15: Even and odd
Last time, we produced some touches of Grandsire Doubles using a mixture of plain and bobbed leads. Such touches are called ‘bobs only’ touches.
We had a false touch of 90: pp-pp-pp-
We had two true touches of 60: p-p-p- and its rotation: -p-p-p
I asked you:
- How do we know, without bothering to write it out, that the 90 must be false?
- Do the two touches of 60 contain the same rows?
The second of these questions can also be answered without writing out all the rows.
To answer these questions (and to show how the two questions are related) we’re going to look more closely at the different rows we can produce, remembering that, on 5 bells, there are exactly 120 in total.
If you go back and look at the rows in all the touches in the last post, you’ll see something interesting: none of them contain the following rows:
21345 13245 12435 12354
Can you see what all those rows have in common? That’s right: they’ve all been produced from rounds by swapping just one adjacent pair (they’re called ‘near-misses‘).
Queens (13524) is also missing from all three touches. Queens is produced from rounds by swapping 3 adjacent pairs, maybe like this:
12345 to 13245 ( swap 2&3) to 13254 (swap 4&5) to 13524 (swap 2&5)
In fact, all the rows produced from rounds by swapping an odd number of adjacent pairs are missing in all three touches. Try and work out some more of these missing changes.
- Changes produced from rounds by swapping an even number of adjacent pairs are called ‘in-course‘ rows.
- Changes produced from rounds by swapping an odd number of adjacent pairs are called ‘out-of-course‘ rows.
In the same way that, when you are counting in maths, there is an equal number of odd and even numbers, in bell-ringing there are an equal number of in-course (even) rows and out-of-course (odd) rows. That means that out of the 120 changes available on 5 bells, exactly 60 must be in-course and the other 60 must be out-of-course.
In maths, if you add 2 to an even number, you get another even number. In the same way, in bellringing, if you swap 2 adjacent pairs from an in-course row, you get another in-course row.
Now, Grandsire Doubles using only bobs and plain leads always swaps 2 pairs between rows, so it can only ever produce in-course rows. That means that:
- Any touch of Grandsire Doubles longer than 60 rows and using bobs only must be false. That, of course, includes our bobs-only touch of 90.
- Any true, bobs-only touch of 60 Grandsire Doubles must contain the same rows as any other, because it must include all the available in-course rows.
We’ve answered our two questions!
To produce 120 of Grandsire Doubles, we need to introduce the idea of a ‘single’. You can probably guess what a single does. There’s a big clue in its name!