Solution¶
I hypothesized that for some max value of brood cycle, the longest stretch going without interference would occur between the largest two odd numbers less than the max value.
I wrote some code and ended up with this graph.
Earlier this year, a new generation of Brood X cicadas had emerged in many parts of the U.S. This particular brood emerges every 17 years, while some other cicada broods emerge every 13 years. Both 13 and 17 are prime numbers — and relatively prime with one another — which means these broods are rarely in phase with other predators or each other. In fact, cicadas following a 13-year cycle and cicadas following a 17-year cycle will only emerge in the same season once every 221 (i.e., 13 times 17) years!
Now, suppose there are two broods of cicadas, with periods of A and B years, that have just emerged in the same season. However, these two broods can also interfere with each other one year after they emerge due to a resulting lack of available food. For example, if A is 5 and B is 7, then B’s emergence in year 14 (i.e., 2 times 7) means that when A emerges in year 15 (i.e., 3 times 5) there won’t be enough food to go around.
If both A and B are relatively prime and are both less than or equal to 20, what is the longest stretch these two broods can go without interfering with one another’s cycle? (Remember, both broods just emerged this year.) For example, if A is 5 and B is 7, then the interference would occur in year 15.
I hypothesized that for some max value of brood cycle, the longest stretch going without interference would occur between the largest two odd numbers less than the max value.
I wrote some code and ended up with this graph.
My hypothesis seems to be correct, and for brood cycles of 17 and 19 years emerging this year, their first interference occurs 153 years later.