The Hall of Lights
RAY: Here's the answer. Let's number all the lights and pick one at random.
TOM: How about 26?
RAY: OK, let's look at light number 26 and figure out if it's going to be on or off. All we need to know are the factors of the number 26. Well, what's a factor? A factor is a whole number that will divide evenly into another number, with nothing leftover.
So, the factors of 26 are 1, 26, 13 and 2.
Here's why that's important. It tells us that light number 26 is going to get its chain pulled four times.
TOM: How did you figure that out?
RAY: Well, when every cord gets pulled it gets turned on, right? Light number 26 gets its cord pulled again at 2, which is a factor of 26.
When every 13th chain gets pulled, light number 26 gets turned on again. And it doesn't get touched again until 26, when it gets turned off forever.
Now it's pretty obvious then that every bulb that has an even number of factors will eventually get turned off for good.
So, which lamps remain on? All those represented by a number with an odd number of factors. And those are, are you ready for this? Light bulbs 1, 4, 9, 16, 25, 36, etc.
All those numbers are called perfect squares. And only they have an odd number of factors, because one of the factors is the square root of the number in question. For example, nine has three factors, 1 and 9 and 3.
Do we have a winner?
TOM: We do have a winner and the winner this week is Laurie Warner from Greer, South Carolina!
[ Car Talk Puzzler ]