Triple metamorphosis of twofold triple systems

Discrete Mathematics 313 (2013) 1872–1883

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Triple metamorphosis of twofold triple systems C.C. Lindner a , M. Meszka b,∗ , A. Rosa c a

Auburn University, Auburn, AL, USA

b

AGH University of Science and Technology, Kraków, Poland

c

McMaster University, Hamilton, ON, Canada

article

info

Article history: Received 12 October 2011 Accepted 6 December 2011 Available online 30 December 2011 Keywords: Twofold triple system Metamorphosis 4-cycle system

abstract In a simple twofold triple system (X , B ), any two distinct triples T1 , T2 with |T1 ∩ T2 | = 2 form a matched pair. Let F be a pairing of the triples of B into matched pairs (if possible). Let D be the collection of double edges belonging to the matched pairs in F , and let F ∗ be the collection of 4-cycles obtained by removing the double edges from the matched pairs in F . If the edges belonging to D can be assembled into a collection of 4-cycles D ∗ , then (X , F ∗ ∪ D ∗ ) is a twofold 4-cycle system called a metamorphosis of the twofold triple system (X , B ). Previous work (Gionfriddo and Lindner, 2003 [7]) has shown that the spectrum for twofold triple systems having a metamorphosis into a twofold 4-cycle system is precisely the set of all n ≡ 0, 1, 4 or 9 (mod 12), n ≥ 9. In this paper, we extend this result as follows. We construct for each n ≡ 0, 1, 4 or 9 (mod 12), n ̸= 9 or 12, a twofold triple system (X , B ) with the property that the triples in B can be arranged into three sets of matched pairs F1 , F2 , F3 having metamorphoses into twofold 4-cycle systems (X , F1 ∗ ∪ D1 ∗ ), (X , F2∗ ∪ D2 ∗ ), and (X , F3 ∗ ∪ D3∗ ), respectively, with the property that D1 ∪ D2 ∪ D3 = 2Kn . In this case we say that (X , B ) has a triple metamorphosis. Such a twofold triple system does not exist for n = 9, and its existence for n = 12 remains an open and apparently a very difficult problem. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The concept of metamorphosis of block designs, as discussed, for example, in [1–4,7–11], is due to the first author. Typically, for a subgraph G′ of G, each block of a G-design of order n and index λ is modified by deleting the edges of G \ G′ , and then reassembling the totality of deleted edges into G′ -blocks so as to form, together with the modified blocks of the original G-design, a new G′ -design of order n and index λ. A typical (and perhaps the first) example of this is the metamorphosis of a Steiner system S (2, 4, n) into a Steiner triple system of order n [9]. The index λ under the metamorphosis may change (i.e. increase). The situation that is the starting point of this paper is a metamorphosis of a (simple) twofold triple system TS(n, 2) into a twofold 4-cycle system 4C(n, 2) (‘‘simple’’ here means that there are no repeated triples; for undefined graph-theoretic terms, see [14]; for undefined design-theoretic terms, see [5]). More precisely, let (X , B ) be a twofold triple system of order n, TS(n, 2). For every x, y ∈ X , x ̸= y, the pair {x, y} is contained in exactly two blocks, say, {x, y, z } and {x, y, w}. Any two blocks B1 , B2 ∈ B such that |B1 ∩ B2 | = 2 form a |B | matched pair. Suppose now that we have a partition of B into 2 matched pairs. If we delete the double edge from a matched pair {x, y, z }, {x, y, w}, we are left with the 4-cycle (x, z , y, w). If C is the collection of 4-cycles obtained by removing the

∗

Corresponding author. E-mail address: [email protected] (M. Meszka).

