1. Activity 2-12: Hikorski Triples www.carom-maths.co.uk

2. What does mean to you?

4. If is the answer, what is the question? Putting this another way:

5. What do the expressions mean to you? We can broaden this out:

6. The Theory of Special Relativity tells us that nothing can travel faster than the speed of light. Suppose we say the speed of light is 1. We can add two parallel speeds like so: So if a train is travelling at 2/3 the speed of light, and a man is travelling at 4/5 the speed of light relative to the train, how fast is he travelling? Task: show that if |a|, |b| < 1, then |(a + b)/(1 + ab)| < 1.

7. You may not have met the functions tanh(x) and coth(x) yet, but when you do you will find that

8. GCSE Resit Worksheet, 2002 How many different equations can you make by putting the numbers into the circles? Solve them!

9. Suppose a, b, c, and d are in the bag. If ax + b = cx + d, then the solution to this equation is x = There are 24 possible equations, but they occur in pairs, for example: ax + b = cx + d and cx + d = ax + b will have the same solution. So there are a maximum of twelve distinct solutions.

10. This maximum is possible: for example, if 7, -2, 3 and 4 are in the bag, then the solutions are:

11. If x is a solution, then –x, 1/x and -1/x will also be solutions. ax + b = cx + d a + b(1/x) = c + d(1/x) c(-x) + b = a(-x) + d a + d(-1/x) = c + b(-1/x)

12. So the solutions in general will be: {p, -p, 1/p, -1/p} {q, -q, 1/q, -1/q} and {r, -r, 1/r, -1/r} where p, q and r are all ≥ 1. Are p, q and r related?

13. It is possible for p, q and r to be positive integers. For example, 1, 2, 3 and 8 in the bag give (p, q, r) = (7, 5, 3). In this case, they form a Hikorski Triple (or HT).

14. Are (7, 5, 3) linked in any way? Will this always work?

15. a, b, c, d in the bag gives the same as a + k, b + k, c + k, d + k in the bag. Translation Law Remember...

16. a, b, c, d in the bag gives the same as ka, kb, kc, kd in the bag. Dilation Law Remember... So we can start with 0, 1, a and b (a, b rational numbers with 0 < 1 < a < b) in the bag without loss of generality.

17. a, b, c, d in the bag gives the same as -a, -b, -c, -d in the bag. Reflection Law (Dilation Law with k = -1)

18. Suppose we have 0, 1, a, b in the bag, with 0 < 1 < a < b and with b – a < 1 then this gives the same as –b, – a, – 1, 0 (reflection) which gives the same as 0, b – a, b – 1, b (translation) which gives the same as Now

19. If the four numbers in the bag are given as {0, 1, a, b} with 1 1, then we can say the bag is in Standard Form. So our four-numbers-in-a-bag situation obeys three laws: the Translation Law, the Reflection Law and the Dilation Law.

20. Given a bag of numbers in Standard Form, where might the whole numbers for our HT come from?

21. The only possible whole numbers here are (b – 1)/a must be the smallest of these. Either one of could be the biggest.

22. Task: check out the following - So the only possible HTs are of the form (p, q, r) where r = (pq + 1)/(p + q), And where p  q  r are all positive integers.

23. We now have that the twelve solutions to our bag problem are:

24. Pythagorean Triples This has the parametrisation (2rmn, r(m 2 - n 2 ), r(m 2 + n 2 )) Hikorski Triples Do they have a parametrisation? Choosing positive integers m > n, r always gives a PT here, and this formula generates all PTs.

25. How many HTs are there? Plenty... All n > 2 feature in at least 4 HTs.

26. Is abc unique for each HT? The Uniqueness Conjecture If (a, b, c) and (p, q, r) are non-trivial HTs with abc = pqr, then (a, b, c) = (p, q, r). On the left are the smallest HTs (a, b, c), arranged by the product abc of their three elements.

27. Why the name? I came up with the idea of an HT by writing my GCSE Equations Worksheet back in 2002. I needed a name for them, and at the time I was playing the part of a bandleader in the College production of They Shoot Horses, Don’t They? The name of the bandleader was Max Hikorski, and so Hikorski Triples were born.

28. With thanks to: Mandy McKenna and Far East Theatre Company. Tom Ward, Graham Everest, and Shaun Stevens. Carom is written by Jonny Griffiths, hello@jonny-griffiths.nethello@jonny-griffiths.net