i have to connect 50 resistors in a series circuit. each resistor has a different value of resistance. i must arrange them in a certain way so that my electrical circuit will be most stabile. although this seems to be an electronical issue, my problem is purely mathematical. the resistors have to be arranged using this equation:
( (R1/sumR - 1/50)^2 + ((R1+R2)/sumR - 2/50)^2 + ((R1+R2+R3)/sumR - 3/50)^2 + ... + ((R1+R2+R3+...+R49)/sumR - 49/50)^2 ) --> min
where R1-R49 are the values of resistance, and sumR is the sum of all 50 resistors
the combination of resistors for which this formula gives the minimum value is the combination i must use.
my problem, obviously, is that i would have to solve this equation for every combination of resistors in order to find the combination i need, and there are 50! possible permutations of 50 numbers.
only thing i've noticed is that the value of the first resistor must be closest to the arithmetic mean of all resistors(sumR/50).