The Hailstone Problem (also known as the Collatz Conjecture) concerns itself with the properties of the series of
numbers which are generated from a given starting number by the application of the following rules:-
- If the current number is even, divide by 2 to generate the next member.
- If the current number is odd, multiply by 3 and add 1 to generate the next member.
1 4 2 1 This is the Hailstone series.
O E E O This is the Hailstone Signature of the Hailstone series.
3 10 5 16 8 4 2 1
O E O E E E E O
7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1
O E O E O E E O E E E O E E E E O
- The Signature of a series is a string of letters which correspond to the numbers of the series, with an O for each
odd number and an E for each even number.
- The members of most hailstone series go up and down in what appears to be a totally random fashion in much the
same way as a hailstone rises and falls in a storm cell until it grows to a such a size and weight that it has no
alternative but to fall to the ground.
- Most commentaries on the Collatz conjecture warn of the unpredictable and haphazard behavior of the number series
it produces. This is undoubtedly a fact, but by the time you have finished studying this tutorial, you will see
that some of the behaviors are reassuringly regular and very predictable.
- The first of the above series appears to be unique. It is the only one which returns to its starting value.
- The number 10 appears in both the second and third examples. This will happen to any number which is half of some
even number, and also one more than three times some odd number.
- All of the series end with the number 1. There is no known example of a series which ends in any other way.
- One of the alternative names for the hailstone problem is the Collatz conjecture, the conjecture being that all
numbers suffer this same fate of terminating at 1. What appears in the following pages doesn't actually prove that
conjecture, but it does offer some compelling evidence in support of its truth.
- To fail the conjecture, a number would have to either enter a loop, or continue its up and down behavior forever,
both of which imply an infinitely long Signature.
- Signatures can vary greatly in length. The longest Signature for any number less than 1017 has 2091
letters. But see Hailstone Tree Outliers for examples
of Signatures which behave in a very strange way, including one with 134,404 letters. There is no upper limit to the
length of a Signature.