The New Sudoku Players' Forum

[Smythe Dakota]

Resistors are connected in an infinite ladder, starting at 0 and going up to infinity. Specifically, if y is any non-negative integer, then there is a 1-ohm resistor between (0,y) and (1,y), another 1-ohm resistor between (0,y) and (0,y+1), and a third between (1,y) and (1,y+1).

What is the total resistance between (0,0) and (1,0)?

Bill Smythe

[Bigtone53]

If the ladder extends up the y-axis to infinity, then adding another 'step' at the bottom will not change the answer.

If the resistance between (0,0) and (0,1) is R, then adding the additional step gives the R between (0,-1) and (1,-1) as a parallel of 1 between (0, 0) and (0,1) directly, and a series of (1+R+1) between (0,1) and (1,1) via the y=1 'step'.

R = 1 in parallel with R+2

= (R +2) / (R+2+1)

= (R+2) / (R+3)

R²+3R = R + 2

R² +2R -2 = 0

R = (-2 ± sqrt(4 +8)) /2 = Sqrt3 - 1