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The hailstone sequence
The rules
The Hailstone sequence is created using the following rules:
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Start with any positive whole number, called the seed.
If the number is even, halve it (½n) to get the next number in the sequence.
If the number is odd, multiply it by 3 and then add 1 (3n + 1) to get the next number
in the sequence.
Continue until you see repetition.
Example
Choosing 17 as the seed:
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, [4, 2, 1, 4, 2, 1 ...]
With 17 as the seed number, the sequence is 13 terms long. It is called a ‘Hailstone’
sequence as the values bounce up and down like hailstones inside a cloud, before falling
to the ground.
The Collatz conjecture
In 1937, Lothar Collatz stated that no matter what number you start with, the Hailstone
sequence will always reach 1. This hasn’t yet been proven mathematically.
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The hailstone sequence
Activity one
Try out different seeds to investigate the Hailstone sequence and see if you agree with the
Collatz conjecture.
How long does it take each of your sequences to reach 1? Is it possible to predict the
length of the sequence from the chosen seed?
Create graphs for three different seeds to show how the numbers bounce around.
Activity two
If all seeds do reach 1, it will be possible to create a ‘Hailstone tree’ to show how different
Hailstone sequences converge together.
The first ‘branch’ occurs at the number
16. This is because 16 could be found
by ...
32 ÷ 2 = 16
3 x 5 + 1 = 16
Explain why there are no ‘branches’ for
the numbers 1, 2, 4 and 8.
Extend the tree below as far as you can
on your page. How high up the tree do
you have to go to find the numbers 1-10?
Extension
The Hailstone sequence has a looped ending, where the terms 4, 2, 1 are repeated.
If we follow different rules, we get different looped endings. Find all the different looped
endings if the rules changed to:
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If the number is even, halve it to get the next number of the sequence.
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If the number is odd, multiply it by 3 and then subtract 1 to get the next number in
the sequence.
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The hailstone sequence
Teaching notes
Activity one
As of 2012, the Collatz conjecture is unproven. There is a research paper that comes close, but
unfortunately contains some incomplete reasoning:
http://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2011-09.pdf
Activity two
There are some hints that may help students create their tree:
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How do you ‘reverse engineer’ the sequence?
o
Every number will have at least one branch coming from it, as they can all be
multiplied by two.
o
A second branch can be found on those numbers where (n – 1) ÷ 3 results in an odd
integer. An interesting question to ask is ‘why do we exclude even integers?’ (because
they would have been halved in the sequence).
What tactics can be used to focus our search for the numbers 1-10?
o
Students may get stuck locating 7 and 9, as they are quite high up the tree. They could
be encouraged to create the Hailstone sequence using 7 and 9 as seeds, to show in
which direction they should extend their tree.
The tree that shows all of the numbers 1-10 is shown as a poster on the following page.
Activity three
The change of rules appears to produce three different looped endings:
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1, 2, 1, 2, …
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5, 14, 17, 20, 10, 5, ...
and the easily missed:
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17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136, 68, 34, 17, ...
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The hailstone sequence
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The hailstone sequence
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