I figured something out on the bus yesterday
Just a quick note.
Say you had a network of resistors. If you find it difficult to reason about what the resistance of the network should be, here's a trick you can use (I think- I think with confidence of about 75% that it will be true for every case.) I have not proved this; it's just my intuition and a few toy examples so far.
Replace every resistor with a switch that could be open or closed. The Kth switch will be closed with probability p_K and open with probability 1-p_K. Figure out what the probability is that there is at least one closed path through the network. This will be some expression in terms of the various p_Ks.
Now, we're going to take your expression and change it so that it tells us about resistances instead of probabilities. Replace every multiplication with addition, and every minus sign with division. Replace every p_K with the resistance of the Kth resistor. Tada! You now have an expression for the resistance of the original network.
This gives us another avenue of attack when trying to understand the most recent xkcd comic.
Say you had a network of resistors. If you find it difficult to reason about what the resistance of the network should be, here's a trick you can use (I think- I think with confidence of about 75% that it will be true for every case.) I have not proved this; it's just my intuition and a few toy examples so far.
Replace every resistor with a switch that could be open or closed. The Kth switch will be closed with probability p_K and open with probability 1-p_K. Figure out what the probability is that there is at least one closed path through the network. This will be some expression in terms of the various p_Ks.
Now, we're going to take your expression and change it so that it tells us about resistances instead of probabilities. Replace every multiplication with addition, and every minus sign with division. Replace every p_K with the resistance of the Kth resistor. Tada! You now have an expression for the resistance of the original network.
This gives us another avenue of attack when trying to understand the most recent xkcd comic.