hailstone (noun): a native English compound. The first element is from the Indo-European root kaghlo- “pebble, hail”; the second element is from the Indo-European root stei- “stone,” so the compound is somewhat redundant. In number theory hailstone numbers are sequences of integers generated in the following way. (a) Choose a positive integer. (b) If it is odd, triple it and add 1, but if it is even, divide it by 2. (c) Reapply the rules of part (b) to each new result. For example, a starting number of 7 produces the hailstone sequence 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, …; the final 4, 2, 1 is bound to repeat forever. These sequences get their name from the fact that the values within a hailstone sequence typically rise and fall like a hailstone inside a cloud. Just as a real hailstone eventually becomes so heavy that it falls to earth, every starting integer ever tested has produced a hailstone sequence that eventually drops down to the number 1, at which time it “bounces” into the small loop 4, 2, 1,. … The first person known to have discussed the hailstone numbers—though not necessarily under that name—is one Lothar Collatz, while he was a student in the 1930's.