Thwaites Conjecture

Cambridge University Puzzles and Conjectures
Read about the prize
Thwaites: "Two conjectures or how to win 1100 Pounds", Mathematical Gazette, 80. (March 1996), pp. 3536. Hear a BBC radio show that poses this question
Think of a positive whole number. Apply the following rules to it:
If the number is even, halve it

If the number is odd, multiply it by 3 and add 1.

Repeat this procedure and you will eventually reach 1.

For example:
Starting with 26, successive results of this iterative procedure are: 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.

Notice that once you hit a power of 2, you cascade down through decreasing powers of 2: 64, 32, 16, 8, 4, 2, 1

But some other numbers can give you quite a run for your money. 31 takes over a hundred iterations before it reaches 1.

However, there is a number smaller than 31 which takes over 100 steps.
Can you find it?

Number to check:

A Hint

Consider a number turning up in the course of a Thwaites iteration. It could, in general, have come from two other numbers. We could have arrived at it by halving the previous number, or by multiplying a number by 3 and adding 1.

Note that only numbers which are 1 more than a multiple of 3 can be arrived at by this second route!

For example, we could have arrived at 31 either from 62 or from 10. But we could only have arrived at 62 by halving 124, since 62 cannot be arrived at by multiplying a whole number by 3 and adding 1. Trace the possible 'ancestors of 31'

The Background and the Prize: Win 1000!
Hear a BBC radio show that poses this question