The Q-Set Parity Law

A Review of its Proofs from Thompson’s of 1886.

(Mathematics of Campanology)

Brian D. Price

January 2006

The Q-Set Parity Law

Brian D. Price

Statement of the Q-set parity law

If the calling of a particular Q-set be altered, all the other Q-sets remaining fixed in state, and the numbers of round blocks before and after the alteration be compared, then
  1. if the Q-set is of odd order, the parity of the number of blocks remains unaltered;
  2. if the Q-set is of even order, the parity of the number of blocks is changed.
It is conditional that there is one-way traffic only along routes.

The Q-set Parity Law, and not the phrase The Q-set Law which I used over fifty years ago, is the title of this paper because some have thought the latter one to be concerned merely with the existence of the Q-set structure. The aim of this paper is to review, hopefully to explain and occasionally to criticise, published material on the proof of it.

What is a Q-set?

If a bob be called at the end of a plain course of Plain Bob Minor, the course 42356 is produced. If this is to be part of a touch, and no call be made beforehand, then a bob must be called at the end of the course 42356, producing the course 34256. A bob at the end of that course will bring the touch round. The effect of the three bobs is to link together three full courses. In composers’ jargon, three P-blocks have been linked by a P/B (Plain-Bob) Q-set. This is the simplest example of a Q-set, one of order 3. Each of the P-blocks has five different lead-ends where a call may be made, the option Plain (i.e. no call by the conductor) at every lead-end signified by the P-. Singles also form Q-sets, generally of order 2 because they swap two bells. If a single be called at the end of a plain course, another single after the second course will bring rounds, and two P-blocks have been linked by a P/S Q-set. If the 120 different P-blocks are written down and every lead-end labelled with its Q-set serial number (there will be 120 of them) the result is difficult to comprehend, and various graphical ways of simplifying the situation have been used. The use of a computer for composition since the 1960s copes with the complexity admirably, but requires mathematical knowledge for the programming of it. A ringer once said to me Why do you use a computer to compose peals, why don’t you do it yourself? which showed a basic misunderstanding of the situation. The computer is merely a tool for handling complexities very rapidly, to the composer’s own instructions.

The Q-set parity law is concerned with a special situation, in which a number of P-blocks have to be linked by just one kind of call, all the lead-heads in the extent being true with one another. It does not apply to any minor method (plain or surprise &c) because owing to the nature of the rows (in-course or out-of-course), a lead such as 23456 in Plain Bob is false with the lead 32546, so only half the possible lead-heads may be used. This problem is discussed later on page 9. However a triples method using bobs only does fit in with the conditions, as the false lead-heads are out-of-course and not reachable by bobs. It is no accident that W.H.Thompson first coined the term Q-set in his work on Grandsire Triples, the most popular of the triples methods. The 360 possible in-course lead-heads of Grandsire form 72 P-blocks. They also form 120 B-blocks (3-lead touches with a bob at every lead-end) and the whole assembly is linked by 72 Q-sets, each with five lead-ends giving, by plain or bob, the same five lead-heads. Separate from his proof of the Q-set Parity Law, Thompson exploited a graphical method of composing peals of Grandsire Triples in parts. [1]

With Grandsire Triples, as in the great majority of other methods, Plaining every Q-set produces an even number of P-blocks, likewise with Bobbing, so that in either way one may start with an even number of round blocks. The Grandsire Q-sets are of order 5, an odd number, hence the Q-set Parity Law establishes the impossibility of just one round block being produced. One cannot get a peal of Grandsire Triples with bobs alone. On a much higher plane of difficulty there are a few cases, using a 21-part or 7-part Group for composition, in which one may start with an odd number of blocks and achieve a peal with bobs. For example, it is not difficult to produce a 21-part peal of spliced Stedman and Erin Triples with bobs only (though the change of method may be thought of as a second type of call). The Manchester prototype computer attempted to solve a problem in change-ringing in the early 1950s trying to produce a peal of Stedman Triples with bobs only, which was found to be impossible in 21 parts [2]. That problem proved much more difficult to solve.

Preamble – Transpositions and Transfigures

I pointed out the duality of transpositions and transfigures in a 1969 article [3], which no doubt was triggered off by my starting programming in Fortran in early September 1968 with a course at Imperial College for their Schools Computing Project. Immediately I was writing change-ringing programs, with their necessity for a clear distinction between the two concepts. This Schools Project started in 1964. Pupils submitted Fortran programs to IC Computer Centre by post on port-a-punch Hollerith cards, and received chain-printed output on continuous stationery by post. They were processed on a vast-sized IBM7904 in the basement of Electrical Engineering (under the outside of the concourse). Before 1964 there was I believe no nation-wide oppportunity for computer programming.

But mathematical literature is at variance with itself over the distinction. Burnside [4] starts off his treatise with a notation for permutations, which is of transfigures (although his use of letters makes this term rather nonsensical). Thus the permutation

a b c d e   to   c b d a e   is given as   (acd)(b)(e)
meaning a is replaced by c, c is replaced by d … etc., whereas Fletcher [5] on p.620 states
2 1 3 6 5 4   to   5 1 4 2 6 3   is given as   (145)(36)

denoting a transposition, the bell in the 1st position goes to the 4th position, the 4th to the 5th etc. In either case, constituent cycles are bracketed. Using Burnside’s notation this second example would be (1)(256)(34), or just (256)(34), rendered more awkward by the fact that rounds is not the first of the two given rows – a situation more likely to be met by the change ringer. Fletcher’s use of the word transition begs the question of the distinction.

