SYNOPSIS OF RESULTS FROM THE PLANS

5-Part Plan (cycle of 2,3,4,5,6)

Apart from Middleton's, no peal-length blocks emerged. A 5,120 in five similar unlinkable touches was found, based on Middleton's courses, with bobs only. The longest touch with singles which emerged was the palindrome of 4,640: 
                B   M   W   H   2 3 4 5 6
                -------------------------
                    -       S   4 6 3 5 2      The search was
                -           S   6 4 5 2 3      completed on
                    -   -   S   2 3 5 4 6      15 February 1988
                -           S   3 2 4 6 5
                        -   -   4 6 3 2 5
                -------------------------
                   Four times repeated

4-Part Plan No.1 (Abelian Group of two pairs crossing among 1,2,3,4)

On the construction of the table of false types, FTYPES.DAT, 12 of the 150 types were found to be self-false, consequently these twelve had to be deleted from the table of calls so that they could not be used. The twelve were:

25, 30, 55, 60, 85, 90, 91, 96, 101, 106, 111, 116.

The tree was run on 16 February 1988, the longest touch produced being 4,224. This came out in various forms, two of which are given below. Both are palindromes. The touches as given form two separate two-part blocks, which are not joined. Had the touches been of peal length, there was hope of finding links, as the blocks came out in various similar forms. 

              4,224                            4,224

      B   M   W   H   2 3 4 5 6       B   M   W   H   2 3 4 5 6
      -------------------------       --------------------------
              -   S   5 4 2 3 6               S   -   4 5 3 2 6
      -           S   4 5 3 6 2       -           -   4 5 2 6 3
          -       S   3 2 5 6 4           S       -   2 3 5 6 4
      -           S   2 3 6 4 5       -           S   3 2 6 4 5
          -   -   S   4 5 6 3 2           -   -   S   4 5 6 2 3
      -           S   5 4 3 2 6       -           -   4 5 2 3 6
      -------------------------       -------------------------

4-Part Plan No.2 (Two pairs swopping independently)

There were four self-false types: 15, 46, 105, 116 which had to be cut out of the calls table, and care had to be taken not to start the tree at types 46 or 116. There were a large number of disposable calls, e.g. type 3 gave type 74 by either B or S, hence as it was a waste of search time to keep the S option, the latter was nullified but might be needed later for shunting the solution. There were 18 such P/B options and 18 P/S options. The reductions restricted the latitude of the tree search considerably. It was noticeable that short courses were frequent among the solutions. The maximum length attained was 3,712. A number of solutions of this length were produced, for instance the palindrome: 
                       B   M   W   H   2 3 4 5 6
                       -------------------------
                               -   S   5 4 2 3 6
                       -               4 3 5 6 2
                       -           S   3 4 6 2 5
                           -*      -   5 6 4 2 3
                           -   -   -   3 2 4 6 5
                       -------------------------
                       Three times repeated, S for B
                       at * in alternate parts.
This was carried out on 17 February 1988.

4-Part Plan No.3 (Cycle of four bells)

There were 8 self-false types, to be cut out of the table of calls. They were 26, 55, 60, 81, 95, 101, 106, 120. When these had been cut out, 54, 59, 94 and 119 had nowhere to go, and were cut out as well.

As there were no linking calls on the plan, a block to be of use had to be truly 4-part, i.e. must have had an odd number of singles in the part.

The solutions tended to be longer when there were more short courses in them. The longest solution with long courses was 3,584, and a peal of 5,120 consisted entirely of short courses. This latter solution turned up repeatedly with its various starts and reversals, but with no actual variation in calling (it will be found in the list of solutions later). It is interesting to note that 8 of the 10 courses per part form an open-ended palindrome, although the full peal is not one. 

               3,584                         4,224

      M    W    H    2 3 4 5 6        B   M   W   H   2 3 4 5 6
      ------------------------        -------------------------
      -    -    S    5 2 6 3 4                -   -   4 5 2 3 6
           -    S    3 6 5 2 4        -           S   5 4 3 6 2
      -    -    S    2 3 4 6 5            S       S   2 3 4 6 5
      -    -    -    5 6 4 3 2        -           -   2 3 6 5 4
      ------------------------            -   -   -   4 5 6 3 2
          3 times repeated            -           -   4 5 3 2 6
                                      -------------------------
                                            3 times repeated
The search was completed on 19 February 1988.

