I dropped in on Ernie a while ago and he told me that there was "good news and bad news". I asked him to tell me about it and he explained as follows: "Remember my Casimir-effect electric motor and how it always used to overheat? The good news is that I have increased its efficiency, so now it won't overheat any more". "And the bad news?", I inquired. "Unfortunately, I got a little carried away and improved things too much. If I ran it at 10 volts and 10 amps, I used to get 99 watts mechanical output and some waste heat, but now I get 101 watts of work out of it". "But that's wonderful!", I exclaimed. "Free energy. How can that be bad news?". "The problem is", Ernie explained, "the extra energy has to come from somewhere. Now as soon as I start the motor it begins to cool down. After a while it gets so cold that it seizes up and stops working. What a disaster!".
"But couldn't you just add a heater?", I asked. Ernie paused for a moment. "Of course!" he cried. "You're a genius. And I can use my new Acme Resist-a-Pak; it just arrived in the mail yesterday". He pulled a brightly coloured box from a drawer and lifted its lid. Inside were a number of components and a sheet of paper labelled "INSTRUCTIONS". I read the following:
Contents: Thirty-one high-precision resistors - each precisely 1.000000 Ohms. One programmable voltage supply - 0.000000 to 9.999999 volts. Sixteen orthogonal connector brackets.
To use: No solder required! Just clip resistors and voltage supply together, using the orthogonal connector brackets at the corners, to produce the resistor network of your choice. Set operating voltage on the power supply and operate as required.
Caution: Orthogonal connector brackets can connect two or more components together, but only at 90 degree angles. Do not try to force them into other orientations or damage will occur!!!
"Hmmm", I mused, "if you take two identical resistors (both oriented the same way) then slide them apart and join their ends with one more resistor and the voltage supply (both oriented orthogonally to the other two resistors), you will end up with a square network with resistance of 3 Ohms. You could use sqrt(3) volts to produce 1 watt of heating - just what your motor needs." Ernie did a few thermal calculations and told me that it wouldn't work. The square network would make the sides of the motor a lot too hot, and the top and bottom would be much too cold.
"Well, I continued, "How about making two identical squares of four resistors each (both oriented the same way) slide them apart and connect their corners with three more resistors and one voltage supply (all oriented orthogonally to the other eight resistors). You will end up with a cubic network and I am sure I can work out its resistance and what voltage you need to get 1 watt of heating." I started to calculate the resistance of the network, and had just got the answer when Ernie told me that it wouldn't work. The cubical network would make the outside of the motor a little too hot and the insides would be a little too cold.
"There doesn't seem to be a solution then", I muttered. "But wait" Ernie replied, "your ideas might just work. If I make two identical cubes of twelve resistors each (both oriented the same way) I can slide them apart and connect their vertices with seven more resistors and the voltage supply (all oriented orthogonally to the other 24 resistors, of course). I calculate that the heat will be even enough to avoid hot or cold spots on the motor - the temperature will be just right everywhere. And there are just enough parts in the Pak to do the job." I was about to point out the impossibility of producing that geometry when I got an emergency phone call from home - the cat was stuck in the tree again - and I had to rush off to help. By the time I had solved that problem, and put away the chain-saw and shot-gun, it was too late to get back to Ernie's shed, but I did sit down in the evening and calculated the resistances of my cubical 11 resistor network and Ernie's weird 31 resistor network (just for fun) - then texted him the results.
To my surprise, I got a message from Ernie the next morning to let me know that 'my' idea had worked perfectly - the motor had run all night without over- or under-heating. This all happened a while ago and I can't remember the two resistances I calculated - maybe you can help.