*Originally posted by sonhouse*

**Clock Confusion puzzle: I came across this today.
**

Clock confusion

Clock

We all know that homemade presents are supposed to be the best, but the clock your Aunty Mabel made you is a little hard to get used to — the hour and minute hands are exactly the same! You can muddle through most of the time but sometimes, say 26 minutes past 2 or just after 12 ...[text shortened]... nd midnight, how many moments are there when it is not possible to tell the time on this clock?

A far more interesting problem than I first thought!

The first thing to note is that we're looking for clock positions that are symmetrical. The trick is, symmetrical with respect to what? In this case, we need the hour and minute hands to be interchangeable. The simplest example of this is 12:00, where it doesn't matter which hand is which since they both point straight up at the 12. Are there any more positions like this? It turns out that there are lots, but the number is not intuitive.

One major wrinkle in this problem is the fact that the minute hand completes twelve full circles before the hour hand completes its first. This is of course obvious, but it complicates the equations somewhat. However, thinking about symmetry lead me to a graphical solution, which I think is the simplest approach here.

If we draw a graph of the rotation of the minute hand with respect to the rotation of the hour hand (hour hand rotation on the x-axis), we get a shark-tooth pattern that looks something like this:

|. / . . . /

| / . . ./

|/____/____

(Sorry about the dots, they're just spacers...)

Basically, the minute hand will rotate from 0 to 2*pi radians by the time the hour hand rotates (1/12)*2*pi = pi/6 radians, and then it goes back to 0 and starts again. As stated above, we need the clock positions to be symmetrical such that we can interchange the hour and minute hands and the clock will read the same. To find these points of symmetry, we simply draw a new graph with the hour hand rotation on the y-axis and the minute hand rotation on the x-axis and superimpose it on the original (if your left hand were the graph, this is this is equivalent to starting with your hand outstretched with your palm facing out, rotating your hand 90 degrees to the right, and then flipping your hand so your palm faces inward - note the position of your thumb in both cases). What you end up with is diamond-shaped grid with numerous points of intersection. Some of these intersections are important and some aren't, but I'll get to all that after lunch.

More to come shortly! ðŸ™‚