Because nearly all the new peals of Cambridge produced by ordinary tree search turned out to be palindromes, or based on palindromic round blocks (the one exception being the 4 part on p.30) there follows an attempt to gain more understanding of them.

Palindromes in Parts

A simple palindrome (not in parts) has two apices, about which the touch or peal has reflective symmetry. If the two tenors are unparted, apices can occur at the Home (plain, bobbed or singled), at a bob Before, or at the centre of the mid-course lead when the treble is lying behind; the two tenors will cross over at each apex, and it is fairly evident that the same pairs of bells must cross over at each apex, and the same working bell make the places.

If a palindrome is in parts, then there will be two apices in each part, and the pairs of working bells crossing, together with the place-making bell, will vary from apex to apex. Apart from the two tenors, in major methods two pairs of working bells will swop at an apex, and one bell make a place, except in the case of a single at Home, when only one pair of working bells will swop.

Here is an attempt to categorise the ways in which the working bells of a palindrome behave:

(a) Palindromes Analogous to the Plain Hunt

The plain courses of all methods with plain bob leadheads form palindromes in parts. Consider, for instance, major methods (plain or treble bob) and the example of Double Norwich. The "plain bob coursing order" is a cycle of seven working bells as on the right. After the first half- lead of the plain course, 4 makes the place and the pairs 26, 38, 57 swop while the treble is lying behind. In the cycle shown, 4 is ringed and the swopping pairs are joined by lines. These three lines are parallel, and this is the condition that the method will have a plain bob leadhead for the second lead.
    3         4

   5           6

      7     8
At the end of the first lead of Double Norwich, 3 will make the place (behind) and the swopping pairs will be 47, 52, 68 which again give parallel lines on the coursing cycle. This structure of swopping pairs is analogous to the following plain hunt on the working bells:
                4  5  7  6  2  3  8
                                     first half-lead
                4  7  5  2  6  8  3
 first leadend
                7  4  2  5  8  6  3
                                     second half-lead
                7  2  4  8  5  3  6
and so on. Note that this plain hunt is notional, and does not exist as actual rows rung. When the method concerned is not a double one, the situation is more confusing at the half-lead, but the above analysis holds for any method with plain bob leadheads.

In the same way, some palindromic touches and peals have working bells performing a notional plain hunt. Middleton's peal of Cambridge Surprise Major is an example:

                   M     W     H     2 3 4 5 6
                   -                 4 3 6 5 2
 (6)(23)(45)(78)   -  *  -           5 6 2 3 4
                         -     -     2 3 5 6 4
                               - *   5 2 3 6 4  (3)(25)(46)(78)
                               -     3 5 2 6 4
                        4 times repeated
The two apices in each part are marked * and the place bell and swops are quoted at the side. The first apex is at the centre of the lead between Middle and Wrong, 6 making the place and 23 swopping, etc. It will be seen that 78 repeat the same work, whereas the five working bells 23456 perform a notional plain hunt. 7-part palindromic peals of major occur when the tenors are working bells.

This type of palindromic touch is not restricted to an odd number of parts. A 4-part palindrome was noted on p.33, and 6-part palindromes are possible. In these, there must be a single at one apex, and one of the working bells making its places must be the one making a place at the other apex, thus being a fixed bell; of the other working bells, an even number perform a notional plain hunt, involving the other two places at the apex single. There may be a pair or pairs of fixed bells crossing at the apices.

(b) Two-part Palindromes with Four Working Bells

If four friends are to play a game of doubles in tennis, there are three different ways of pairing them off: AB/CD, AC/BD, AD/BC. If two pairs of bells among four are to swop, the result is a Group of order 4, {2345, 3254, 4523, 5432}. The three distinct transpositions (or the transdigits) between rows are all of the "two pairs swopping" type, and the result of applying any two of the three in succession is the remaining one.

