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#46 OhmWork

Set by Phil Lloyd

When electrical resistors are connected in series, the total resistance is the sum of the individual resistances. When they are connected in parallel, the total resistance is found by taking the reciprocal of the resistance of each resistor, adding them together, and taking the reciprocal of the result. This circuit, using three 1-ohm resistors, has a resistance of 3/2 ohms:

There are three more circuits using three 1-ohm resistors. What are their resistances?

A circuit uses 10 1-ohm resistors – three in series, then seven in parallel. What is its resistance? (It is an approximation to a well-known constant.)

Can you find a way to produce this value using only nine 1-ohm resistors? Hint: In this circuit, pairs of resistors in series occur four times, so there are effectively four 2-ohm resistors and one 1-ohm.

Solution next week

#45 Square differences

Solution

You can express the number 9 as a difference of two squares like this: 5² – 4² = 25 – 16 = 9. The white squares in the table show numbers between 1 and 30 that can also be written as the difference of two squares (some in more than one way). The shaded numbers are impossible, and in general, numbers that are 2 greater than a multiple of 4 (2, 6, 10, 14, 18…) cannot be written in this way.