# Infinite staircase where each step is a 1 ohm resistor [closed]

There is an infinite staircase where each step is a 1 ohm resistor. How do you calculate the equivalent resistance between any of the two points on the staircase?

• In other words, an infinite sequence of resistors connected in series? Count the number of resistors between your two points and multiply by $1\,\Omega$. – hmakholm left over Monica Jul 15 '16 at 10:46
• @HenningMakholm I think he's refering to a circuit like =|=|=... A circuit scheme would be greatly welcome – Bosoneando Jul 15 '16 at 10:59
• Please would you post a diagram or sketch of a few sections of the "infinite staircase," and the points between which resistance is to be calculated.... Also please note that this is not a homework site : you are expected to show some effort to work through the problem yourself, and to ask about a conceptual difficulty. Please show your attempt to solve this problem. – sammy gerbil Jul 15 '16 at 11:26
• Hey, did you read what I wrote below before commenting? If you could pose this question as a stats/prob/optimization problem this would be helpful... – PT272 Jul 15 '16 at 11:41
• @PT272, you just said the resistors are arranged in an infinite staircase fashion. We are not aware such an arrangement and you haven't provided us with any picture using which we can understand. If its a series of resistors connected one after other infinitely long, then the equivalent resistance is infinity. – Yashas Jul 15 '16 at 11:55

it depends on how you connect the resistors

if you arrange them in a grid, where they connect both going sideways along the stairs and up/down the stairs, then I think it would tend towards zero

but if you have separate lines of resistors going up/down the stairs then it should be infinite?

If you're considering the following:

then the resistance between the ends is $1 + \frac{1}{2} + \frac{1}{3} + \cdots$ which doesn't converge.

• It is the Harmonic series with $ln(n) + \gamma$ where $\gamma$ Euler–Mascheroni constant. But I meant more of a ladder as suggested above: =|=|=. Would the resistance be just the number of ladder rungs between them? – PT272 Jul 15 '16 at 14:51

I'm imagining this infinte staircase:

         +--+ -->
|  |
+--+--+
|  |  |
+--+--+--+
|  |  |  |
+--+--+--+--+ -->
|           |
V           V


If the staircase has infinite underfill, by symmetry, the resistance of two points on the edge will be twice that between the same two points on the infinite resistor grid plane.

• Thanks, I am not so familiar with physics terminology, do you mind to write the equation for it. – PT272 Jul 15 '16 at 11:04
• Regarding your diagram it could be a staircase with steps or a letter with rungs, don't know whether this would make any difference. – PT272 Jul 15 '16 at 11:07
• I don't recall the equation, but it's a common problem, a web search should find it. – Jasen Jul 15 '16 at 11:08
• xkcd.com/356 won't help with the solution, but worth a look, – Jasen Jul 15 '16 at 11:10
• mathpages.com/home/kmath668/kmath668.htm may be more helpful – Jasen Jul 15 '16 at 11:12