6-part Plan No.1 (2, 3 and 4 permuting, 5 and 6 fixed)

There were a large number of disposable calls on this plan, which cut down the number of options on the call table. There were 6 places where P, B or S give the same type, and 12 each for P/S and B/S options. There were no self-false types.

The whole search took about half an hour, and the longest touch produced was the following palindrome, which appeared a number of times: 

                         4,224

                 B   M   W   H   2 3 4 5 6
                 -------------------------
                     S       S   6 4 3 5 2
                 -           S   4 6 5 2 3
                         S   -*  5 2 6 4 3
                 -           -   5 2 4 3 6
                 -------------------------
                 Five times repeated, with
                 S for -* half way and end

6-part Plan No.2 (2,3 and 4 rotate, 5 and 6 swop independently)

Call options consisted of six each of P/B (3-part), B/S and P/S varieties. There were no self false types. A tree search took half an hour to complete, without any significantly long touches being produced, on 20 February 1988.

6-part Plan No.3 (2, 3 and 4 permute, with 5, 6 keeping parity)

There were six call options of P/B variety (two of which were unusable because of self falsity). Eight types were self-false: 1, 5, 6, 10, 81, 86, 95, 100 with two more (4, 9) unusable because they had nowhere to go.

The complete tree search took about an hour, on 20 February 1988. The hope that 2-part blocks would emerge, to be linked by options, did not materialise. The only long touches were the following two palindromes, which would be difficult to use as they are 3-part and links of two pairs crossing would be difficult or impossible to find. 

                  4,224                        4,608
        B   M   W   H   2 3 4 5 6        B   H   2 3 4 5 6
        -------------------------        -----------------
            -       S   4 6 3 5 2        -       3 5 2 6 4
        -           S   6 4 5 2 3        -   S   5 3 6 4 2
                -   -   5 2 6 4 3        -       3 4 5 2 6
        -           -   5 2 4 3 6        -   -   3 4 2 6 5
        -------------------------        -   S   4 3 6 5 2
                                         -   -   4 3 5 2 6
                                         -----------------

3-part Plan (2, 3 and 4 rotate cyclically)

This tree search took much longer, and was completed on 22 February 1988. There were twelve call options of P/B, four each in M, W and H positions. The longest touches produced were of length 4,800, some palindromic. An example is given below. During the search, many excursions were made of over peal length using all short courses, but none materialised as round blocks. 
                 4,800
      B    M    W    H    2 3 4 5 6
      -----------------------------
           -              4 3 6 5 2
           -    S    S    5 2 3 6 4
      -              -    5 2 6 4 3   (the last call, the bob at
                     -    6 5 2 4 3    home, is at an apex of the
                     -    2 6 5 4 3    palindrome and is a
      -              S    6 2 4 3 5    disposable call, P or B, but
           S    -         3 5 4 2 6    a plain lead brings rounds)
                -    -    4 2 3 5 6
      -----------------------------
              Repeat twice

2-part Plan (Two pairs of bells swopping, in-course Group)

Here there were no disposable calls, but there were eight self- false types. Overnight and weekend runs on a Zenith micro were started on 3 March 1988 and an Apricot was found which could be run continuously. The Apricot micro was started on 7 April 1988 and ran for months. It was discovered that the types could be marshalled into sets of four (with the swopping bells in the same pairs of positions) which, with the 2-part structure, corresponded to the dihedral group of order 8. A start from each of a set of four gave the same touches. Hence, supposing that a Type 1 start were barren, it would be useless to start with the other three types of the set of four - Types 11, 181, 191. The Apricot was left running on Type 1 start, while the Zenith available overnight and at weekends was started (arbitrarily) on Type 26 after Types 1, 11, 181, 191 had been removed from CALLS.DAT; this strategy ensured that the same ground was not covered twice. It was found that some of the sets of four types started trees which were completely barren; the reason for this must be connected with false course heads. On return from the Bank Holiday weekend on 3 May 1988 a 5,120 was found on the screen, and there were further related blocks of the same length shortly afterwards.

Four distinct sets of peals were produced:

  1. A family of peals of 5,120 in two round blocks, giving irregular 2-part peals (3 May et seq)
  2. A family of peals of 5,184 in exactly 2 parts, related to other palindromic peals (8 June et seq)
  3. A family of peals of 5,056 in two round blocks, giving irregular 2-part peals (17 June et seq)
  4. Short-course peals: Two palindromic 2-parts, and a 4-part which had already been produced by the 4-part search.