In any case, these bracketed cycles do not make for easy computer programming.

Using Fletcher’s two stated rows:  
2 1 3 6 5 4
5 1 4 2 6 3

  it is simpler to say:

Transfigure:    1 5 4 3 6 2 (figure 1 remains 1, figure 2 is replaced by 5, figure 3 by 4, … )

Transposition:  4 2 6 5 1 3 (1st figure becomes the 4th, 2nd figure remains 2nd, 3rd figure becomes 6th …)

Using Pascal notation, if a row (a one-dimensional array) X is said to be altered to row Y, using the transfigure/transposition Z, then

{Z a transfigure} for i:= 1 to n do Y[n]:=Z[X[n]];

{Z a transposition} for i:= 1 to n do Y[n]:=X[Z[n]];

illustrating clearly the dual relationship [6] between the two different treatments.

In the remaining part of this section, in order to avoid the ambiguity of row, I use perm for permutation. The above dual relationship surfaces in the distinction between the left-cosets and right-cosets of mathematical literature [7]. For instance White [8] p.727 gives Bob Minimus:

1 2 3 4
2 1 4 3
2 4 1 3
4 2 3 1
4 3 2 1
3 4 1 2
3 1 4 2
1 3 2 4
1 3 4 2
3 1 2 4
3 2 1 4
2 3 4 1
2 4 3 1
4 2 1 3
4 1 2 3
1 4 3 2
1 4 2 3
4 1 3 2
4 3 1 2
3 4 2 1
3 2 4 1
2 3 1 4
2 1 3 4
1 2 4 3

which is a microcosm of the principle of part composition. The three columns are left-cosets of the Dihedral Group D4 (including D4 itself in column 1) whereas the eight rows are the right-cosets of the first row, which is the cyclic Alternating Group A3. In simpler parlance, the perms in the top row form a group (though this has been shown to have rare exceptions) which defines transfigures, and the remaining rows of perms exhibit the same transfigures with each other (in this case the cyclic rotation of bells 2–3–4); whereas the first column defines a series of transpositions (dictated by the method, principle or calls) which same transpositions are found correspondingly in the other columns. The whole structure is an extension of the duality of transfigures and transpositions.

In the above example, both the first row (Group A3) and the first column (Group D4) are Groups, and this is true in cases where the plain hunt (a Dihedral Group of perms) is involved, but generally in composition it is the top row of perms (the part-heads) which (pragmatically always) forms a Group, whereas the first column is selected by some method so its transpositions contribute to a true peal. There are more difficult cases, including Stedman, in which the first column is a Group (e.g. the Stedman six) which is capable of various forms with the same content of perms.

But as to which are left- and which right-cosets depends on (a) how a permutation is defined and (b) the convention of multiplication of operations, whereas the distinction between the more pedestrian terms transfigure and transposition is more easily understood.

The idea of a Group in change ringing is not all that simple, as it is the relations between the perms (either as transpositions or as transfigures) which form the Group, and not the perms themselves. Rounds itself is an arbitrary start-and-finish for ringers, but 123456 on six bells, regarded either as a transposition or as a transfigure, is technically the identity opertion, i.e. that which leaves a perm unaltered, and a Group necessarily contains an identity operation. Technically the first column above regarded as a set of transpositions forms the Group D4, and the other two columns are cosets of D4 in the extent, but in a looser sense the other columns are forms of the same Group, in that their own perms are inter-related by operations forming the same Group, the identity operation being the relation of any perm with itself. As Fletcher says in [5] p.620, The [perms] … by numbering the bells … which result could be regarded as the elements of a … group, but it is more logical to regard the transitions between [perms] as constituting the group … and he proceeds to explain a transition as a transposition. Here I edit Fletcher’s wording to perms in order to avoid the confusion of the usually-accepted meaning of change as the restricted transposition of one perm to the next – Fletcher is implying the transitions between any two of the perms.

I state in the preceding paragraph (either as transpositions or as transfigures) because the fundamental relation between the two, together with the closure property of a Group, ensure that either implies the validity of the other.

Proofs of the Q-set Parity Law – (1) by W.H. Thompson, 1886 [9]

By a process of exhaustion, Thompson considered all the topological ways in which the five exits of a particular Q-set of order 5 may be connected (by means of a circuit through other Q-sets) to its five entrances. In each case, the no-change in parity of the number of circuits involved in that Q-set was checked when the calling of the particular Q-set was altered. There are indeed 28 essentially different situations of a Q-set of order 5, of which 5 are rotationally symmetrical. It is not particularly easy to enumerate them. This enumeration would be increaingly complex for higher-order Q-sets, and on Thompson’s proof method each order would have to be checked separately. Having constructed the 28 logically, I predict that the number of topologically different Q-set situations of order seven is 7 + (6! - 1) = 726.