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double edge from each matched pair, and if the collection F of deleted double edges can be reassembled into a collection of 4-cycles D then (X , C ∪ D ) is a twofold 4-cycle system which is said to be a metamorphosis of the twofold triple system (X , B ). For this metamorphosis to exist, it is necessary that both the twofold triple system and the twofold 4-cycle system of order n exist, that is, one must have n ≡ 0, 1, 4 or 9(mod 12). It was shown in [7] that this necessary condition is also sufficient, that is, the spectrum for twofold triple systems having a metamorphosis into a twofold 4-cycle system is the set n ≡ 0, 1, 4 or 9(mod 12), n ≥ 9 (clearly no such metamorphosis is possible for n = 4). In this paper, we extend the concept of metamorphosis as follows. Suppose that a twofold TS(n, 2), say, (X , B ), has three distinct metamorphoses into twofold 4-cycle systems (X , Ci ∪ Di ), i = 1, 2, 3, where Ci is the set of 4-cycles obtained by deleting the double edges from the matched pairs of blocks of B , and Di is the collection of 4-cycles reassembled from the collection of the deleted double edges Fi . If the double edges of Fi , Fj are mutually disjoint, Fi ∩ Fj = ∅, i ̸= j, i, j = 1, 2, 3, and their union is the twofold complete graph 2Kn , we speak of a triple or complete metamorphosis of the twofold triple system (X , B ) into twofold 4-cycle systems. We abbreviate this as TM3,4 (Kn ) or more briefly as TM(Kn ) (since this is the only kind of metamorphosis we are going to consider in this paper). A similar problem of complete metamorphosis of twofold 4-cycle systems into twofold 6-cycle systems is considered in [2]. Clearly, the necessary conditions for the existence of a triple metamorphosis TM(Kn ) are the same as those for a (single) metamorphosis: n must be of the form n ≡ 0, 1, 4 or 9(mod 12). It is the purpose of this article to show that these necessary conditions are also sufficient, with one exception (n = 9) and one possible exception (n = 12). In spite of an extensive computational effort, we have so far not been able to decide the existence or nonexistence of a TM(K12 ). 2. The case of n = 9 There are exactly 13 nonisomorphic simple twofold triple systems TS(9, 2) of order 9 [5,6]. We have conducted an exhaustive search of each of these systems for metamorphoses. Only two systems (systems No.30 and 33 in the numbering of [5,6]) admit a metamorphosis into a twofold 4-cycle system. (The metamorphosis presented in [7] is that of the system No. 33.) Only one of the systems (No. 33) admits a double metamorphosis. The blocks of the system are A: 0 1 2; B: 0 1 3; C: 0 2 4; D: 0 3 4; E: 0 5 6; F: 0 5 7; G: 0 6 8; H: 0 7 8; I: 1 2 5; J: 1 3 5; K: 1 4 6; L: 1 4 8; M: 1 6 7; N: 1 7 8; O: 2 3 7; P: 2 3 8; Q: 2 4 6; R: 2 5 8; S: 2 6 7; T: 3 4 7; U: 3 5 6; W: 3 6 8; X: 4 5 7; Y: 4 5 8 The pairing of the blocks is as follows (for each pair, the first two letters indicate the blocks of the matched pair, and the next two digits indicate the double edge): I. metamorphosis: AB01, CD04, EG06, FH07, IJ15, KM16, LN18, QS26, PR28, UW36, OT37, XY45. II. metamorphosis: AC02, BD03, EF05, OP23, IR25, JU35, KQ46, TX47, LY48, MS67, GW68, HN78. Partition of the double edges into 4-cycles: I. (0,6,3,7) twice, (0,1,5,4) twice, (1,6,2,8) twice. II. (0,2,3,5), (0,2,5,3), (0,3,2,5), (4,6,7,8), (4,6,8,7), (4,7,6,8). An exhaustive computer search establishes that none of the 13 simple TS(9, 2)s admits a triple metamorphosis. 3. Triple metamorphosis: the case of n ≡ 1 (mod 12) Let n = 12t + 1, and let (ai , bi , ci ), i = 1, 2, . . . , 2t be a solution to the I. Heffter’s difference problem (cf. [5,6]), that is, for each i = 1, . . . , 2t, we have either ai + bi = ci or ai + bi + ci ≡ 0(mod 12t + 1), and {(ai , bi , ci ) : i = 1, 2, . . . , 2t } is a partition of the set {1, 2, . . . , 6t }. Then the base blocks {0, ai , ai + bi }, {0, bi , ai + bi }(mod 12t + 1), i = 1, 2, . . . , 2t define a simple cyclic TS(12t + 1, 2). We will show that each such cyclic TS(12t + 1, 2) admits a triple metamorphosis. Consider j ∈ {1, 2, . . . , 6t }, and suppose j ∈ {ax , bx , cx }. Then there are two triples containing the pair 0, j, say, {0, j, m} and {0, j, p}, and it must be that {j, d(m, j), d(p, j)} = {ax , bx , cx } where d(u, v) = min(|v − u|, 12t + 1 − |v − u|). Take these two triples to form a matched pair Pj . Omitting the double edge {0, j} results in the 4-cycle (0, m, j, p) whose edge-lengths contain each of the two differences in (ax , bx , cx ) other than j exactly twice. So, each triple (ai , bi , ci ) will yield three matched pairs Pai , Pbi , Pci . Selecting now for each i = 1, 2, . . . , 6t one of Pai , Pbi , Pci yields the first metamorphosis, selecting next, for each i = 1, 2, . . . , 6t, one of the remaining two matched pairs yields the second metamorphosis, and finally the remaining (third) matched pairs yield the third metamorphosis. It is clear that the double edges removed cover 2Kn . It remains to be shown that the removed double edges can, in each metamorphosis, be reassembled into 4-cycles. This can easily be seen as follows. Partition arbitrarily the set {1, 2, . . . , 6t } into pairs. If {x, y} is one of these pairs, consider the four block-orbits corresponding to the two triples (ax , bx , cx ), (ay , by , cy ) in the solution of I. Heffter’s difference problem, and consider, for either of the three metamorphoses, the matched pairs

