# [r-t] Decisions / Algorithms for generating the extent

Richard Smith richard at ex-parrot.com
Tue Jun 20 15:09:39 UTC 2006

```Philip Earis wrote:

> It's very interesting to see how the 'algorithm' of plain
> changes is introduced as an iterative way of generating
> the extent on n bells.
>
> What other, more dynamic, ways would people use to do the
> same thing?

Well, there's another one in Tintinnalogia, though
Stedman/Duckworth isn't explicit that it works on arbitary
stages.

The starting point of the series is the extent on three
bells, Original Singles.

123   3
213   1
231   3
321   1
312   3
132   1
---
123

All bells hunt throughout and this is enough to get the
whole extent.  If you try this on four bells, though, you
only get a third of the extent:

1234  x
2143  14
2413  x
4231  14
4321  x
3412  14
3142  x
1324  14
----
1234

However, we can take this as one lead of a three lead extent
back three bells:

1234
1324
----
1342
1432
----
1423
1243
----
1234

This is Plain Bob Minimus.  Now let's repeat the operation.
and ends produce the extent on the back four.  This gives an
extent of Plain Bob Doubles using 123 singles with the
second as observation.  It is called Old Doubles in
Tintinnalogia.

This can be continued indefinitely, with each stage
introducing a new type of call.  On six bells this gives an
touch of PB6 whose calls are 1236 and 1234 singles; this is
described in the "Trebles and Doubles on six Bells" section
of Tintinnalogia.  It seems pretty clear from the last
paragraph of that chapter (and elsewhere) that this
principle was understood back in 1671:

| In this Peal is contained the Six-score changes of Old
| Doubles on five bells, and also the Twenty-four changes
| Doubles and Singles on four bells; the half hunt, the
| quarter hunt, and three extream bells, makes the Six-score
| changes of Old Doubles in a perfect course; and the
| quarter hunt, and three extream bells, do make the
| Twenty-four changes Doubles and Singles. Every time the
| Six-score; and every time the whole hunt leads, and the
| half hunt lies next to it, there are two changes mades in
| the Twenty-four.

In general, you can take any n bell extent and use it as the