W.H.Thompson’s 28 Cases of Q-set Disposition

It is worthwhile examining the 28 cases of the disposition of a single Q-set, which Thompson enumerated as part of his proof [9]. They were later enumerated, in an explanation of Thompson’s proof, by Rev. C.D.P.Davies in [11]. Refer to the page 13 of diagrams. Thompson was concerned with the Q-sets of Grandsire Triples of order 5. In the top left-hand corner is a conventional diagram of such a Q-set, which has five entrances (distinguished by heavy dots on the circle) and five exits. Logically, the 360 in-course lead-heads identify with the routes connecting the Q-sets, the heavy dots at the Q-set entries focusing attention on them. Any Q-set has either all its entrances Plained, or all Bobbed. One-way traffic is assumed. Next to that is a diagram of the kinds of outside circuits from the exits to the entrances, to be catalogued by the Pascal program below. These outside circuits will be involved in other Q-sets also, but we are concerned here only with the dispositions of the round blocks through the sample Q-set under consideration (and finally, how many there are for parity checking). There are likely to be other round blocks which do not traverse the sample Q-set at all.

To interpret the Pascal output, in all the other 28 diagrams on page 13 a start is made at the dot of the uppermost entrance. Proceeding clockwise, the near exit is considered first, and has five different ways of linking externally to an entrance. These five are labelled 1 to 5 in clockwise order of entrances. To avoid confusion in the diagrams, No.3 (which links two diametrically-opposed points) is given as two short straight lines, to be notionally linked. Nos.4 and 5 appear to be anti-clockwise, as again if they were drawn clockwise the lines would be longer and the diagrams more confusing. Similarly for clarity, the five internal Bob links of each Q-set are omitted, so that the number of round blocks may more easily be counted when the Q-sets are Plained.

Program Qset (input,output);
Type  row = array[1..5] of integer;
Var   x,y: row;   i,j: integer;   xfile: text;

Procedure IndArr (c: integer; var b: row);
  var d: row;  e: array[1..6] of integer;  m,n: integer;
    d[1]:= (c + 23) div 24; c:= c - 24*d[1] + 24;
    d[2]:= (c + 5) div 6; c:= c - 6*d[2] + 6;
    d[3]:= (c + 1) div 2; d[4]:= c - 2*d[3] + 2;
    for m:= 1 to 6 do  e[m]:= m;
    for m:= 1 to 4 do
        b[m]:= e[d[m]]; for n:= d[m] to 5 do e[n]:=e[n+1]
    b[5]:= 15 - b[1] - b[2] - b[3] - b[4]
{procedure converts indices, in quasi-numerical order, to arrays}

  assign(xfile,'a:out.dat'); rewrite(xfile);
  for i:= 1 to 120 do   {i to be index of all 120 permutations}
      IndArr(i,x);  {array x runs through all 120 permutations}
        {x will denote the five fixed re-entry points}
      for j:= 1 to 5 do y[j]:= (x[j] + 5 - j) mod 5 + 1;     {CALCULATION}
        {y becomes the route-spans, from five exits in cyclic order}
      for j:= 1 to 5 do write(xfile,y[j]:2); writeln(xfile)

The output from this program was a series of 120 sets of external links:

  1 1 1 1 1,  1 1 1 2 5,  1 1 2 5 1, 1 1 2 2 4,  1 1 3 5 5 …

In further explanation, consider the fifth solution, 11355. This catalogues the external linkage of the Q-set diagram in the top right-hand cell of page 13. Starting at the top dot and moving clockwise, the first two exits are 1, 1 that is, they are linked to the next clockwise entrance; but the third exit (at the lowest point) must have an external link of type 3, diametrically across to the starting dot. The last two exits are 5, 5 both of which are external links to the fifth dot clockwise, i.e. the preceding dot. The net result is three round blocks passing through the Q-set when it is in state Plain, and this number is recorded by the large 3 in the centre of the Q-set. The 28 external dispositions are all of a Q-set in state Plain.

Five of these 120 solutions had rotational symmetry. The remaing 115 duplicated cyclically (for instance, 11224 is identical with 12241, 22411 &c) and I reduced these 115 to 23 different kinds, choosing the lowest of each set of five (regarding them as 5-digit quasi-integers), and sorting them by hand. The resulting 28 were:

   11111,  22222,  33333,  44444,  55555.
   11125,  11224,  11314,  11355,  12223,  12313,  12345,  12525,
   13344,  13353,  13524,  14145,  14244,  14253,  14555,  22335,
   22425,  23334,  23424,  24455,  25355,  34445,  34535.

These 28 occupy the twenty-eight cells of page 13 of diagrams. A further relation was found between them, on considering the change occuring when the Q-set is Bobbed. When the internal linkage of each Q-set assemby is altered from Plain to Bob, the resulting diagram is either the same assembly, or topologically another of the 28. To check this, one has to resort to pencil-and-paper. The eight diagrams along the top and right-hand side of page 13 result in themselves, and the other twenty pair off (as on the page across fainter lines) in that Bobbing one gives the other.

A curious fact is that nearly all of the 28 diagram have an axis of bilateral symmetry. The exceptions are the two labelled asymmetric, and they are mirror images of one another, each qualifying as giving itself topologically (by reflection) when bobbed, but the two forming a pair, because altering the calling of either gives the other directly. The topological comparisons usually reverse the direction of tracing round blocks, and of course swap Plain for Bob, but the important parity of the number of round blocks is unaffected. Five of the diagrams have also 5-part rotational symmetry, as one might expect.