2-part Plan (One pair of bells swopping, mixed Group)

The search was conducted on a Zenith microcomputer, and took (during nights and some weekends) from 28 July to 12 September 1988. There were a large number of optional calls producing the same types, 18 of the B/S variety and 18 of the P/S, but no self-false types. Cancelling the second option in 36 different places in the data file of calls produced a significant shortening of the time to search the tree completely, compared with the other 2-part search. It was found that types formed sets of six, any one of a set producing the same touches, so that there were ten distinct starts to explore (corresponding to the 10 combinations of five positions taken two at a time). After each start had been searched exhaustively, the corresponding set of six types was removed from the data file thus avoiding duplication of results and shortening the total search time.

Compared with the other 2-part search it proved disappointing. The longest length produced with some full courses was 4,736, and the only peal produced was the 5,120 in short courses composed previously by N.J. Diserens. This peal appeared twelve times in all, with no variations. Fourteen of its twenty courses (per part) form an open- ended palindrome and the singles, one per part, are integrated into the composition, and cannot be regarded as linking two round blocks.

The Idea of a Palindromic Search

It was truly remarkable how frequently the various tree searches produced palindromic touches and peals. Early on in the project, a version of the program SEARCH was written to construct palindromes, with immediate success. Long blocks were produced in a small fraction of the time taken by the full tree search for any part plan, but the process was not without its teething troubles.

During a palindromic tree search, a 'backwards' branch of the tree is explored at the same time as a 'forwards' one. An extra data file, the inverse call matrix, has to be computed, recording which types will give which. Starting at an arbitrary type, the tree search builds up a touch backwards as well as forwards - if a bob at Middle is chosen for the forwards branch, then a bob at Wrong has to be made in reverse on the end of the backwards branch, i.e. the section type has to be found which will produce the type already on the end of the branch by a bob at Wrong. As a result of this double process, each decision made (P, B or S) results in two new types on the chain instead of one, thus cutting down enormously the time taken for an exhaustive tree search.

There is a possible further economy of time in the testing of falsity. After checking that the two new types in the chain (the ones on the ends) are not the same (which would otherwise give a round block) or are false with one another, they must be checked against all the existing types in the chain. Is it necessary to check both? The answer is yes, and no! Yes, it is necessary to check them both if a search in parts is being carried out, but no, it is not necessary if the search is a general (one-part) search. At first it was thought unnecessary to test both, until a 2-part search produced a wealth of results and rounds was observed half way through the second part of a "peal"!

It was also discovered by comparing results that the removal of options such as one of a B/S pair in a palindromic part search was too restricting - the existence of an option at one end of the chain does not imply an option at the other.

This experimenting suggested the idea of a general (one-part) search for palindromes. The program SEARCH was rewritten in Fortran for use on a large mainframe computer, and a trial run produced Middleton's peal in a twinkling of an eye! The necessary three data files (calls, inverse calls, falsity) were calculated on a micro by the simple expedient of setting the identity Group (order 1) on the existing BASIC programs, and they were subsequently transferred to the mainframe.

The palindromic version of SEARCH was programmed to cope with five types of apex for start and finish - plain, bob or single at Home, bob at Before, and the mid-lead between Middle and Wrong. Five separate tree searches were carried out, one for each start, and five different tests for a round block were programmed. Every distinct result came out twice, in mutually 'half-reverse' pairs.

As an heuristic way of producing peals, the palindromic search has proved successful, but the use of a method for which logical reasons are not forthcoming is slightly disturbing. Why are peals 'likely' to be palindromes? Some peals are not palindromes, and palindromes can easily be produced which are false, so there is no simple explanation. Palindromes appear to be a feature of the structure of treble bob (including surprise) methods.

The answer may possibly be found in a consideration of probability. A palindromic structure may increase the chance of a randomly-produced block being true for the same reason that it is not necessary to test for the truth of both new additions to the palindromic tree.

Palindromic 1-part Plan

A total of nearly three hours of central processor time was used in completing the search of the five trees, made possible by the use of a mainframe Cyber 855 on Sundays when the unit charge was very low. The solutions found have been classified below according to the nature of their apices:
Both Apices Singles Home
There were eight different solutions, all peals of 5,120 in 40 short courses. Two were the 2 part palindromes which had already been produced by the 2-part search.

