a set of capacitors. As you may recall, capacitors in parallel sum the capacitance of each element; in series, they are the inverse of the sum of the inverses of each element.

Put graphically:

Capacitors in Parallel:Capacitors in Series:
cap_parallelcap_series2
60 µF + 60 µF = 120 µF1/(1/60 µF + 1/60 µF) = 30 µF

There’s only one way to “wire in” a single 60 µF capacitor, and three different ways to wire in up to two 60 µF capacitors – one by itself, two in series, and two in parallel.

This week’s GeekDad Puzzle of the Week is to determine how many different capacitance values can be created by wiring together up to 10 identical 60 µF capacitors. Note that not all 10 need to be used for each circuit, and that subsets can be used in parallel and in series to each other.

This week’s puzzle was straightforward, as long as you didn’t get too wrapped up in the math. If you think of each incrementally larger circuit as a circuit one capacitor smaller plus the new capacitor either in series or parallel, you will be on the right track.

For example:
circuit_3_01 in series with circuit_1_01 = circuit_4_01

and

circuit_3_01 in parallel with circuit_1_01 = circuit_4_05

For the first capacitor counts, we get:

1 @ 60 µF Capacitor:

caption
60 µF

2 @ 60 µF Capacitor:

caption
30 µF

caption
120 µF

3 @ 60 µF Capacitor:

caption
20 µF

caption
90 µF

caption
40 µF

caption
180 µF

4 @ 60 µF Capacitor:

caption
15 µF

caption
36 µF

caption
24 µF

caption
45 µF

caption
80 µF

caption
150 µF

caption
100 µF

caption
240 µF

caption
60 µF

caption
60 µF

Of course, with the last two examples, you can see that not each value of overall capacitance is unique, as
circuit_3_01, circuit_3_01, and circuit_3_01 all provide 60 µF of capacitance.

This provides us:

nNumber of distinct wiring diagrams using exactly n @ 60 µF capacitorsNumber of distinct capacitance values using exactly n @ 60 µF capacitorsNumber of distinct capacitance values using up to n @ 60 µF capacitors
1111
1223
1447
110915
1242235
1665377
1180131179
1522337429
115328691039
1426422132525

There are 2,525 different capacitance values can be created by wiring together up to 10 identical 60 µF capacitors.

Congratulations to David Pelletier, this week’s winner of the $50 ThinkGeek Gift Certificate. Thanks to everyone that submitted an entry.

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