Under each of the 28 diagrams is given also the Davies labelling of the case. For instance,

11314  AC/BOOO

signifies, to the right of the Pascal output 11314, that in the Plained state the two resulting round blocks are A and C, of which A in the key-diagram of Rev. C.D.P.Davies’s paper [11] — reproduced on page 15 — denotes using an adjacent pair of Q-set entrances, and C a triangle of three consecutive entrances. In the Bobbed state the round blocks are B, a non-adjacent pair of entrances, plus OOO, three loops each containing but one Q-set entrance, making four round blocks in all (and conserving parity). This tallies with the case’s opposite number of 25355 (C and D differing only in the rotational sense of visiting the triangle corners). G and H signify trapezia, KLMN crossed trapezia, etc. The notation is involed with the order of occurence in the round block, linked to directional arrows on the geometrical figures of the key-diagram. All related pairs preserve the parity of the number of round blocks through a Q-set.

Davies’ paper gives geometrical diagrams (reproduced on page 15) to help explain the different 28 cases, whereas Thompson’s paper does not. The orders in which the cases are tabulated differ a little. Thompson’s paper is more austere and illustrates his legal mind, whereas Davies’s style make one wonder about the length of his sermons!

Proofs of the Q-set Parity Law – (2) by Dr. R.A. Rankin, 1948 [12]

Rankin proves a more general theorem, of which a special case is the Q-set parity law of interest to change ringers. The article is well-clothed in the jargon of advanced mathematics, in fact it is double-wrapped in that the indexes of letters, representing members of a more generalized Q-set under consideration, themselves have suffixes. Dickinson [14] has extracted for us the crux of Rankin’s argument (he says so in his article) as it affects ringers, in a more understandable form less formidably wrapped. His article was a reply to Fletcher’s [5] in the same Journal, as apparently Fletcher was unaware of any proof of the Law other than Thompson’s one of 1886. Dickinson stated the proof as for Grandsire, but Rankin’s proof is generalized, for all Q-set orders and with further deductions not of ringing interest. The editor of the Proceedings [12] received Rankin’s paper for publication on 20 November 1946.

Rev. C.F.D. Moule was Dean of Clare College, Cambridge 1944-1951 and a Fellow from 1944, whereas Robert Alexander Rankin was a Research Fellow of Clare College 1939-1947. The two must have known one another. Rev. Moule was a member of the Cambridge University Guild of Change Ringers (we met regularly in his rooms for handbell ringing, though he was so busy he hardly ever rang with us) and possibly Moule had interested Rankin in the change-ringing parity problem.

Proofs of the Q-set Parity Law – (3) by B.D. Price, 1949

I proved the law by mathematical induction. Assuming the law is true for Q-sets of order n, a Q-set of order n+1 is then considered as a combination of a Q-set of order n, together with one of order 2. The behaviour of a Q-set of order 2 has to be found by inspection, there being only one possible configuration of circuits (refer to Figure 1 on page 14). Parity relations are deduced, and the law proved by induction. Though given only in print, the proof is a diagrammatic one well with the capability of an A-level student to understand:-

The Ringing World leader (half of the front page) of 10 June 1949, No.1992.


The term Q-set was given to us by W.H. Thompson in his pamphlet (1886) on Grandsire Triples. In this pamphlet (reproduced in Snowdon’s Grandsire, 1905, p.197) he proves the Law for Q-sets of order 5. Dr. Rankin, in his paper on Group Theory (Proc. Camb. Phil. Soc., Vol.44, Pt.1, p.17), proves the Law, but the method of his proof is beyond the reach of most of us.

In order to clarify ideas, we may define a Q-set as follows. A, A′, B, B′, C, C′ etc. as far as N, N′ are points in order on a circle. The number of points may be considered as 2n, there being n points A, B, C, etc., separated by the n points A′, B′, C′, etc. Any letters may be supplied between C′ and N, and there are not necessarily 28 points in all.

Routes lead to points A, B, C, etc., and away from points A′, B′, C′, etc. The problem of composing a peal is reduced to that of finding one continuous circuit around a number of interlaced Q-sets, where routes lead from dashed letters A′, B′, etc., on each Q-set to plain letters A, B, etc., on the same or on different Q-sets.

When a Q-set is in the state plain each letter is linked to its partner — A to A′, B to B′, etc. When in state bob each is linked to its other partner — N′ to A, A′ to B, etc. The order of a Q-set is the number of such links (equal to n).

Each Q-set in the extent may be plain or bob at will. If the state of each is fixed, then the aggregate of routes will form a number of circuits (round blocks). The Q-set Law concerns the oddness or evenness (parity) of this number of circuits, as follows:—

If a single Q-set of odd (even) order is altered in state, the others being fixed in state, the parity of the total number of circuits is not (is) changed.

There are two complementary parts to the Law — either miss out, or substitute, the words in parentheses. Two important conditions are, firstly, every point must be used (this cuts out arguments about Stedman Triples where all six-ends are not used), and secondly, every route must have one-way traffic (this disqualifies Treble Bob Minor, where a lead may be used backwards).

The Law may be proved by induction. We assume the Law to be true for all Q-sets up to order n, and then show it to be true for the next order (n+1). Since the Law is obviously true for n=1 or n=2, it must therefore be true for all values of n.

Suppose, therefore, that the Law is true for the Q-set order n:—

A, A′, B, B′, C, C′, …… N, N′.

