Hey mathematicians, would you like to help me find patterns between rolling ball clocks?
Asked by
phaedryx (
6137
)
March 5th, 2013
I recently wrote software to model a rolling ball clock. You can see what a rolling ball clock is here: http://www.youtube.com/watch?v=-3hNCinIBdk and read a description here: http://en.wikipedia.org/wiki/Rolling_ball_clock
The problem I was solving was to figure out how many days it takes for the balls to return to their original order once you start the clock, starting with a given number of balls. Here’s a list of the results: https://gist.github.com/phaedryx/b6ce1df06714c8e66634
I’m wondering if there is some kind of pattern between them, for example, if I know the results of a 27-ball rolling ball clock does that tell me anything about the 54-ball rolling ball clock? Could I calculate a subset of the clocks and extrapolate the results for the rest of the clocks somehow?
I’ve looked at it a bit, but it’s beyond my ken.
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3 Answers
I read the article & watched the video, but I don’t get it. It takes 27 balls to display 12:59 (12 balls on the hours rack, 11 on the 5-min, 4 on the 1-min). But then to roll over to 1:00 would seem to require at least one additional ball “in play,” representing the next minute to trigger the mechanism to dump the racks. That makes 28 balls minimum. Your simulation starts at 27 balls, hence my confusion.
When you increase the number of balls where do you put them? Does it simply increase the length of the queue of balls waiting to be scooped up?
I think it’s very cool, @phaedryx, that you wrote a simulation program to determine periodicity in the system for variable number of balls. My intuition is that the machine is sufficiently complicated and quirky that there are no simple rules among the numbers you’ve compiled. In other words, your algorithm might be its shortest description!
@gasman
1. There are only 11 spots on the hours rack, if the hour rack is empty it is 1 o’clock (that threw me a bit at first when coding it). The particular clock in the video has 12 balls in the hour track, but the first (larger) ball is glued in place.
2. yeah, it just increases the size of the initial queue
Another feature that might not be obvious: when the balls are emptied from a track they are added to the end of the queue in reverse order. For example, the 6-minute mark of a 27-ball clock would look like this (balls numbered 1–27):
queue: 7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,4,3,2,1
minute track: 6
five-minute track: 5
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