In the east and midwest United States there is a noisy summer insect called a "periodical cicada". These insects are noted for three things. They are annoyingly loud during their 4-6 week mating season; the larva spend either 13 or 17 years maturing underground; and on years in which they emerge, there are literally millions of them per acre of woodland. There are 30 "broods" of periodical cicada located in different areas of the country. Broods I-XIV have a 17 year cycle and broods XV-XXX have a 13 year cycle. Last year (2004), brood X (one of the largest 17 year broods) emerged from Washington D.C. to Michigan. In 2002 brood XXIII, a 13 year brood emerged in some of the same region. ( More details are available.)

1. How often over the next 100, 200, 500, 1000 and 10,000 years will the 17 year brood X and the 13 year brood XXIII emerge in the same year?
2. It is curious that the two periods commonly observed are 13 and 17 years — both prime numbers. How often would broods with periods of 12 and 16 years emerge in the same year?
3. Suppose the next time the 13 year brood and the 17 year brood emerge together a new hybrid subspecies with a 15 year period develops. How often over the next 10,000 years will all three of these broods appear at the same time?

Can you develope a general rule to predict how often, on average, broods with periods N and M will emerge together? Try the same question for three broods with period L, M, and N. What about four broods...?

Communicated by G. Hall.