## Monday, June 13, 2016

### Solution to Clock Puzzle

On a wall clock, the minute hand (black) and the sweep second hand (red) are the same length, while the hour hand (grey) is shorter. If the clock starts at noon, then at approximately what time after that do the tips of the three hands first become collinear?

During the first minute, the hour hand will barely move, and the minute hand will move a bit, so there arises a bit of angular separation between them. The line through their tips intersects the far edge of the clock somewhere to the left of the 6. Thus, there will be a time, somewhat after 30 seconds have passed, when the tips of the clock hands are collinear. Here's a picture of that moment:

If we take the hour hand to be three-fourths as long as the other hands, as suggested in the previous post, then we can give a more precise answer: collinearity first occurs after 33.6 seconds have passed.

Between noon and midnight, there are 708 different moments when the tips of the three hands are collinear.

(This count does not include the trivial instances of collinearity that occur when the minute hand and second hand coincide—there are 708 of those also.)

Here's a movie that animates all 708 of the moments when the hand-tips are nontrivially collinear.

If you want to know how I found these 708 cases, click here. (I think this is probably the first time I've ever factored a 1,438th-degree polynomial!)