Consider the Q-set of order n+1:—

A–A′, B–B′, C–C′, …… N–N′, X–X′.
It is in the state plain. We proceed to bob it by several steps.
First, connect up the letters as follows:—

A′–B, B′–C, …… M′–N, N′–A, X–X′.
The pair X–X′ has been left alone. The change otherwise is the same as bobbing a Q-set or order n. Now consider the four letters in order: A, N′, X, X′. For the purpose of proof they may be considered as forming a Q-set of order 2. At present they are in the state N′–A, X′–X. Alter the state of this Q-set to X–N′, A–X′. The whole situation is now:—

A′–B, B′–C, …… M′–N, N′–X, X′–A,
which is the Q-set of order n+1 in state bob. The bobbing of this Q-set has been carried out in two steps, equivalent to bobbing Q-sets of order n and 2. Since bobbing a Q-set of order 2 changes the parity of the total number of circuits, it will be seen that the Law then follows for Q-sets of order n+1. For example, if n is even, then bobbing the Q-set of order n changes the parity of the number of circuits, but so does altering the Q-set of order 2. Therefore the effect of bobbing the Q-set of order n+1 is not to change the parity. A similar argument holds if n is odd. By application of this argument, starting from n=2 (a case which must be proved by inspection), the Law is proved for any value of n.

Comment on the above Proof

Roger Bailey has pointed out to me that the term partner is confusing, and I agree that it would have been better expressed as neighbour, each letter in the circle having two neighbours. That was the only time since my proof was published over fifty years ago in which anyone has commented on it, verbally or in writing.

In a previous letter to the Ringing World editor [15] I pointed out that one-way traffic along routes was essential to the proof of the Q-set parity law, as I had been aware of the cases of two-way traffic through Q-sets in which the law did not apply. Such a practical change-ringing matter would not have suggested itself to Rankin, who was not a change-ringer.

Proofs of the Q-set Parity Law – (4) by D.J. Dickinson, 1957 [14]

Dickinson offers the proof as a special case of Rankin’s more general conclusions, using his logic. He considers a particular Q-set for altering, the others remaining undisturbed. The extent will be in a number of circuits, some of which involve the particular Q-set, which has its entries labelled 1 to 5 in some cyclic order. (So far the proof is identical to Thompson’s.) The involved circuits through a Q-set create a permutation by the way in which they link the exits to the entries, each circuit creating a constituent cycle in the transfigure. The Q-set is then altered in calling, the result being expressed as the product of that permutation with the cyclic permutation (12345), which in turn is broken down into four single swaps. Hence the no-change in parity is deduced. As expressed, it is for (Grandsire) Q-sets of order 5 only; but unlike Thompson’s proof, the technique as given applies for any Q-set order.

As an illustration of Dickinson’s (simplifying Rankin’s) logic, consider the pair of Q-sets given in Figure 3 on page 14. They represent a Q-set of order 5 at random from page 13, in the states Plained and Bobbed. The five routes external to the Q-set remain unchanged – Rankin’s segments of chains. The internal change from Plained to Bobbed produces a re-linking of the segments, and we are interested in any parity change of the number of routes. Here, each has 2 routes.

For each of the two states of the Q-set, a permutation is constructed which gives the succession of the five segments among the several routes [round blocks] with which it is involved. (These routes do not necessarily comprise all the routes covering the extent of Triples.) The upper diagram, the Q-set in Plained state, of kind 14244 [notation of Pascal output] gives the substitution

as from   [ 1 2 3 4 5 ]
to        [ 2 1 5 3 4 ]

or to put it in Burnside’s notation (12)(354), in terms of the two cycles of digits which it contains. Clearly this is the transfigure form of a permutation – figure 1 proceeds to 2, 2 to 1, 3 to 5 …. The digits 1 to 5 refer to Rankin’s five segments (which form two chains) outside the Q-set. In a similar way, the Q-set with its external segments unaltered, but now Bobbed, gives kind 22425 (that is, reckoned topologically, with a reflection of route direction, and a rotation) with substitution

as from   [ 1 2 3 4 5 ]
to        [ 4 2 1 5 3 ]

The crux of the proof is in the nature of the substitution which

transforms [ 2 1 5 3 4 ]
into       [ 4 2 1 5 3 ]

and is a way of expressing the cyclical result of the internal Bob routes replacing the internal Plain routes of the Q-set. Dickinson quotes this as the permutation (12345) – Rankin as the more general (123…n) – and both actually use the term transposition to describe it. If the transformation last-quoted above is analysed, as a transfigure it is 24351 and as a transposition it is 23451. Its duty as a cyclic interchange is indeed as a transposition.

The effect of the 5-fold cyclic transposition on the parity of the number of routes is explained by dissociating the transposition (12345) into four separate transposition steps (12), (13), (14), (15). With regard to the above example,

                                         2  1  5  3  4
Swap the numbers in positions 1 and 2    1  2  5  3  4
Swap the numbers in positions 1 and 3    5  2  1  3  4
Swap the numbers in positions 1 and 4    3  2  1  5  4
Swap the numbers in positions 1 and 5    4  2  1  5  3

The graphical effects of these separate transposition are illustrated in Figure 4 on page 14, where the dotted internal lines are the routes to be substituted in the ensuing alteration. Each simple swap is in fact the alteration of a Q-set of order 2. As Dickinson explains, the transposition [swap] increases the number of cycles [routes] by one if p and q are in the same cycle, and diminishes the number of cycles by one if p and q are in different cycles. This is precisely the action of the Q-set of order 2 (Figure 1 on page 14) and is easier to comprehend. As the number of swaps is four, an even number which is one less than the order of the Q-set, the Parity Law follows.

