An infinite grid of 1ohm resistor

[3gp]

http://xkcd.com/356/

"I first saw this problem on the Google Labs Aptitude Test. A professor and I filled a blackboard without getting anywhere. Have fun."

Discuss.

 

[The Electrician]

This is an old chestnut.

It's not really indicative of someone's smarts, in my opinion. I don't think anybody who hasn't seen it before is going to be able to come with the solution on the spur of the moment. See: http://www.geocities.com/frooha/grid/node2.html

The "across a diagonal" problem is the difficult one. The "across one of the resistors" problem can be solved with a few moments thought.

 

[strokebow]

Question:

What would be the total resistance... the equivalent resistance of the entire grid?

i.e.) If you have a grid that is very large (assume infinite) but yet is still finite how would one being to calculate the total resistance of the grid?

Anyone?

 

[Averagesupernova]

Question:

What would be the total resistance... the equivalent resistance of the entire grid?

i.e.) If you have a grid that is very large (assume infinite) but yet is still finite how would one being to calculate the total resistance of the grid?

Anyone?

Ummmm, infinite or not? Pick one. The ohmeter will settle on a value and that is it. If you carried out a very very very large number of places to the right of the decimal point and had the remaining hardware built precise enough to get a sensible measurement you would see the resistance fall but never approach zero. It is like rolling out a tape measure to 100 feet and then cut it off at 50, then cut it off at 25, then 12.5, etc. The tape continues to get smaller but never will get to zero.

 

[strokebow]

How so?

From what I have read it appears the opposite to what you say.

The resistance will increase but at a very slow rate. And when one gets to the boundary (obviously there is no boundary to infinity - but for a huge grid) you would expect that actually the resistance would make bigger jumps to a large finite resistance.

 

[Averagesupernova]

You put ohmeter leads across a single resistor or a pair of series resistors that are in parallel depending on how you position the probes. Tell me how the resistance can increase. Unless we continually insert resistance in series with the resistors that are directly across the probes, the resistance cannot increase. If you don't believe me, start with one resistor across the ohmeter leads and then start adding to it in parallel. Do the math with each resistor addition.

 

[strokebow]

On a very large -> infinite grid.

If you have a reference point at node x and you measure the resistance further and further away from the reference towards infinity you will find that your resistance increases and also tends towards infinity.

 

[Averagesupernova]

I misunderstood your first question.