Now consider the following palindrome:

                  2,560 Superlative Surprise Major

                    B   M   W   H    2 3 4 5 6
                        S            6 3 4 5 2
                        S       S    2 4 3 5 6
     (24)(35)(6)  * -           S    4 2 5 6 3
                            S        6 2 5 4 3
                            S   S    4 5 2 6 3
                    -           - *  4 5 6 3 2  (23)(45)(6)
                    -           S    5 4 3 2 6
The first apex is at a bob Before, with 6 making the bob and pairs 24, 35 swopping. At the second apex the swopping pairs are different, but the place bell the same. The result of applying the two transdigits of the apices in succession is the transdigit (25)(34)(6) and the palindrome is in two parts. Such a palindrome may be called a "pairs of pairs" palindrome.

This touch was produced by a 4-part tree search, the particular Group used being the cyclic Group. The touch is one-half of a peal of 5,120 rows, the other half being produced by starting from 35246 instead of 23456. The two half-peals may be joined irregularly by singles, but there is no way of obtaining a regular 4-part peal.

A happier example of the same class of palindrome is:

                3,456 Superlative Surprise Major

                    B   M   W   H    2 3 4 5 6
                    -                3 5 2 6 4
                        -   S        6 5 4 2 3
                            S        2 5 4 6 3
                        S            3 5 4 6 2
     (35)(46)(2)  * -             #  5 6 3 2 4
                            S        2 6 3 5 4
                        S            4 6 3 5 2
                        S   -        5 2 3 6 4
                    -             *  2 6 5 4 3  (34)(56)(2)
Note how the overall transdigit (2)(36)(45) of the first part is the result of the separate transdigits of the apices. This block was produced by a different 4-part search, using the Group of two independently swopping pairs, the partends being 23456, 23546, 26453, 26543. To produce a peal of 6,912 it is necessary to reverse one pair of (45) or (36) somewhere in the block, and there are ten different places where a single substituted for plain will do this. None of these are at a point of symmetry (only the apices are that) but the Home position labelled # has the appearance of being so. Two singles Home at # are needed, in alternate parts of a 4-part peal.

(c) Other Palindromes with Singles Home at an Apex

The two short-course peals of 5,120 on page 31 are examples of this category. Referring to the one on the left, the first apex of the part has the three bells 2, 5, 6 making places and only (34) swopping, apart from the tenors. At the second apex 2, 3, 4 make places and (56) swop. The overall transdigit for the part is (2)(34)(56) giving a two-part peal. This arrangement is radically different from the first two categories.

The example below was produced by a 4-part palindromic search for peals of Superlative Major, and is the basis of a peal by A.J. Cox. The generating Group has two pairs swopping independently:

             3,584 Superlative Surprise Major

                 M    W    H    2 3 4 5 6
                      -         5 2 4 3 6
                 - a            4 2 6 3 5
                 -              6 2 5 3 4
                 - b  S    S *  3 4 2 5 6  (2)(3)(4)(56)
                 S    - a       5 6 2 4 3
                      -         4 5 2 6 3
                      - b       6 4 2 5 3
                 -         - *  3 2 4 5 6  (23)(4)(56)
The first apex is at a single Home, and this single cannot be altered. It crosses one pair of working bells, (56). The other apex also crosses (56) but in addition another pair (23) swop, and this call is an optional B/S on the part plan, but singling it makes the basic block issue into rounds in one part. The overall part transdigit is just one pair of bells swopping, (23).

The block is curious in that it has two distinct ways of being doubled into a peal of 7,198. These appeared as different peals in the tree search. If one of the bobs marked a be singled in alternate parts, (46) are swopped and the partends have (23) and (46) swopping independently. Alternatively, singles for bobs marked b (the images of aa in the apices) will swop (45), giving a different set of partends generated by (23) and (45) swopping independently.

This duality of the peals derives from the symmetry of the palindrome. Tony Cox's peal of 7,616 utilises singles at b; it has in addition singled-in courses in alternate parts, which the computer search could not discover because they would be false with each other if introduced in all parts.

