**From New Scientist #1550, 5th March 1987** [link]

I asked Electrophorus what he was working on.

“You know that joining unlike terminals of a pair of batteries produces a voltage across the two free terminals equal to the sum of the voltages of the separate batteries. And connecting unlike terminals produces a voltage equal to the difference of the voltages of the separate batteries?”

“Yes”, I replied. “With a battery of 2 volts and one of 5 volts one obtains 3 volts (sources opposing) or 7 volts (sources reinforcing).”

“Well, before lunch I had three batteries, none of which had zero voltage, and a voltmeter with a holder that would accommodate only two batteries. So I measured the voltages across the free terminals of all possible pairwise combination of these three batteries both in the case where the voltages reinforced and where they opposed. I wrote on my blackboard the resulting six positive numbers in order of increasing magnitude.”

“When I returned from lunch eager to calculate the ratings of the three batteries, I found the three batteries gone and my blackboard wiped clean. I remember that the second smallest reading occurred twice. It was either 13 or 17 volts, I forget which. I had noticed, rather inconsequentially perhaps, that reversing the digits of this double reading produced another reading which occurred in my measurements.”

What were the ratings of the three batteries?

[enigma400]

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The pairwise differences of the three numbers are all non-zero, so the numbers must all be different.

This Python 3 program considers the three numbers by looking at increasing values for the total sum, and then breaking down that value into three different numbers, until it finds a set that satisfies the conditions.

It runs in 66ms.

Solution:The three batteries are 4V, 9V and 22V.The fact that the doubled reading occurs in the list with its digits reversed is not inconsequential. Without this fact there are 14 solutions.