356: Nerd Sniping
Title text: I first saw this problem on the Google Labs Aptitude Test. A professor and I filled a blackboard without getting anywhere. Have fun.
Nerds have a way of getting distracted easily and focusing one thing and ignoring the rest. Black Hat has decided to make this into a disturbing game of getting nerds, in this case a physicist, to stop in the middle of a street by showing a problem for the victim to solve.
The problem Black Hat shows us is an electronics engineering thought experiment to find resistance between two points. In normal wiring, a one ohm resistor would result in one ohm of resistance. Two resistors connected in a series, where electricity has to go through each has two ohms of resistance. Two resistors in parallel give the circuit only half an ohm since you average the resistance of the path (1 ohm of resistance over 2 paths).
With an infinite grid of resistors, you have an infinite number of paths to take, and for each path an infinite number of both series and parallel paths to consider, so much more advanced methods are needed. The exact answer to the question is 4/π − 1/2.
- [Hat Guy is sitting on a chair, the Normal Guy is standing next to him. Across the street another man is coming from a building.]
- Hat Guy: There's a certain type of brain that's easily disabled. If you show it an interesting problem, it involuntarily drops everything else to work on it.
- [The man across the street is about to enter a crosswalk]
- Hat Guy: This has led me to invent a new sport: nerd sniping. See that physicist crossing the road?
- [Hat Guy holds up a sign]
- Hat Guy: HEY!
- [There is an image of a grid with resistors on every connection, two nodes a knight's move apart are marked with red circles.]
- The sign reads: On this infinite grid of ideal one-ohm resistors, what's the equivalent resistance between the two marked nodes?
- Physicist on the street: It's... Hmm. Interesting. Maybe if you start with... No. Wait. Hmm... You could--
- [A truck is zooming past, apparently where the physicist just stood]
- Normal guy: I will have no part in this.
- Hat Guy: C'mon, make a sign. It's fun! Physicists are two points, mathematicians three.