ArthurDent, that was very ingenious of you and I thank you! I sheepishly admit that I should have thought of that because I have seen similar heaters but instead of the elements being enclosed they are open, of the spiral wire that gets red hot type, and they have the configuration you discovered. I don't know why I didn't think of it. Sometimes intuition and ingeniousness beats raw math and algebra.
Doing it by algebra brute force should have yielded an extremely high or infinite value for the assumed fourth element.
The problem with the values as they are is that the different combinations of switching sometimes results in repeated values of resistance and power so of 16 possible combinations of the four switches, 6 result in zero power, six result in unique values and three result in repeated values.
Two things that come to my mind now is that it would be interesting to use a thermal camera to see the distribution of heat on the hot plate as the different elements are used and also that for combinations that result in the same power it may be preferable to use the one that uses more resistors so that the heat is more distributed and it would prolong the life of the heater.
All very interesting. Thank you!
E.T.A. I did a quick calculation of the resistance and power for each combination of switches but chances are there might be some mistake because my head hurts. I will review it again tomorrow.
By the way, this was/is a hotplate that was controlled by a rotary switch that failed and I could not find a replacement so I fitted four switches as shown and I have a card with a table showing the switch combinations and corresponding power. The most common combinations I know by heart now.
