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

Richard Smith richard at ex-parrot.com
Mon Jun 26 14:00:14 UTC 2006

```I wrote:

> Another way is to use Plain Changes on (n-r) bells as the
> coursing order for a singles-only touch of an n-bell method
> with r fixed bells which get into each position relative to
> each other in the plain course.

rather more subtle than I first thought.  When we apply
Plain Changes on (n-1) to the coursing order of Original on
n, we not only visit every coursing order, but also every
coursing order backwards.  As these two courses are false
against each other, what prevents this from making the touch
false?  And what, for that matter, prevents part of the
course from being omitted?

In Plain Changes on three or more bells, the pair of bells
swapped to reach any given row is different from the pair
swapped to leave that row.  This means that when applied to
the coursing order of Original, we enter and leave the
course in different places -- so, assuming only a single
type of single is used in the Original, we necessarily omit
part of the course.

Quoting from my original example of Plain Changes on 3
starting from the coursing order 243:

>   243   swap 3 & 4
>   234   swap 2 & 3
>   324   ...

This generates the following fragment of Original:

>     1234
>   s 2134 < enter course 234 by swapping 3 & 4 in c.o.
>     2314
>     3241
>     3421
>     4312
>     4132
>   s 1432 < leave course 234 by swapping 2 & 3 in c.o.
>     ----
>     1342

I.e. we've only had six of the eight rows from the 234
course.  The missing section is the bit around the lead end:

1423
1243

This has to be reached and left via a pair of singles
(otherwise it's false against the fragment above), and it
has to be rung backwards so that the coursing order is 432
(otherwise we have the same coursing order twice, which
would imply Plain Changes is false and we know that it
isn't.)  So:

2143
s 1243  enter course 432 by swapping 3 & 4 in c.o.
----
1423
s 4123  leave course 432 by swapping 2 & 3 in c.o.

Clearly both fragments have to be entered and left by
swapping the same pairs of bells in coursing order --
otherwise they would either overlap or omit part of the
course.

If we look at the preceding and superseding coursing orders
for the two fragments, we see that they are also reverses of
each other, and that the (plain) changes that join them are
also reverses of each other.

Fragment #1    Fragment #2

2  4  3        3  4  2
|   \/          \/   |
|   /\          /\   |
2  3  4        4  3  2
\/   |        |   \/
/\   |        |   /\
3  2  4        4  2  3

We can now apply the same logic to the adjacent coursing
orders, and get eventually get two mutually-reverse sections
that join together to form a glide symmetric round block.
As Plain Changes are glide symmetric (in fact, they are
maximally symmetric -- i.e. they have palindromic, glide and
rotational symmetry), they can be used on the coursing order
of Original to produce an extent.

More generally, any glide symmetric extent of single changes
on (n-1) bells can be used to produce an extent of Original
on n bells, and no extent of singles changes on (n-1)
bells without glide symmetry can produce an extent of
Original on n bells.

>From Ander's list, we can see that there are only four
extents of minimus that only use single changes:

#1   3PGR    I|n|O           (Double Court)
#2   3PGR    In=|O           (Double Canterbury)
#9   2--R    O|n|n|u|O
#22   1P--    In=|u|=n=|n|u|I

(See http://www.math.ubc.ca/~holroyd/min29.txt for notaton.)

Double Court and Double Canterbury are variants of Plain
Changes using near and far extremes respectively.  As an
example of these, we can generate a fixed-treble extent of
Original Doubles using 345 singles in place of 5.  As 34
(the bells swapped by the single) are not adjacent in the
standard coursing order, we need to choose an alternative
coursing order by choosing every other bell from the usual
one (exactly as for the extent of Bob Minor in the previous
email).  This gives a starting coursing order of 4325 from
which we can ring Double Court with the 5 as the hunt bell:

4325    2435    3245
----    ----    ----
4235    2345    3425
4253    2354    3452
4523    2534    3542
5423    5234    5342
5243    5324    5432
2543    3524    4532
2453    3254    4352
2435    3245    4325

This gives a three part extent of doubles through which the
treble plain hunts.

12345    15243    12543    14253
21435    51423  s 21543  s 41253
24153    54132    25134    42135
42513    45312    52314    24315
45231    43521    53241    23451
s 54231  s 34521  s 35241  s 32451
52413    35412    32514    34215
25143    53142    23154    43125
21534    51324    21345    41352
s 12534    15234    12435  s 14352
-----    -----    -----    -----
15243    12543    14253    13425

Twice repeated

But if the same is tried with a non-glide symmetric extent,
it fails.  For example, extent #9 has rotational, but not
the required glide symmetry:

4325    2543
3425    5243
3245    5423
3254    5432
3524    5342
3542    5324
3452    5234
4352    2534
4532    2354
4523    2345
4253    2435
2453    4235
----    ----
2543    4325

Attempting to fit this to Original only lasts 39 rows before
becoming false.

12345    15234*   14532     15432
s 21345    51324    41352   s 51432
23154    53142    43125     54123
32514    35412    34215     45213
35241    34521    32451     42531
s 53241  s 43521  s 23451   s 24531     False rows marked
52314    45312    24315     25413     with a *.
25134    54132    42135     52143
21543    51423    41253     51234
s 12543  s 15423    14523   s 15234*
-----    -----    -----
15234    14532    15432

The idea of apply Plain Changes to the coursing order is
given in Knuth's "The Art of Computer Programming"

<http://www.ex-parrot.com/~richard/papers/fasc2b.ps>.

as Exercise 69 on page 32, and attributed (page 48) to E.S.
Rapaport [Scripta Math. 24 (1959), pp51-8].  I've not
managed to lay my hands on a copy of this paper, though it
seems to be cited quite frequently.  (Interestingly,
Rapaport is quoted as being "particularly motivated by bell