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corresponding to the two pairs of these orbits. If the two orbits of double edges corresponding to this selected pair of blockorbits (in the particular metamorphosis) consist of edges of length, say, r and s, then reassemble these edges into 4-cycles (0, r , r + s, s). This can be done independently for each pair {x, y}, and the proof is complete.  We illustrate the above construction for n = 25. One possible solution to the I. Heffter’s difference problem is

(1, 11, 12), (2, 6, 8), (3, 7, 10), (4, 5, 9). The corresponding simple cyclic TS(25, 2) is given by base triples {0, 1, 12}, {0, 11, 12}, {0, 2, 8}, {0, 6, 8}, {0, 3, 10}, {0, 7, 10}, {0, 4, 9}, {0, 5, 9} mod 25. The matched pairs are: From the first pair of orbits: {0, 1, 11}, {0, 1, 12}; {0, 11, 12}, {0, 11, 24}; {0, 1, 12}, {0, 11, 12}. From the second pair of orbits: {0, 2, 8}, {0, 2, 19}; {0, 6, 8}, {0, 6, 23}; {0, 2, 8}, {0, 6, 8}. From the third pair of orbits: {0, 3, 10}, {0, 3, 18}; {0, 7, 10}, {0, 7, 22}; {0, 3, 10}, {0, 7, 10}. From the fourth pair of orbits: {0, 4, 9}, {0, 4, 20}; {0, 5, 9}, {0, 5, 21}; {0, 4, 9}, {0, 5, 9}. Select one matched pair from each of the four rows above for the first metamorphosis, another matched pair for the second metamorphosis, and the remaining matched pair for the third metamorphosis. For example, we may choose {0, 1, 11}, {0, 1, 12} from the first row, {0, 2, 8}, {0, 2, 19} from the second row, {0, 7, 10}, {0, 7, 22} from the third row and {0, 4, 9}, {0, 5, 9} from the fourth row. If we delete the double edges with edge-lengths 1 and 2 from the first and second row, we can reassemble these edges into 4-cycles (0, 1, 3, 2) mod 25. Similarly, the double edges from the third and the fourth row, with the edge-lengths 3 and 4, can be reassembled into 4-cycles (0, 3, 7, 4) mod 25. The remaining two metamorphoses are seen to be obtained in exactly the same way. Summarizing this section, we have the following. Theorem 3.1. For all n ≡ 1(mod 12), n ≥ 13, there exists a triple metamorphosis TM(Kn ) of a TS(n, 2) into a 4-cycle system 4C(n, 2). 4. A triple metamorphosis of a transversal design We start this section with an example. Example 4.1. A triple metamorphosis TM(K16 ). Let X16 = {1, 2, 3, 4} × {1, 2, 3, 4}, and let (X16 , B ) be an affine plane of order 4, a.k.a. a Steiner system S (2, 4, 16) one of whose parallel classes consists of blocks {(i, 1), (i, 2), (i, 3), (i, 4)}, i = 1, 2, 3, 4. Replacing each block {a, b, c , d} with the four blocks {a, b, c }, {a, b, d}, {a, c , d}, {b, c , d} of a TS(4, 2) results in a simple TS(16, 2). For the first metamorphosis, form the matched pairs {(i, 1), (i, 2), (i, 3)}, {(i, 1), (i, 2), (i, 4)} and {(i, 1), (i, 3), (i, 4)}, {(i, 2), (i, 3), (i, 4)}, i = 1, 2, 3, 4 (so that the double edges are {(i, 1), (i, 2)} and {(i, 3), (i, 4)}), and for each block {(1, j1 ), (2, j2 ), (3, j3 ), (4, j4 )} ∈ B , form the matched pairs {(1, j1 ), (2, j2 ), (3, j3 )}, {(1, j1 ), (2, j2 ), (4, j4 )} and {(1, j1 ), (3, j3 ), (4, j4 )}, {(2, j2 ), (3, j3 ), (4, j4 )} (so that the double edges are {(1, j1 ), (2, j2 )} and {(3, j3 ), (4, j4 )}. The totality of double edges comprises the edges of a ‘‘doubled’’ complete bipartite graph 2K4,4 on {1, 2} × {1, 2, 3, 4} (with parts {1} × {1, 2, 3, 4} and {2} × {1, 2, 3, 4}) together with the edges {(i, 1), (i, 2)} and {(i, 3), (i, 4)}, i = 1, 2, and of a disjoint 2K4,4 on {3, 4} × {1, 2, 3, 4} together with the edges {(i, 1), (i, 2)} and {(i, 3), (i, 4)}, i = 3, 4. The former can be viewed as a union of a 2K4 on {1, 2} × {1, 2}, a 2K4 on {1, 2} × {3, 4}, and of a 2K2,2 on {(1, 1), (1, 2)} (one part) and {(2, 3), (2, 4)} (second part), and of a 2K2,2 on {(1, 3), (1, 4)} (one part) and {(2, 1), (2, 2)} (second part). Each of these is easily seen to be decomposable into 4-cycles. For the second metamorphosis, form instead the matched pairs {(i, 1), (i, 2), (i, 3)}, {(i, 1), (i, 3), (i, 4)} and {(i, 1), (i, 2), (i, 4)}, {(i, 2), (i, 3), (i, 4)}, i = 1, 2, 3, 4, and for each block {(1, j1 ), (2, j2 ), (3, j3 ), (4, j4 )} ∈ B , form the matched pairs {(1, j1 ), (2, j2 ), (3, j3 )}, {(1, j1 ), (3, j3 ), (4, j4 )} and {(1, j1 ), (2, j2 ), (4, j4 )}, {(2, j2 ), (3, j3 ), (4, j4 )}. Finally, for the third metamorphosis, form the matched pairs {(i, 1), (i, 2), (i, 4)}, {(i, 1), (i, 3), (i, 4)} and {(i, 1), (i, 2), (i, 3)}, {(i, 2), (i, 3), (i, 4)}, i = 1, 2, 3, 4, and for each block {(1, j1 ), (2, j2 ), (3, j3 ), (4, j4 )} ∈ B , form the matched pairs {(1, j1 ), (2, j2 ), (4, j4 )}, {(1, j1 ), (3, j3 ), (4, j4 )} and {(1, j1 ), (2, j2 ), (3, j3 )}, {(2, j2 ), (3, j3 ), (4, j4 )}. For a definition of a (twofold) transversal design TD2 (3, n) and of a (twofold) group-divisible design 3-GDD2 , see [5]. The concept of metamorphosis (or of triple metamorphosis) of a twofold triple system into a twofold 4-cycle system is easily extended to a (twofold) TD2 (3, n) or a (twofold) 3-GDD2 . We recall a result by Sotteau [13] concerning the existence of a decomposition of the complete bipartite graph Km,n into 4-cycles. Lemma 4.2. A complete bipartite graph Km,n is decomposable into 4-cycles if and only if both m and n are even. Next we record an easy lemma. A (twofold) divisible 4-cycle system is simply a decomposition of a twofold complete multipartite graph into 4-cycles.

