Change RingingMethod of ringing bells in towers

Generally 8 bells or less are used.

http://www.youtube.com/watch?v=kzLZBdas6ck

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

The Goal

Play all possible permutations of the bells without repetition.

How many permutations are there for n bells?

The Goal

Play all possible permutations of the bells without repetition.

How many permutations are there for n bells?

n!

Think about it:

For 8 bells, that’s 40320 permutations.

That’s a lot.

• 7!=5040 (more than 5000 changes = “peal”) •6!=720

•5!=120•4!=24

How do Ringers Keep Track

• Methods

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

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)

(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 2 3 4

4 2 1 3

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)

Four Bells

• The only possible transpositions on four bells are – (1 2)– (2 3)– (3 4)– (1 2)(3 4)

Even and Odd Permutations

• Of the symmetric group Sn (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.

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.