It therefore transpires that as far as the Q-set Parity Law of ringers is concerned, Rankin’s proof reduces to the relatively simple matter of dissociating the 5-part cyclical transposition of the Grandsire Q-set into four steps of altering Q-sets of order 2. Indexes with their suffixes are not required, neither are permutations of chain segments. The reduced proof applies to Q-sets of any order, including even orders in which parity is changed, and depends ultimately on the behaviour of the Q-set of order 2, as does also Price’s proof which however uses mathematical induction.

The Q-set Assembly of Twin-bob Composition in Stedman Triples

The book on Stedman [18] explains at length the twin-bob way of composing peals of Stedman Triples discovered by William Hudson in 1832. By far the majority of peals of Stedman Triples rung since have been on this plan. The twin-bob calls at S and at L are linked by Q-set assemblies, as are the calls at H and Q. The basic assembly is illustrated in Figure 5 on page 14. This assembly is formed of two simple Q-sets of order 2, linked for truth in the situation either – or so that the parity of the number of round blocks is not disturbed, hence always even. A peal must resort to singles for linkage. This fact must have been quite evident to the early composers, but [18] p.265 makes only woolly references connecting it for truth with Thompson’s Q-set paper on Grandsire Triples of 1886.

An Example of the Q-set Parity Law not Applying

The sixty in-course leadheads of Plain Bob Minor may be placed on the face-corners of a dodecahedron (or its dual the icosahedron) symmetrically, so that the three leadheads of a B-block form a clockwise rotation about a vertex, and the five of a P-block are on a face, rotating anti-clockwise as a pentagram. Each vertex has then a P/B Q-set on it. (This is one of four different dispositions of all minor methods.) However, these sixty leads are false in pairs, each pair being diametrically opposite (antipodal) on the solid, their rows being mutually in reverse order. Hence in composing a true 360 with bobs only, one has to avoid falseness by selecting only 30 of the leads, one from each antipodal pair, and the Q-sets are not fully used. (In the parallel case of Treble Bob Minor – including Surprise Minor – the full 720 results.)

However, the diagram on the dodecahedron may be (notionally) folded [19] to bring the two leadheads of each pair into coincidence. The simple surface of the dodecahedron becomes distorted into a multiply-connected surface (has Henry Moore holes in it), but of course it is not necessary to make a model of it – a flat diagram will suffice, with broken routes around its perimeter to be linked by letters to the ones diametrically opposite. All the B-block, P-blocks, Q-sets come together in mutual pairs, so that there are then just 10 B-blocks, 6 P-blocks, 10 Q-sets in the diagram. But the routes are no longer uni-directional because pairs of directed routes amalgamated were orientated in opposite directions. In finding a round block by experimenting with the calling of the Q-sets, one does not pay any regard as to which way round a lead is being used, until finally just one block is pricked out. And crucially, although there are an even number of P-blocks when all the Q-sets are Plained (similarly for being Bobbed), it is possible to find a single round block. The Q-set Parity Law does not apply, because the paired-off routes are not uni-directional.

White, in [8] pp.730–732, explains this folding starting with Plain Bob Doubles.

This folding removes the difficulty of avoiding false leadheads, as mutually-false pairs become just one which may be used in either direction, and composition is now a simple matter of how to dispose the Q-sets, each Plained or Bobbed. The same argument applies to the composition of Original TriplesMinor with bobs only. With the in-course leads of Plain Bob Minor:

2 3 4 5 6 3 4 2 5 6 4 2 3 5 6 This is the usual 3-part 360 of Plain Bob Minor, to be doubled to a 720 by two singles. The five underlined leads are all part of one plain course, but the isolated lead is reversed in its direction relative to the other four (this is true twice more cyclically).
--------- --------- ---------
- 2 3 5 6 4 - 3 4 5 6 2 - 4 2 5 6 3
3 6 2 4 5 4 6 3 2 5 2 6 4 3 5
6 4 3 5 2 6 2 4 5 3 6 3 2 5 4
4 5 6 2 3 2 5 6 3 4 3 5 6 4 2
- 4 5 2 3 6 - 2 5 3 4 6 - 3 5 4 2 6
- 4 5 3 6 2 - 2 5 4 6 3 - 3 5 2 6 4
5 6 4 2 3 5 6 2 3 4 5 6 3 4 2
6 2 5 3 4 6 3 5 4 2 6 4 5 2 3
2 3 6 4 5 3 4 6 2 5 4 2 6 3 5
3 4 2 5 6 4 2 3 5 6 2 3 4 5 6
--------- --------- ---------


