### The Geyser Problem

#### by Aiona

Question

There are three art pieces in the museum just outside the office that are designed to mimic the behavior of a geyser. The geysers each have a different period; one goes off every 6 minutes, another goes off every 7, and a third goes off every 8. The question is, when will all three geysers go off at the same time.

Four Part Answer

Part I: Watch the 6 minute geyser and write down the time it goes off. Now count how many minutes until the 8 and 7 minute geysers go off. Call these two time intervals *a *and *b* respectively.

Part II: Calculate how many minutes before the 6 and 8 minute geysers will coincide using the equation *x*=3(8-*a*). (note: if *a* is odd the geysers will never synchronize.)

Part III: Calculate how many minutes before the 6 and 7 minute geysers will coincide using the equation *y*=6(7-*b*).

Part IV: Calculate how many minutes after the next 6 by 8 minute synchronism, a 6 by 8 synchronism will coincide with a 7 by 8 synchronism. Call this *m*, and use the following table to look it up.

*y*–*x * *m*

-36 48

-30 96

-24 144

-18 24

-12 72

-6 120

0 0

6 48

12 96

18 144

24 24

30 72

36 120

All three geysers will synchronize at (the original time you wrote down)+*x*+*m*.

Explanation

This problem turned out to be much more complicated than I originally anticipated. I spent all day with a scratch sheet working on it. Basically, in parts II-IV of the solution there are two rhythms happening at the same time and you want to know when they will come together. For example, in part II, the 6 and 8 minute geysers will synchronize every 24 minutes. I drew out the entire 24 minute cycle on a piece of paper, and for each 6 minute geyser within it I wrote down both the time interval from the 6 minute geyser to the 8 minute geyser (*a*) and the time from the 6 minute geyser to the synchronization point (*x*). Then I looked for a relationship between *a* and *x* in order to generate the equation in part II. That equation is basically just a map that tells you where you are in the pattern. The chart in part IV serves the same function. It’s all a little tricky unless you write it out, but next week I plan on visiting the geysers with a watch to verify that this works. Please add corrections, simplifications, or better explanations if you have any. I got sucked into a geeky math whirlpool and struggled to make this at all understandable when I came out. As a side note, similar rhythmic patterns are used by drummers, in African polyrhythms and in Jazz.

Wow, messy! :-)

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I’m not a mathematician so I wouldn’t know where to start to solve it more cleanly (you really can’t use plain old equations as there’s a constraint that the result must be an integer, somewhere, I think).

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Just a little note: 7-6 = 1, so every time you add a seven to another seven you go one unit farther than when you add a six to another six. I.e. 7-6 = 1 but 14-12=2, 21-18=3 all the way to 63-54=9. So is not surprising that, whatever the initial delay, you can always line up the two series (i.e. the multiplication tables for 7 and for 6).

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Otoh, 8-6=2 so no odd number allowed (16 will be four units far from 12, 24 will be 6 units far from 18 and so on), to line up the series you must have even increments, as you noted… it makes sense :)