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:

2 3 4 5 6 7 8At 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 6and 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 repeatedThe 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.

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 -------------------------- repeatedThe 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) -------------------------- repeatedNote 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.

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) ------------------------ RepeatedThe 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.

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 300The 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 ApexThis 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 ApexThe 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 ApexHere, 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 starting 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 starting 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 B435N=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.