Odd and Even - The Nature of the Rows

Let's start with some call changes. For simplicity in counting them, we will call a single swap each time.

Imagine we want to get from the row 213456 back to Rounds. Of course, only one call is needed; "2 to 1" (or "1 lead", depending on your tower). But say we've got a few minutes to fill in before service and want to have few more calls. Perhaps include Queens (135246) before getting back to Rounds? Or Tittums? Can we do it with any number of calls?

Go on, have a go. It's great, as there's no right or wrong answer, as long as you get back to rounds.

I should leave it there, really, and see what answers you send in...

Anyway, I'll have a go myself:

4 to 5 213546  
3 to 5 215346  
1 to 5 251346  
2 to 5 521346  
1 to 3 523146  
2 to 3 532146  
2 to 1 531246  
5 to 3 351246  
5 to 1 315246  
3 to 1 135246 Queens
5 to 2 132546  
3 to 2 123546  
5 to 4 123456 Rounds

That took me 13 calls. And hopefully all of you got an odd number too. It is impossible to do with an even number of calls (even swapping a non-adjacent pair at each call).

Like the recent page on music, I don't want to include any technicalities about maths here, as there are many better qualified in the District to discuss this than me. But the key point is that some rows can only be reached with an odd number of calls and some rows can only be reached with an even number of calls. In fact, exactly half the rows can only be reached with an odd number of calls and exactly half can only be reached with an even number of calls.

To save me writing "can only be reached with an odd number of calls" every time, it's going to be much easier just to refer to those rows as being "odd" and the rows that can be reached with an even number of calls as being "even". This property is sometimes called "the nature of the rows" or "parity".

Rounds itself is an even row, as it can be reached in 0 calls, or 2 calls, or 4 calls, and so on. The row 213456 that we started with above is an odd row, as you can only get to or from rounds in an odd number of calls. And in my list of call changes above, they must be alternately even and odd rows.

This might seem rather academic, but it is actually a surprisingly powerful concept. It can tell you that some touches are false and some are true and that some touches are impossible to find.

Here's another example that I like, that involves parity. This pattern has 62 squares, so if I give you 31 dominoes, each of which cover exactly two squares, can you entirely cover the pattern?

If the pattern were cut from a chess board, it might look like this:

Of the 62 squares, we find that 32 are black and 30 are white. But think what squares each domino will cover. Wherever you put it, it will cover a black square and a white square, as blacks and whites alternate.

So 31 dominoes, however they are placed, will cover 31 black squares and 31 white squares. We can immediately tell that the puzzle is impossible, without having to try every possible layout.

We shall now move on from call changes, to apply the notion of parity to change ringing. We have seen that every row will be either odd or even. We start and finish with rounds, which is an even row. Each time we swap over a pair of bells, the parity of the row changes, from odd to even or back again.

Let's start with Plain Hunt Doubles.

Rounds is an even row. Then, to get to 21435, two pairs have swapped over; 1 and 2, as well as 3 and 4. If we had done it as call changes, it would have taken two calls. Each swap changes the nature of the row. Swapping 1 and 2 first, say, would have given an odd row, then swapping 3 and 4 would get us to an even row. So 21435 is an even row.

Next we go from 21435 to 24153. Again, two pairs have been swapped over, so 24153 will be another even row.

In fact, at every row, two pairs swap over. That's why it's called "Doubles".

So every row in Plain Hunt Doubles will be an even row.

This is also true in a plain course of Stedman Doubles. It only contains place notations 1, 3 and 5, each of which involves two pairs swapping. The plain course contains 60 rows, out of the extent of 120. Remember we said that half of all the rows (on any number of bells) are even and half are odd. This means that the plain course of Stedman Doubles contains all the possible even rows. We can only reach the odd rows by using a Single.

Again the name "Single" is significant; at a Single in Stedman Doubles, only one pair of bells swap, whether it's place notation 145 (as we use now) or 123, as was used in Stedman's time. In the first case, the bells in 2nds and 3rds place swap over; in the second example, the bells in 4ths and 5ths swap. Either way, we will move from an even row to an odd row. And every row after that will be an odd row, so you can never get back to rounds until there's another Single called. All touches of Stedman Doubles must contain an even number of Singles. And if you're asked to call a touch of Stedman Doubles, as long as you call an even number of Singles, it will come round (although it may not be true).

Grandsire Doubles also just uses double changes in its plain course - and at the Bobs too. So all the rows will be even, however many Bobs you call. As there are only 60 even rows available, the longest true touch using Bobs only is just 60.

As with Stedman, to get a 120 of Grandsire you need to use Singles. Again, it doesn't matter much which particular change you use, as long as it just swaps over one pair of bells. Usually the Single includes a 123 place notation, where the bells in 4ths and 5ths places are the only pair to swap. And, in the same was as with Stedman, you need another Single to ever get back to rounds again.

The standard 120s of Grandsire are PBPBPS PBPBPS or PSBS PSBS PSBS. They must have an even number of Singles, to get back to rounds. The first touch has 2 Singles, the second touch has 6 Singles. In the first touch, the whole of the first half contains even rows, then the second half has all the odd rows, before the final Single in needed to get back to rounds, which is an even row. In the second touch, there are two leads of even rows, then two leads of odd rows, and so on.

