# 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. 35Ð36.
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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?

### 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!

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