Admittedly, this third category is a rag-bag of palindromes which do not conform to either of the clearly-defined first two.

An Analysis of Two-Part Palindromes

A systematic two-part search for palindromic blocks of Cambridge Surprise Major led to some unexpected discoveries.

The kind of transdigit of two separate pairs of working bells swopping, and one working bell staying put, interacts with the similar transpositions at the apices of surprise major palindromes.

In order to understand the process, it is necessary to enlarge on the logical course of events carried out by computer. What follows has been carried out exhaustively with Cambridge, but similar operations may be done for any surprise major method; the fact that the falsity of the method may be different means that the results will vary, but the general conclusions reached for Cambridge will apply, and will assist in the composition of peals of other methods.

The tenors are kept unparted. The available calls cause the plain course of a method to fall into five distinct sections; in Cambridge the first two leads form the first section, the 3rd, 4th and 5th leads the next three, and the last two leads of the plain course form the fifth section. The first section can either give the second by a plain, or the fifth section of another course by a bob Before, and so on. The fourth section can only give the fifth by a plain, but the sections have to be considered separately because a bob Before will reach the fifth without the preceding fourth.

The logical start is to declare the part plan, by stating the Group to be used. The one used is {23456, 32546}. A checklist of all 120 permutations of 23456 is created, in rising order when they are considered as 5-digit numbers (23456, 23465, 23546, 23564, 23645 etc). The initial pair 23456 and 32546 are checked off, and the lowest of the 118 remaining taken; this is 23465. The same transdigit (32)(54)(6) is applied to it, giving 32564. This creates a coset {23465, 32564} which is checked off as No.2. Then the lowest unchecked permutation is taken, and so on. The result of this process is to marshal the 120 course heads as follows:

Course     Coset        Section     Course     Coset        Section
  No                     Types        No                     Types

   1    23456, 32546     1 to 5       31    43256, 52346   151 to 155
   2    23465, 32564     6 to 10      32    43265, 52364   156 to 160
   3    23546, 32456    11 to 15      33    43526, 52436   161 to 165
   4    23564, 32465    16 to 20      34    43562, 52463   166 to 170
   5    23645, 32654    21 to 25      35    43625, 52634   171 to 175
   6    23654, 32645    26 to 30      36    43652, 52643   176 to 180
   7    24356, 35246    31 to 35      37    45236, 54326   181 to 185
   8    24365, 35264    36 to 40      38    45263, 54362   186 to 190
   9    24536, 35426    41 to 45      39    45326, 54236   191 to 195
  10    24563, 35462    46 to 50      40    45362, 54263   196 to 200
  11    24635, 35624    51 to 55      41    45623, 54632   201 to 205
  12    24653, 35642    56 to 60      42    45632, 54623   206 to 210
  13    25346, 34256    61 to 65      43    46235, 56324   211 to 215
  14    25364, 34265    66 to 70      44    46253, 56342   216 to 220
  15    25436, 34526    71 to 75      45    46325, 56234   221 to 225
  16    25463, 34562    76 to 80      46    46352, 56243   226 to 230
  17    25634, 34625    81 to 85      47    46523, 56432   231 to 235
  18    25643, 34652    86 to 90      48    46532, 56423   236 to 240
  19    26345, 36254    91 to 95      49    62345, 63254   241 to 245
  20    26354, 36245    96 to 100     50    62354, 63245   246 to 250
  21    26435, 36524   101 to 105     51    62435, 63524   251 to 255
  22    26453, 36542   106 to 110     52    62453, 63542   256 to 260
  23    26534, 36425   111 to 115     53    62534, 63425   261 to 265
  24    26543, 36452   116 to 120     54    62543, 63452   266 to 270
  25    42356, 53246   121 to 125     55    64235, 65324   271 to 275
  26    42365, 53264   126 to 130     56    64253, 65342   276 to 280
  27    42536, 53426   131 to 135     57    64325, 65234   281 to 285
  28    42563, 53462   136 to 140     58    64352, 65243   286 to 290
  29    42635, 53624   141 to 145     59    64523, 65432   291 to 295
  30    42653, 53642   146 to 150     60    64532, 65423   296 to 300
The cosets are numbered arbitrarily as they are generated. Each course has five sections, and the sections of the cosets are numbered 1 to 300, Nos 1 to 5 being in the first coset, 6 to 10 in the second, and so on. These section numbers are types and it is with these types that we will be juggling. The particular way in which they are enumerated is arbitrary, but henceforth the computer will identify them by their numbers. Each type represents two actual sections, to be in corresponding parts of 2-part solutions.

