The Life of a Natural Number

The life of a natural number can have its ups and downs. If you don’t believe it, just ask one who has lived through a hailstorm. This particular storm tosses the numbers about in the following way: if N is even, it drops to N / 2; if N is odd, it is lifted to (3N + 1) / 2. The choppy ride continues like this forever unless the number 1 is reached; 1 is the ground. Some numbers fall to the ground immediately; others are buffeted all over the place. For example, 8 falls to 4, then to 2, and then to the ground. In contrast, 13 takes quite a ride: 13 -> 20 -> 10 -> 5 -> 8 -> 4 -> 2 -> 1.

We say that 8 has a trip of length 3; 13 has a trip of length 7.
The number 8 did not exceed its original height, whereas 13 reached an altitude of 20.

During this hailstorm, N is less than or equal to 10,000:

  1. Which number reaches the highest altitude, and what is this record height?
  2. Which number experiences the longest trip, and how long is it?
  3. Numbers tend to get quite ill when raised to altitudes well beyond what they are used to. If we find a measure for air sickness by dividing the maximum altitude attained by the number itself, which number experiences the greatest air sickness?
  4. Which number experiences the most consecutive —
    • downdrafts? How many are there?
    • updrafts? How many are there?
  5. Comment about the following theories concerning hailstorms:
    • No number N larger than 100 rises above N2.
    • There is no longest trip.
    • There is no maximum airsickness quotient.
    • All numbers eventually hit the ground.

(Reprinted with permission from Exploratory Problems in Mathematics, copyright 1992 by the National Council of Teachers of Mathematics. All rights reserved.)

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