Resistor Network Puzzle

Thanks a billion for the references. This has been keeping me awake nights for two decades now. (O.K., not EVERY night.) I understand the mechanism in the examples given, which seem to necessitate the use of non-linear elements - the string combination or the zener diodes - but I thought it was also possible to do with only resistors. I might have been remembering this part incorrectly, but if anyone has that solution, I would be interested. Now that I see these examples though, it seems that a purely linear network is incapable of this behaviour.

The presence of elements other than resistors in a "resistor network" turns this into an unfair trick question/puzzle.

Of course, in the traffic world, it is easy to see how a new one-way street could aggravate congestion.

But then, how about this sequence of signs: Right turn only; U-turn not permitted; dead end street? (I think I've seen a cartoon like this.)

It's not a %%$#& cartoon. Here in Pittsburgh, I used to cut up a small alley behind the post-office so that I could use the letter drop. I hadn't used it for years, so didn't know. I started up the street (which is no longer a real street). Halfway up, where I used to turn, the alley was gone. Dead ahead was a "Do not enter" sign. To my left, a sign said "Buses only". In the middle of the street was a sign saying "No U-turn". To make it worse, there was a cop sitting on the other side, so I stopped, got out, and gave him a shrug. He yelled, "Turn around and get the %%# outta here, you moron!". So, I did.

Art imitates life!

For better or worse.

And/or vice versa.

Ah give this guest a break, Tornado. The OP mentioned that this was from memory and that it was counterintuitive. He brought this paradox here with enough information that TVP45 quickly sniffed it out. (Great job TVP45, and a GA from me.) We need more people to visit us with this type of perspective.

Guest, please sign up with us. I think you'll enjoy the wacky community we've formed here.

Maybe, but I'm not sure.

I tried checking out the link to the Braess paradox, but got into a daisy chain of translations and article citations, none of which laid out an example of this concept. It was easy to imagine a resistor network in which an additional conductive path would not change the overall resistance at all. (Such as any conductive path between equipotential nodes.)

I'm with you in encouraging the OP to join, and in the designation as a "wacky group." To that I heartily plead guilty.

Hey! I do my best/worst as the case may be. I actually beat up on a few of MoronicBumble's earliest posts, but there have been some really good ones lately. Now I'll have to check out Magwer.

I found this to be informative on Braess' Paradox http://en.wikipedia.org/wiki/Braess's_paradox

Thanks! Now I can scratch my head for a resistor analogue to this traffic example.

It might look like and act like a decade box.

Ken.

http://www.google.co.uk/search?q=resistance+box&sourceid=ie7&rls=com.microsoft:en-gb:IE-SearchBox&ie=&oe=&rlz=1I7DKUK_en-GB&redir_esc=&ei=vv0BTMODA4L-0gTNs7X3Ag

http://en.wikipedia.org/wiki/Wheatstone_bridge

http://www.google.co.uk/search?q=resistor+bridge&sourceid=ie7&rls=com.microsoft:en-gb:IE-SearchBox&ie=&oe=&rlz=1I7DKUK_en-GB&redir_esc=&ei=k_4BTO-KL4z-0gTykZ33Ag

http://www.google.co.uk/search?q=super+resistor+bridge&sourceid=ie7&rls=com.microsoft:en-gb:IE-SearchBox&ie=&oe=&rlz=1I7DKUK_en-GB&redir_esc=&ei=Mv8BTImmK4Si0gSw85T3Ag

http://www.google.co.uk/search?hl=en&rls=com.microsoft:en-gb:IE-SearchBox&rlz=1I7DKUK_en-GB&q=wheatstone+bridge+proof&revid=919118303&ei=Xf8BTPb_JYvy0gTLqqTMAw&sa=X&oi=revisions_inline&resnum=0&ct=broad-revision&cd=4&ved=0CEAQ1QIoAw

I suspect you have to really define this kind of puzzle pretty well. Braess made a big deal outta 42nd Street - how if you closed it, traffic flowed faster. Well, to where? Can you stand at Fort Lee and watch the cars coming out/going in the tunnel and tell whether 42nd is closed?

If you really wanted to try this for traffic, I'd say take 5 000 NY cabs at 2:00 am, turn off all the traffic sugnals (not that they obey them anyway), take all the cops off the streets and see what happens. My guess is those cabs are the closest thing to a "sea of electrons" there is in traffic.

I can remember pulling up to a light somewhere around 50th and Madison one morning. Madison goes uptown, but was just about empty. A cabbie, wanting to go downtown, backed into Madison so that he was facing the right way and then went flying backwards down the Avenue at about 50 mph. Now that's the path of least resistance!

The effect is, by the way, easy to see if you add a time component (in effect doing impedance rather than resistance). If you have a major highway (like Rte 5 in CA) and close one lane for maybe 30 miles, traffic slows slightly, but that's all. If you close the same lane for only 500 yards, traffic almost comes to a standstill (impedance mismatch = reflected waves).

A very good puzzle, though. Please join, guest.