Exactly the same process would be carried out if a more complex Group were chosen. There is a great deal of difference, from the human point of view, between composing a peal with a simple cycle of working bells, and using a complex Group like J.W. Parker's "Mysteries Unveiled" Group of order 168, but there is no fundamental difference, and a computer will apply the algorithms whatever the generating Group. However, if a mistake were made and a set of transdigits were given which did not form a Group, then the process would lead to chaos.

Note that we are dealing above with transdigits, in that we are exchanging bells 2 and 3, 4 and 5 irrespective of the positions in which they happen to be.

A great deal of calculation has then to be carried out, in order to work out the tables of falsity and of calls. Each of the 300 types has up to 14 types (including itself) with which it is false. When we say, for example, that type 11 is false with type 14, this implies that each of the two sections constituting type 11 is false with one of the two sections type 14. The same concept applies to calls - if type 60 gives type 111 by a bob, then each of the sections of type 60 gives one of the sections of type 111. The underlying truth behind this process is the basic lemma concerning the inter-relation of transdigits and transpositions; the structure of a method involves transpositions, and the part-plan involves transdigits.

Since we are searching for palindromes, it is necessary to calculate the inverse call table, which "goes back" on plain, bob or single.

A palindromic tree search starts at an apex, and by trial and error goes through every possible call at each type, building up a touch in both directions at once. Suppose we decide to start at a "bob Home" apex, using type 61. 61 is the first section of a course, and by referring to the inverse call table we discover that it is given from type 125. The first step of the tree search will be to take a "plain" from type 61, which gives type 62 (the relevant course being 61, 62, 63, 64, 65) and in order to preserve the palindrome, an "inverse plain" from type 125 gives type 124. We now have:

                  124   -P-  125  -B-   61  -P-   62
This reads from left to right, and in fact represents two open-ended touches of six leads (some of the sections have two leads each) related to one another by a 2-part transdigit. If during the course of the tree search a single is chosen for type 62, then an "inverse single" has to be applied to type 124, giving:
         153  -S-  124  -P-  125  -B-   61  -P-   62  -S-  278
The hope is that the two ends of the chain will eventually meet at a far apex. When this happens, it is a matter of "luck" as to whether a 2-part round block is produced, or two separate identically-called blocks. If 5000 or more changes are produced in two blocks, then human ingenuity is required in order to link them. Such a process builds up blocks much more rapidly than a non-palindromic search, as two types are added to the chain for each call option. Before two types are added to the ends of the chain, they must be checked for truth against all the types already assembled, as well as against one another, and most of the run time is occupied in this checking process. As the chain gets near peal length, a much larger proportion of trials are rejected.

Is it necessary to check the truth of both new additions to the chain? In general when composing in parts it is, but there are exceptions; it is not necessary when building up a one part peal (in which case one is handling the leads themselves, not types) but a case will be noted below in which it is also thought not necessary (though the full check was in fact carried out).

If a mid-lead apex (between Middle and Wrong) is selected as start, then the same type starts the two horns of the chain off, for instance:

           21  -P-  22  -S-  148  148  -S-  179  -P-  180
Here, type 148 was chosen for mid-lead starting apex and the first- level call option was a single, followed by a plain. The result in an eventual peal would be a pair of courses called single Middle, single Wrong.

