[r-t] Ringable Alternating Groups On Five Bells

Joe Norton super.joey.norton at gmail.com
Tue Aug 25 15:20:57 BST 2020

Some time ago I started looking into ways of ringing the alternating 
group. Obviously, this is essentially the orderings of the in-course 
changes on a given number of bells which are legal to ring. Stedman 
Doubles is a classic example of a ringable ordering of A5.

My intention is to look into orderings of subsets of the alternating 
group at higher stages and the combinatorics thereof. But as an aside I 
have some interesting results on A5 and the number of ways of ringing 
it. These are particularly interesting to me because any pair of such 
blocks can be used to create an extent by swapping a pair halfway and end.

I’m sure I’m not the first to generate these blocks but, just in case 
there is part of it that is useful, I thought I’d share my findings thus 



There are 8608 ringable orderings of A5. After removing rotations, this 
number is reduced to 152.

Of these 152 blocks:

4 blocks have order 5 (length 12). These are Stedman, Reverse Stedman, 
Carter and Reverse Carter. These blocks are all palindromic.

8 blocks have order 3 (length 20). None of these blocks is palindromic. 
They fall into two subsets of four. One subset is the Grandsire half 
hunt, along with its reflections and reversals. The other is possibly 
spliced Grandsire and Double Grandsire, with its reflections and reversals.

Interestingly, all blocks of order 3 have a hunt bell. (Is that a 
manifestation of some result in group theory?)

The remaining 140 blocks have order 1 (length 60). 20 of these are 
palindromic and 120 are non-palindromic.

The 20 palindromic blocks fall into 10 subsets of 2 reflections.

The 120 non-palindromic blocks fall into 30 subsets of 4 reflections and 

Here are all 152 blocks. I have used “REVERSE” to mean reflection, as in 
the reverse method. I have used “BACKWARDS” to mean reversal, as in 
reading the changes from last to first. Other than methods which have 
been named previously, I haven't chosen any particular rotation of any 

_*Blocks of order 5 (length 12)*_

(Block 41a) Stedman
(Block 41b) Reverse Stedman

(Block 42a) Carter
(Block 42a) Reverse Carter
(Carter and Stedman are PALINDROMIC)

_*Blocks of order 3 (length 20)*_

(Block 43a) Grandsire half hunt
(Block 43b) Backwards Grandsire half hunt
(Block 43c) Reverse Grandsire half hunt
(Block 43d) Backwards Reverse Grandsire half hunt
43a is BACKWARDS 43b
43b is REVERSE 43d
43d is BACKWARDS 43c
43c is REVERSE 43a

(Block 44a) Possibly bobbed lead of Grandsire plus bobbed lead of Double 
(Block 44b)
(Block 44c)
(Block 44d)
44a is BACKWARDS 44b
44b is REVERSE 44c
44c is BACKWARDS 44d
44d is REVERSE 44a

_*Blocks of order 1 (length 60)*_

(Block 1a)
(Block 1b)
1a is REVERSE 1b
1a and 1b are PALINDROMIC

(Block 2a)
(Block 2b)
2a is REVERSE 2b
2a and 2b are PALINDROMIC

(Block 3a)
(Block 3b)
3a is REVERSE 3b
3a and 3b are PALINDROMIC

(Block 4a)
(Block 4b)
4a is REVERSE 4b
4a and 4b are PALINDROMIC

(Block 5a)
(Block 5b)
5a is REVERSE 5b
5a and 5b are PALINDROMIC

(Block 6a)
(Block 6b)
6a is REVERSE 6b
6a and 6b are PALINDROMIC

(Block 7a)
(Block 7b)
7a is REVERSE 7b
7a and 7b are PALINDROMIC

(Block 8a)
(Block 8b)
8a is REVERSE 8b
8a and 8b are PALINDROMIC

(Block 9a)
(Block 9b)
9a is REVERSE 9b
9a and 9b are PALINDROMIC

(Block 10a)
(Block 10b)
10a is REVERSE 10b
10a and 10b are PALINDROMIC

(Block 11a)
(Block 11b)
(Block 11c)
(Block 11d)
11a is REVERSE 11b
11b is BACKWARDS 11c
11c is REVERSE 11d
11d is BACKWARDS 11a

(Block 12a)
(Block 12b)
(Block 12c)
(Block 12d)
12a is REVERSE 12b
12b is BACKWARDS 12c
12c is REVERSE 12d
12d is BACKWARDS 12a

(Block 13a)
(Block 13b)
(Block 13c)
(Block 13d)
13a is REVERSE 13b
13b is BACKWARDS 13c
13c is REVERSE 13d
13d is BACKWARDS 13a

(Block 14a)
(Block 14b)
(Block 14c)
(Block 14d)
14a is REVERSE 14b
14b is BACKWARDS 14c
14c is REVERSE 14d
14d is BACKWARDS 14a

(Block 15a)
(Block 15b)
(Block 15c)
(Block 15d)
15a is REVERSE 15b
15b is BACKWARDS 15c
15c is REVERSE 15d
15d is BACKWARDS 15a

(Block 16a)
(Block 16b)
(Block 16c)
(Block 16d)
16a is REVERSE 16b
16b is BACKWARDS 16c
16c is REVERSE 16d
16d is BACKWARDS 16a