[1] Snowden, William (et al.): Grandsire: the method, its peals and history, 2nd edn. London 1905. This edition contained a further paper on Grandsire Triples by Thompson, giving a number of part-diagrams with their Q-sets for peal composition. The copy in the British Library was destroyed in World War 2. The 1st edition 1888 was by the late Jasper W. Snowdon, edited by his brother William (a copy in the British Library).
[2] Mentioned in Andrew Hodges, Alan Turing, the Enigma (Vintage, 1992 edn.) p.445. Refer to a letter published in The Ringing World on 3 April 1953 p.219, from B.D.Price.
[3] Price, B.D.: Mathematical Groups in Campanology, Math. Gaz., Vol.53 (1969) pp.129-133.
[4] Burnside, W.: Theory of Groups of Finite Order, (CUP 2nd ed. 1911; reprinted Dover 1955)
[5] Fletcher, T.J.: Campanological Groups, American Mathematical Monthly, Vol.63 (1956) p.619. In the references, there is poor editing of Thompson, W. H. A Note on … (failure to discriminate by font between name and title) which has led White in [8] to style Thompson’s name erroneously as W.H.A. Thompson.
[6] A dual relationship in mathematics denotes a similarity between two entities or ideas, such that a comparison of the differences of one with the other reveals a correspondence between them. Another duality, concerned with graphical representation of permutations discussed below, is the duality of the icosahedron (a geometrical solid with twenty triangular faces) and the dodecahedron (twelve pentagonal faces). If the centres of the faces of either solid be joined to the centres of the adjoining faces, the result is the other. They have the same set of symmetry axes, and the 60 +ve changes on 5 bells (mathematically, the group A5) may be placed on either. W.H.Thompson must have used the dodecahedron to arrange the 360 +ve changes on six bells (leadheads of Grandsire Triples) symmetrically as his papers mention it, but exactly how he did it is apparently lost as there seem to be no known surviving models of his work. There is a six-figure permutation Group isomorphic with the dodecehedron/icosahedron which Thompson might have used (the Group used by Hudson’s courses of Stedman Triples).
[7] Though I myself studied mathematics for a degree 1942-1945, there was no mention of Group theory (or, for that matter, statistics!) in my course, and my initiation was by studying smatterings of Burnside [4] as part of my ringing interests. Consequently I myself have never thought in terms of left- or right-cosets. It is curious to realise in hindsight that while I was an undergraduate, the boffins at Bletchley Park were decoding German intelligence using much the same mathematics as in ringing composition; and when I was making a change-ringing machine just after the war, the second-hand telephone relays I was using were very probably from dismantled Enigma-decoding equipment.
[8] White, Arthur T. (Western Michigan University): Ringing the Cosets; American Mathematical Monthly, Vol.94 No.8, October 1987 pp.721-746.
[9] Thompson, W.H.: A Note on Grandsire Triples, London 1886. Thompson’s proof of the Q-set parity law for Grandsire Triples. There is a copy in the Central Council Library, but the British Library copy was destroyed in World War 2.
Thompson was born 26 September 1840. He gained admission to St. John’s College, Cambridge in 1858 but gained a scholarship to Gonville and Caius College in 1859. He studied mathematics and was 23rd wrangler (first class honours) in 1863. He worked as an assistant magistrate for the East India Company at Dacca for five years, then studied at Inner Temple, London for the Bar and became a barrister in 1872. From 1872 he working in the Punjab, retiring in 1897/98. A batchelor, he retired to Brighton where he lived in lodgings, and died 10 July 1934 [10]. Obituary in The Times on 12 July.
[10] Alumni Cantabrigienses, British Library HLR 378.42659 (open shelves in Humanities 1).
[11] Davies, Rev. C.D.P.: Odds and Ends of Grandsire Triples, 1929. This booklet of pp.22 + folding end-diagram, re-stated [9] with an explanation of the 28 different Q-set cases checked (pp.1-15 with end-diagram). Davies also made a revue of Thompson’s proof [9] in the two editions of [1].
[12] Rankin, R.A.: A Campanological Problem in Group Theory, Proc. Camb. Phil. Soc. Vol.44, 1948, pp.17-25 (received by the editor 20 November 1946). Robert Alexander Rankin was a foundation scholar of Clare College, Cambridge in 1934. In 1936 he was a wrangler (Mathematical Tripos Part II, 1st Class) and in 1937 gained special credit in Part III. He was a Research Fellow of Clare 1939–47 and Official Fellow 1947–51 [13].
[13] Harrison, W.J.: Notes on the Masters, Fellows, Scholars and Exhibitioners of Clare College, Cambridge, British Library ref. X.529/19550.
[14] Dickinson, D.J.: On Fletcher’s Paper Campanological Groups, American Mathematical Monthly, May 1957, pp.331-332.
[15] The Ringing World leader (half of the front page) of 30 April 1948, No.1934.


… … …
        Mr Brian D. Price, while welcoming Dr. Rankin’s work with enthusiasm, writes that it would be a mistake to think that it is the first thorough approach to Group theory in change ringing that has been made. Rather it is that such advances have not been made public simply because there has not been a demand for such knowledge among ringers.

His letter continues:After a study of W.H. Thompson’s papers, it is clear to me that he knew much more of Group Theory than Dr. Spice’s article would suggest. The so-called Thompson Papers in Grandsire are not the originals, but rather a much-simplified extract of them from the point of view of the practical composer. Last year I made efforts through his relations and associates to find any remaining papers and to preserve his unpublished works (he died in 1934 at Brighton). The Central Council ought to have these in its library. Where are the genuine Thompson Papers?