During the operation of the tree search, tests have to be made to detect the four different kinds of far apices in order to link the two ends of the growing chain of types. When two new types are created they must be checked for equality before being tested for truth against one another; if they are equal, then a mid-lead far apex has been reached. After the block has been noted and the tree search continues they must be rejected. If however one of the other three kinds apex is found, after the block has been noted the tree search continues without rejection.

Every possible palindrome will be produced twice, in "half- reverse" forms, its apices being "near" and "far" in turn.

The results are condensed in the following table:

Courses                           Starting Apex
with type Plain Home   Bob Home   Centre Lead Bob Before  Single Home

1+   11-  3072        3456        3264        3968        2816
181+ 191- N13   M1969 N20   M2224 N6    M2283 N17   M2150 N10   M2074
  XXYYF   T10    B355 T24    B687 T8     B241 T18    B593 T24    B747

6-   16+  3072        3456        3136        4288      * 2816
186- 196+ N8    M1824 N20   M2224 N19   M2068 N281  M2765 N10   M2074
  XXYFY   T4     B167 T24    B687 T26    B739 T161  B3599 T24    B747

21+  26-              5120     ** 2752                    3712      #
201+ 206-    FALSE    N292  M2853 N8    M2144    FALSE    N15   M2406
  XXFYY               T410  B8275 T15    B445             T23    B597

31-  61+  3072        4480        3136        4224        3072
131- 161+ N8    M1824 N20   M2150 N19   M2068 N39   M2654 N11   M2630
  XYXYF   T4     B167 T23    B657 T26    B739 T55   B1375 T23    B701

Courses                           Starting Apex
with type Plain Home   Bob Home   Centre Lead Bob Before  Single Home

36+  66-  3072        4480        3264        4480     ** 3072
136+ 166- N13   M1969 N23   M2354 N6    M2283 N434  M3136 N11   M2630
  XYXFY   T10    B355 T26    B737 T8     B241 T481 B10025 T23    B701

41+  71-  3456        4480        2688        3584        4608
121+ 151- N14   M2117 N23   M2354 N10   M1888 N13   M1876 N11   M2735
  XYYXF   T15    B519 T26    B737 T20    B619 T10    B405 T14    B435

46-  76+  3456        4480        2688        4288      * 4608
126- 156+ N14   M2117 N20   M2150 N10   M1888 N291  M2738 N11   M2735
  XYYFX   T15    B519 T23    B657 T20    B619 T162  B3569 T14    B435

51-  81+  3072        3840      * 2752        4224        3072
146- 176+ N9    M1948 N169  M2574 N12   M2027 N18   M2780 N5    M2099
  XYFXY   T4.5   B183 T101  B2169 T9     B271 T20    B601 T7     B263

56+  86-  3584        4032      * 3136        3968        3072
141+ 161- N21   M2219 N163  M2603 N8    M1608 N17   M2360 N5    M2099
  XYFYX   T21    B617 T79   B1771 T7     B247 T14    B469 T7     B263

91-  96+                          2752        4224
231- 236+    FALSE       FALSE    N8    M2144 N39   M2654    FALSE
  XFXYY                           T15    B445 T55   B1375

101+ 111- 3584        4480        3136        3968        2560
216+ 226- N21   M2219 N19   M1984 N8    M1608 N17   M2150 N6    M2005
  XFYXY   T21    B617 T18    B565 T7     B247 T18    B593 T13    B401

106- 116+ 3072        4480        2752        3584        2560
211- 221+ N9    M1948 N19   M2092 N12   M2027 N13   M1876 N6    M2005
  XFYYX   T4.5   B183 T24.5  B691 T9     B271 T10    B405 T13    B401

241+ 246- 5120     **             4672     ** 3968        5120      #
291+ 296- N203  M3229    FALSE    N168  M3094 N17   M2360 N18   M3634
  FXXYY   T247  B5349             T334  B6463 T14    B469 T29    B663

251- 261+ 3840      * 4480        4288      *             2304
276- 286+ N89   M2581 N19   M2092 N224  M2622    FALSE    N9    M2076
  FXYXY   T69   B1473 T24.5  B691 T155  B3215             T15    B435