You can call a touch of Grandsire with just one Bob (PBPP), but you can never have a touch with just one Single.

There's an interesting 120 of spliced Stedman and Grandsire, that relies on the even and odd nature of the rows. Start by ringing a plain course of Stedman, but call a Grandsire Single at the end - so the bells at the back are unaffected, while 2nds and long 3rds are made at the front. That puts you to an odd row. Then change to Grandsire and ring PBPBPS; this generates all the rest of the odd rows, swapping back to even to come into rounds. All the even rows are rung as Stedman, while all the odd ones are Grandsire and seeing as each half is true to itself, the whole thing must be true.

What about Plain Bob Doubles? We don't need Singles in Plain Bob Doubles. Why not?

It's because there are already single changes in the plain course. The lead end change is 125, where only the bells in 3rds and 4ths places swap.

Even rows are sometimes shown by a '+', while odd rows are shown by a '-', so the plain course of Plain Bob Doubles is like this:

We get alternate odd and even leads, first 10 even rows, then 10 odd rows, then 10 even rows, and so on. And Bobs don't have any effect on this pattern, as the place notation for a Bob is 145, instead of 125. Both are single changes, swapping over a single pair of bells, so each has the same effect on the nature of the rows.

This is why you can't get touches of Plain Bob Doubles (using these Bobs) of length 30, 50, 70 etc. And it's why you don't need Singles, as any 120 using Bobs will already contains 60 even rows and 60 odd rows, due to the alternating even and odd leads.

Moving on to Minor, things get more interesting, as we can use triple changes. Let's start with Plain Hunt Minor.

The first change swaps over three pairs of bells, so the row 214365 must be odd.

Then we're back in more familiar territory, with two pairs swapping at the next change, while the bells at front and back stay in the same place. Two swaps means that the parity of the next row will be the same again. So 241635 will be odd.

Then the triple swap again means that the parity will be changed for the next row; 426153.

So we're getting both odd and even rows, in pairs.

Moving on to Plain Bob Minor, the only change comes at the lead end, where we get a 12 place notation, instead of the 16 that we had in Plain Hunt. But this involves two swaps, so has the same effect on parity as the 16, so the pattern of the Plain Hunt is continued. And a Bob gives a 14 place notation, which is also a double change, so the pattern is continued even then. So we get:

You can see this pattern is the same in every lead. And will carry on the same, even with Bobs.

But, although we are getting some even rows and some odd rows, look at the lead ends. Every row where the treble leads is an even row. This pattern can only be broken by the use of Singles, which, once again, are single changes; with place notation 1234, only the bells in 5ths and 6ths places swap over. So we will get to a lead head with odd parity, if a Single is called.

And thereafter, the pattern of odds and evens is the other way round, so whenever the treble leads from then onwards, the row will be odd. Until another Single is called, to get back to an even lead head. So we get the same situation that we had with Stedman and Grandsire Doubles, where you need Singles to get the extent (720 rows). And any touch must have an even number of Singles (remember zero is an even number). The rows (particularly the lead ends and lead heads) are said to be "in course" when they are of even parity, and "out of course" when they have odd parity.

If you have any true touch of Plain Bob Minor with Bobs only, you can double it in length, by calling a Single at some point in the touch and repeating the whole thing. The most common thing is to call the Single just as it is about to come round, then repeat. And you know it will still be true, as the section between the Singles is true, from the truth of the original touch, and the in-course leads can't be false against the out-of-course leads, as they are of opposite parity (when the treble is in each particular position, not just leading).

Finally, we'll look at a Cambridge Surprise Minor and see why you don't need Bobs to get a 720 there.

You can see that we get the same alternating pattern of two odds, two evens that we had in Plain Hunt and Plain Bob. But we have to look a bit more carefully. Yes, the lead end and the lead head rows are both always even, but we also have to look at the other rows where the treble is leading - and both of these are odd. So we're keeping an even balance. The same is true when the treble is in any other place - there are two odd rows and two even rows in every lead. For instance, look at the rows where the treble is in 4ths place - 624153 and 456123 and both odd, while 264153 and 546123 are even. And that pattern isn't changed by the use of Bobs.

Admittedly, this doesn't ensure a given touch will be true, but it does provide access to all the rows that we need.

Compare Cambridge with Canisp Surprise Minor. Not only is it not a right place method - you can see that it doesn't have a X place notation every other change - but it has a much more varied mix of double and triple changes. Here is a lead with the parity of each row marked:

Look at the nature of the rows where the treble is in 4ths place now, compared with Cambridge. We get 325164, 235146, 324156 and 234165, all of which are even rows. So, even with Bobs, we can't get any odd rows when the treble is in 4ths place. We know the "standard" 720 (Bobs at Wrong, Home, Wrong, three times) isn't going to work. In fact, I don't think a 720 is possible, even with Singles, although you can ring a 1440, containing each row twice.

If you want to do some more, have a look at Plain Bob Triples and see if you can see why you can get an extent with Bobs only...