(Block 17a)
(Block 17b)
(Block 17c)
(Block 17d)
17a is REVERSE 17b
17b is BACKWARDS 17c
17c is REVERSE 17d
17d is BACKWARDS 17a

(Block 18a)
(Block 18b)
(Block 18c)
(Block 18d)
18a is REVERSE 18b
18b is BACKWARDS 18c
18c is REVERSE 18d
18d is BACKWARDS 18a

(Block 19a)
(Block 19b)
(Block 19c)
(Block 19d)
19a is REVERSE 19b
19b is BACKWARDS 19c
19c is REVERSE 19d
19d is BACKWARDS 19a

(Block 20a)
(Block 20b)
(Block 20c)
(Block 20d)
20a is REVERSE 20b
20b is BACKWARDS 20c
20c is REVERSE 20d
20d is BACKWARDS 20a

(Block 21a)
(Block 21b)
(Block 21c)
(Block 21d)
21a is REVERSE 21b
21b is BACKWARDS 21c
21c is REVERSE 21d
21d is BACKWARDS 21a

(Block 22a)
(Block 22b)
(Block 22c)
(Block 22d)
22a is REVERSE 22b
22b is BACKWARDS 22c
22c is REVERSE 22d
22d is BACKWARDS 22a

(Block 23a)
(Block 23b)
(Block 23c)
(Block 23d)
23a is REVERSE 23b
23b is BACKWARDS 23c
23c is REVERSE 23d
23d is BACKWARDS 23a

(Block 24a)
(Block 24b)
(Block 24c)
(Block 24d)
24a is REVERSE 24b
24b is BACKWARDS 24c
24c is REVERSE 24d
24d is BACKWARDS 24a

(Block 25a)
(Block 25b)
(Block 25c)
(Block 25d)
25a is REVERSE 25b
25b is BACKWARDS 25c
25c is REVERSE 25d
25d is BACKWARDS 25a

(Block 26a)
(Block 26b)
(Block 26c)
(Block 26d)
26a is REVERSE 26b
26b is BACKWARDS 26c
26c is REVERSE 26d
26d is BACKWARDS 26a

(Block 27a)
(Block 27b)
(Block 27c)
(Block 27d)
27a is REVERSE 27b
27b is BACKWARDS 27c
27c is REVERSE 27d
27d is BACKWARDS 27a

(Block 28a)
(Block 28b)
(Block 28c)
(Block 28d)
28a is REVERSE 28b
28b is BACKWARDS 28c
28c is REVERSE 28d
28d is BACKWARDS 28a

(Block 29a)
(Block 29b)
(Block 29c)
(Block 29d)
29a is REVERSE 29b
29b is BACKWARDS 29c
29c is REVERSE 29d
29d is BACKWARDS 29a

(Block 30a)
(Block 30b)
(Block 30c)
(Block 30d)
30a is REVERSE 30b
30b is BACKWARDS 30c
30c is REVERSE 30d
30d is BACKWARDS 30a

(Block 31a)
(Block 31b)
(Block 31c)
(Block 31d)
31a is REVERSE 31b
31b is BACKWARDS 31c
31c is REVERSE 31d
31d is BACKWARDS 31a

(Block 32a)
(Block 32b)
(Block 32c)
(Block 32d)
32a is REVERSE 32b
32b is BACKWARDS 32c
32c is REVERSE 32d
32d is BACKWARDS 32a

(Block 33a)
(Block 33b)
(Block 33c)
(Block 33d)
33a is REVERSE 33b
33b is BACKWARDS 33c
33c is REVERSE 33d
33d is BACKWARDS 33a

(Block 34a)
(Block 34b)
(Block 34c)
(Block 34d)
34a is REVERSE 34b
34b is BACKWARDS 34c
34c is REVERSE 34d
34d is BACKWARDS 34a

(Block 35a)
(Block 35b)
(Block 35c)
(Block 35d)
35a is REVERSE 35b
35b is BACKWARDS 35c
35c is REVERSE 35d
35d is BACKWARDS 35a

(Block 36a)
(Block 36b)
(Block 36c)
(Block 36d)
36a is REVERSE 36b
36b is BACKWARDS 36c
36c is REVERSE 36d
36d is BACKWARDS 36a

(Block 37a)
(Block 37b)
(Block 37c)
(Block 37d)
37a is REVERSE 37b
37b is BACKWARDS 37c
37c is REVERSE 37d
37d is BACKWARDS 37a

(Block 38a)
(Block 38b)
(Block 38c)
(Block 38d)
38a is REVERSE 38b
38b is BACKWARDS 38c
38c is REVERSE 38d
38d is BACKWARDS 38a

(Block 39a)
(Block 39b)
(Block 39c)
(Block 39d)
39a is REVERSE 39b
39b is BACKWARDS 39c
39c is REVERSE 39d
39d is BACKWARDS 39a

(Block 40a)
(Block 40b)
(Block 40c)
(Block 40d)
40a is REVERSE 40b
40b is BACKWARDS 40c
40c is REVERSE 40d
40d is BACKWARDS 40a

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