One Apex at Mid-lead (between Middle and Wrong)
In addition to Middleton's peal (both the full version of 5,600 and the 5,152 with three Homes omitted) there was but one solution, a 5,024 with the other apex at Bob Home. This peal had no variations.

One Apex at Bob Before
In addition to Johnson's variation of Middleton's, a family of four palindromic peals of 5,120 in 40 short courses issued, with bobs Before at both apices.

One Apex at Plain Home
As well as Middleton's peal with three Homes omitted, two distinct families of peals were produced. One family of 32 peals of length 5,120 had 16 long and 12 short courses, all having a bob at Home at the other apex. Of the other family of ten peals of length 5,184, all with 14 long and 16 short courses, four had plain Home at both apices, and six a bob Home at the other apex, but all were variations of the 2-part peals of 5,184 already produced by the 2- part search.

Both Apices at Bob Home
Two distinct peals were produced. A family of 12 palindromes of length 5,184 with 22 long and 2 short courses, all variations of the same calling; and two solutions with the same statistics which were mutually half-reverse and had some non-palindromic variations.

Two-Part Peals with a Coda or Prelude

The success of the two-part search (with two pairs swopping) suggested a further approach to the problem, which turned out to have very limited success.

Occasionally the search produced solutions in the 4900s. Would it be possible to add a true coda of four leads and two changes, ending in rounds just after the Wrong, or a prelude of two leads and 30 changes starting at the treble's snap lead??

The simplest way of adding a coda is as follows: if a touch ends in a bob or single at Home, a coda may be added by changing bob to single or vice versa, and the touch will then come round without a further call. This might be done to a touch of 4,928 or 4,992 (154 or 156 leads) giving a peal of 5,058 or 5,122 which would be virtually in two parts (other more complex codas, extending back into the previous course, were found to be unworkable).

The search program was modified to carry out the following steps:

  1. After loading the call table, the four self-false types 91, 96 231 and 236 were reinstated in the table, having previously been cut out, in a 'disabled' form (500 being subtracted from the value) so that they could be used in the coda where they would appear once only.

  2. The four types 1, 11, 181 and 191 were disabled because it was found previously that these types did not occur in any touches approaching the 4900s.

  3. The program assumed a bob Home at the end of the block, a single substituted giving the coda. A reversal of this would be made for further runs.

  4. The starting type was stated arbitrarily, as a multiple of 5 giving a type at the end of a course. This gave fourteen possible distinct starts (omitting one self-false group of types).

  5. The program calculated the three types forming the coda. It was not necessary to check these against each other, as they would be in the same course, neither was it necessary to check the extra two rows against the main block for truth, as their false type was found to be a false type of the preceding section of the coda (this is not true if there is a call at the Wrong).

  6. The program then removed from the call table all types false with the starting type (except itself) and with the coda types.

  7. The program then progressively deleted from the call table all types which had 'nowhere to go', thus avoiding exploring abortive twigs in the tree.

  8. The program then searched for 2-part touches of 4,928 or 4,992.

  9. A procedure was found which considerably reduced the run time for the tree search (and which might have been used previously). An arbitrary break in the tree at nodes 15/16 was taken. When the tree search went over this break to No.16, the call table was copied and all the types false with the first fifteen were deleted, making it unnecessary to check the truth of further types against the first section. When the search backtracked to No.15, the original call table was reinstated. Although it took about 10 seconds to copy and delete the call table, a considerable saving in time resulted.
Alternatively, a prelude of 94 rows commencing at the treble's snap lead just after the Wrong position in course 32456 was a possibility. A two-part touch of 4928 or 4992 would be constructed, having first decided on a prelude ending at a particular Home position. The fact that the prelude has to enter the block at the same Home as the final Home call means that the three Home calls (unmodified 2-part touch, prelude, finish) have to be all different, selected from plain, bob, single. The three possibilities are:
  1. Unmodified 2-part touch has bob at Home, 24356 to 32456 Prelude ends in plain Home, peal ends at single Home

  2. Unmodified 2-part touch has single at Home, 34256 to 32456 Prelude ends in plain Home, peal ends at bob Home

  3. Unmodified 2-part touch has plain Home, course 32456 Prelude ends in single at Home, peal ends at bob Home
Of these ways, only the first is viable and that gave no results at all, after protracted searches. Only one peal with a coda was found, the 5,122 on page 28 based on palindromic blocks of 4,992. In fact, no 2-part 4,928 was ever discovered, and there is probably only the one distinct 4,992.

Next section The Resulting Peals
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