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The hailstone sequence The rules The Hailstone sequence is created using the following rules: 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. © www.teachitmaths.co.uk 2014 17644 Page 1 of 5 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: If the number is even, halve it to get the next number of the sequence. If the number is odd, multiply it by 3 and then subtract 1 to get the next number in the sequence. © www.teachitmaths.co.uk 2014 17644 Page 2 of 5 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: 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: 1, 2, 1, 2, … 5, 14, 17, 20, 10, 5, ... and the easily missed: 17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136, 68, 34, 17, ... © www.teachitmaths.co.uk 2014 17644 Page 3 of 5 The hailstone sequence © www.teachitmaths.co.uk 2014 17644 Page 4 of 5 The hailstone sequence © www.teachitmaths.co.uk 2014 17644 Page 5 of 5