There are three good reasons for knowing that Thompson was conversant with Group Theory. Firstly, he used the dodecahedron for plotting the lead-ends of Grandsire Triples. This in itself has a very deep connection with Groups. Secondly, he almost completely explored part composition of Grandsire Triples (I have found only one plan that he missed). This involves an exploration of sub-groups among the lead-ends. Thirdly, from his handling of Q-sets, not only in the well-known case of the extent of Grandsire Triples, but also in the part-diagrams of his, and in using the various singles and other calls, it is clear that he knew the general [parity] law of Q-sets.

In 1944 I proved the general [parity] law of Q-sets as follows: When an odd-numbered Q-set is altered in calling it does not alter the oddness or evenness of the total number of round blocks. Similarly, substituting even-numbered and does alter. The law is only operative for one-way traffic through the Q-set. For instance, one can compose the extent of Original Minor with bobs-only. This may be what Dr. Rankin has proved. I am convinced that W.H.Thompson had proved it too.
… … …

… It is necessary and sufficient for the part-ends of a composition in similar parts to form a sub-group …

The final quote from B.D.P. asserts the word necessary, which Roger Bailey has, years later, demonstrated as untrue. In [16] I gave examples of peals of Bob Triples with part-heads not forming a Group. A further case is provided in a ten-part peal of Erin Triples by Andrew Johnson, Erin peal No.8 in [17], which though based evidently on the Group [7.14] of ten part-heads, conceals with a singled-in block a twenty-part structure whose part-heads do not form a Group, but rather twenty rows of the forty-part Group [7.11]. The Group references such as [7.14] refer to my Catalogue of Permutation Groups on up to 7 Bells (1996) available as [16][20] below.

[16] Price, B.D.: The Structure of Palindromic Peals (1998) available on
[17] Saddleton, P.A.B.: A Collection of Compositions of Stedman Triples and Erin Triples (Central Council publication 1999).
[18] Davies, Rev. Charles D.P. et al., Stedman (1st. edition, Leeds 1903), Jasper Snowden Series.
[19] The term fold does not necessarily mean a process like folding a bed-sheet, or creasing and folding a piece of paper with an ink blot to produce an ink-shape with an axis of bilateral symmetry. Here, the folding is the application of half the dodecahedron-diagram on to the other half, so that mutually false leads coincide; it can be thought of notionally in this case as a flattening of the 3-dimensional diagram into two equal pancakes, slitting around the circumference, rotating one relative to the other through 180° then fusing together. This would bring together false pairs of leads which were at mutual antipodes of the original three-dimensional diagram. As the under-pancake became face-down, the routes on it were in the opposite direction to the fused routes of the upper pancake, making a non-directional diagram.
[20] Price, B.D.: The Composition of Peals in Parts (1996) available on

Key to the following three pages of diagrams

I apologise for dissociating text and diagrams. This is to keep down the cost of photocopying.

Page 13 Diagrams illustrating Thompson’s 28 kinds of Q-set disposition. See page 4.
Page 14 Miscellaneous diagrams
Figure 1 The simple behaviour of a Q-set of order 2, which is the basis of general proofs. It illustrates a uni-directional Q-set, and parity is altered, 1 ↔ 2. It is logically the same as the intermediate switch of three-way (or more) light control.
Figure 2 A Q-set of order 2, not uni-directional. Parity here is not altered.
Figures 3, 4 To illustrate Dickinson’s proof, on pages 7 and 8.
Figure 5 The Q-set assembly of Hudson’s twin-bob courses of Stedman Triples. It contains two interlocked Q-sets of order 2, uni-directional, hence parity is unaltered.
Figure 6 A Q-set of order 3, not uni-directional. Parity is altered.
Figure 7 A third option enables a Q-set of order 3 to be used atypically.
Page 15 Reproduction (enlarged) of Thompson’s 28 cases of Grandsire Q-sets, taken from the Rev. C.D.P.Davies’s folded end-paper of [11].
Copies of this paper were originally available from the author, B.D.Price, who died in 2012.
HTML version by Andrew Johnson, 2017, available from

Thompson’s 28 kinds of Q-set disposition Thompson’s 28 kinds of Q-set disposition
Figure 1   Uni-directional Q-set of order 2, parity is changed Uni-directional Q-set of order 2, parity is changed with the letters of Price’s proof
Figure 2   Not uni-directional Q-set of order 2, parity is not changed Not uni-directional Q-set of order 2, parity is not changed
Figure 5   Twin-bob Q-set assembly Twin-bob Q-set assembly for Hudson’s 60 Courses of Stedman Triples
Figure 7   A normal Q-set of order 3 with atypical use of extra call A normal Q-set of order 3 with atypical use of extra call
Figure 3   Dickinson’s proof:
The two transfigures Dickinson’s proof: The two transfigures Permutations of the five external segments
Figure 6   Plain Bob Minor,
not a uni-directional Q-set Plain Bob Minor, not a uni-directional Q-set Q-set order 3, parity changed
Figure 4   Dickinson’s Proof:
The sequence of four swaps Dickinson’s Proof: The sequence of four swaps

Davies’s interpretation of Thompson’s Q-set dispositions

Unfolding end-paper of Rev. C.D.P.Davies’s pamphlet Odds and Ends of Grandsire Triples [11] (enlarged by a factor of ×1.4)

Davies’s interpretation of Thompson’s Q-set dispositions

HTML version by Andrew Johnson, October 2017

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