User:Iainscott/changeringing

First, bear in mind that a "method" is not a single permutation, but rather a particular sequence of permutations, which, (just to add intrest!), can also be altered at specific points by additonal predfeined permutations. So to take the example of "Plain Bob Doubles", which is a method on 5 bells (hence with 5! = 120 possible changes). We start with the permutation (2143) which I am going to call "5" (ringers use "place notation", where each permutation is named after the elements which arnt affected by the permutation: as we only have nieghbour-swaps between each row, this is unique.), followed by the permutation (3254) (similarly called "1"), followed by 5 again, followed by 1, etc. untill we have had the 5th 5, at which point we have the permutation (43) (again called "125" - usually writeing place notation saves space!)

Starting from rounds we now have

12345     5 21435      1 24153      5 42513      1 45231      5 54321      1 53412      5 35142      1 31524      5 13254      125 13524

this is called a "lead", and is the basic building block of most methods. If we repeat the lead 4 times, we should get back to rounds (12345), after ringing 40 different changes (proof that they are diferent is more complex in the general case, and there are different techniques for simplifying the process in different classes of method, but here the brute force technique of actually writeing them out and checking them is fairly a simple, if time consumeing, process). Obviously 40 different changes is not 120 different changes. to get 120 that we add in some of the aforementioned "alterations". One example of how this can be done is to have a "bob" at the 4th, 8th and 12th leads. A bob is the replacement of the (43) permutation at the end of the lead with a (32) permutation (called "145"). The first two bobs move you to a different portion of the group of possible changes, and hence the next 40 changes rung are different to the previous ones (the 3rd bob brings you back to rounds).