1. Change Ringing Method of ringing bells in towers Generally 8 bells or less are used. http://www.youtube.com/watch?v=kzLZBdas6ck

2. The Problem Because of the bell mechanism, the order of the bells can only change slightly within one round Each bell can only move one place in the order for each “change” 1 2 3 4 5 6 7 8 2 1 4 3 6 5 8 7

3. The Goal Play all possible permutations of the bells without repetition. How many permutations are there for n bells?

4. The Goal Play all possible permutations of the bells without repetition. How many permutations are there for n bells? n!

5. Think about it: For 8 bells, that’s 40320 permutations. That’s a lot.

6. 7!=5040 (more than 5000 changes = “peal”) 6!=720 5!=120 4!=24

7. How do Ringers Keep Track Methods

8. My Project: Figuring out how to construct methods / bell-change-algorithms Generalizing rules for methods of 4 bells, and finding out what are possible and impossible patterns to play

9. Permutation Cycles In order to describe a permutation, mathematicians use permutation cycles. Basically, permutation cycles show which elements of a set map to which positions for a specific permutation. Ex: (4 1 3) (2)

10. (4 1 3) (2) What it means: Take the set {1 2 3 4}, and put the 4 where the 1 is, 1 where 3 is, 3 where 4 is, and 2 where 2 is. 1 234 4 213

11. Transpositions Permutation cycles of two elements--any permutation cycle can be written as a product of transpositions What this has to do with bells: The allowed changes on bells are all transpositions, and methods are compositions of transpositions (Composition= when you compound functions)

12. Four Bells The only possible transpositions on four bells are –(1 2) –(2 3) –(3 4) –(1 2)(3 4)

14. Even and Odd Permutations Of the symmetric group S n (all possible permutations of n elements), n!/2 permutations are even, n!/2 of the permutations are odd Odd permutations are those which are the product of an odd number of transpositions. Even=even number.

15. Properties of odd/even permutations: Odd * Odd = Even Even * Even = Even Even * Odd = Odd This strongly affects the construction of a method, because of the 4 transpositions you can perform, 3 of them are odd.