256+ 266- 4032      * 4480        4288      * 4224        2304
271+ 281- N167  M2571 N19   M1984 N282  M2833 N18   M2780 N9    M2076
  FXYYX   T79   B1793 T19    B565 T222  B4475 T20    B601 T15    B435
N=No.of solutions M=Mean length T=Time in minutes B=Backtracks

The left-hand column of the table gives the sets of four courses by the types of their first sections. When a general (non- palindromic) search was carried out on this part plan, it was found that different types produced the same results. The code such as XYFYX gives the positions of the fixed bell F of the generating transdigit and the swopping pairs XX and YY, as appearing in the course-ends, pairs XX and YY being reckoned as interchangeable. This structure of the 60 course-types falling into 15 sets of four is equivalent to an 8-part plan with the generating Group {23456, 32546, 23546, 32456, 45236, 54326, 45326, 54236}.

At the top of each window is the longest length produced.
** cases where the fixed bell of the transdigit of the Group coincides with the fixed bell at the starting apex, and the swopping pairs also coincide.
* cases where the fixed bells coincide, but the swopping pairs do not.
# cases where the one pair of bells swopping is a transdigit pair.
FALSE marks cases where self-false types are necessarily involved. These arise because Cambridge happens to have self-false section types; other methods might not have these.
N gives the number of solutions of all lengths in the palindromic search, whether or not the solutions happen to have one or two round blocks making up the length.
M gives the mean ("average") length of all N solutions.
T gives the duration in minutes, to the nearest minute, of the search
B gives the number of backtracks of the search.

For all starts marked **, because the pairs of bells swopping at the starting apex, and the place bell, are the same as in the transdigit of the part plan, they will be the same at the far apex, and the solution will be in two separate round blocks, which can be linked only by asymmetric calls (if such exist). A special relationship exists in these cases, all types having unique image types, and any pair of types occur in a solution only if both are present, and are equidistant from the apices. If one of the two blocks be inverted, then it contains the same sequence of rows as the other, excepting that 7-8 are reversed. It is probable that in these cases it is unnecessary to check the truth of both new types on the chain during a palindromic search. The increased probability of both types being true (if one is, then the other must be) results in a more productive search, and accounts for the fact that of the four distinctly different 2-part peals produced (excluding the short- course solutions) three are based on palindromes on this plan, and unfortunately require irregular linkages.

For the starts marked *, the place bell at the starting apex is the same as in the transdigit of the part-plan, but the swopping pairs are different. For any solution, it is a matter of chance as to whether at the far apex the pairs are the same or not. If they are different, then a two-part palindrome is produced of the "pairs of pairs" kind. The remaining exact 2-part peal produced is based on such a 2-part palindrome, supplemented by other palindromes.

For all the starts without asterisks, the solutions cannot be in two parts, for place bell at the starting apex of a two-part palindrome must also be the place bell at the far apex, and it is not the fixed bell of the part plan. Hence the solutions will all have two separate round blocks.

Examination of the statistics in the above table shows how more productive the asterisked searches were. It is thought that a logical approach to the theory of palindromes must be made on the grounds of probability; how likely is it that a random round block, produced irrespective of truth, will be true? The palindromes will conjecturally be more likely to be true than the others.

A sample * start was made for Superlative Surprise Major (the blocks and calls were identical with Cambridge and only the false table had to be recalculated) with the Plain Home starting apex, type 256. The results were spectacular! There were 6319 solutions, of which 552 were in the range 5000-5376 and were recorded. The run took 6338 minutes, there were 121281 backtracks, the maximum length was 7040 and the mean length 4359. Of the 552 recorded solutions, 260 gave two-part peals (indicating a probability of 0.5) and of these, 10 solutions had 68 CRUs. A ** start for Superlative would have been even more productive, but every solution would have been in two separate blocks.

Next section Conclusions
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