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.