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:
- Which number reaches the highest altitude, and what is this record height?
- Which number experiences the longest trip, and how long is it?
- 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?
- Which number experiences the most consecutive —
- downdrafts? How many are there?
- updrafts? How many are there?
- 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.)