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Lemma 4.3. Let n ≡ 0(mod 2), n ≥ 2. There exists a simple twofold transversal design TD2 (3, n) admitting a triple metamorphosis into a twofold divisible 4-cycle system. Proof. A TD(3, n) is equivalent to a latin square of order n, and a simple TD2 (3, n) is equivalent to a pair of disjoint latin squares of order n (two latin squares on the same symbol set are disjoint if for each i, j, the (i, j)-entries in the respective squares are different). Let Q1 = (X , .), Q2 = (X , ∗) be two disjoint quasigroups corresponding to the two disjoint latin squares of order n, and let the set of elements of the corresponding TD2 (3, n) be X × {1, 2, 3}. For the first metamorphosis, take the matched pairs {a1 , b2 , (a.b)3 }, {a1 , b2 , (a ∗ b)3 }, a, b ∈ X , a ̸= b. Deleting the double edge {a1 , b2 } leaves the 4-cycle (a1 , (a.b)3 , b2 , (a ∗ b)3 ). The deleted edges comprise the edge-set of the twofold complete bipartite graph 2Kn,n on (X1 , X2 ) which is decomposable into 4-cycles by Lemma 4.2 since n is even. For the second metamorphosis, take the matched pairs {a1 , b2 , (a.b)3 }, {c1 , b2 , (a.b)3 } where c is such that c ∗ b = a.b. Deleting the double edge {b2 , (a.b)3 } leaves the 4-cycle (a1 , (a.b)3 , c1 , b2 ). The deleted edges comprise the edge-set of the twofold complete bipartite graph 2Kn,n on (X2 , X3 ) which is decomposable into 4-cycles by Lemma 4.2. Finally, for the third metamorphosis, take the matched pairs {a1 , b2 , (a.b)3 }, {a1 , d2 , (a.b)3 } where d is such that a ∗ d = a.b. Deleting the double edge {a1 , (a.b)3 } leaves the 4-cycle (a1 , b2 , (a.b)3 , d2 ). The deleted edges comprise the edge-set of the twofold complete bipartite graph 2Kn,n on (X1 , X3 ) which is decomposable into 4-cycles by Lemma 4.2.  We denote the triple metamorphosis of a TD2 (3, n) by TM(Kn,n,n ). Lemma 4.4. Let X be a set, |X | = n, and let (X , M , L) be a decomposition of the complete graph Kn on X into complete subgraphs Km where m ∈ M, and complete tripartite graphs Kl,l,l where l ∈ L. If there exists a TM(Kmi ) for each mi ∈ M and a TM(Kli ,li ,li ) for each li ∈ L then there exists a TM(Kn ). Proof. Just put on each complete subgraph Km a copy of TM(Km ) and on each complete tripartite graph Kl,l,l a copy of a TM(Kl,l,l ).  Corollary 4.5. Let (V , B ) be a pairwise balanced design PBD(n, K , 1). If there exists a TM(Kk ) for each k ∈ K then there exists a TM(Kn ). 5. Triple metamorphosis: the case of n ≡ 0(mod 12) We start with an example. Example 5.1. A triple metamorphosis TM(K24 ). The following is a twofold TS(24, 2)(V , B ) : V = Z8 × {1, 2, 3}; base triples mod (8, 8, 8) are as follows: A. {01 , 11 , 52 }, B. {01 , 11 , 02 }, C. {01 , 21 , 32 }, D. {01 , 21 , 02 }, E. {01 , 31 , 52 }, F. {01 , 31 , 72 }, G. {01 , 41 , 62 }, H. {02 , 12 , 63 }, I. {02 , 12 , 13 }, J. {02 , 22 , 43 }, K. {02 , 22 , 13 }, L. {02 , 32 , 63 }, M. {02 , 32 , 03 }, N. {02 , 42 , 73 }, O. {01 , 33 , 43 }, P. {01 , 03 , 13 }, Q. {01 , 53 , 73 }, R. {01 , 03 , 63 }, S. {01 , 13 , 43 }, T. {01 , 23 , 53 }, U. {01 , 23 , 63 }, V. {01 , 12 , 33 }, W. {01 , 32 , 73 }. Matched pairs of orbits in the three respective metamorphoses (here ipj means ‘pure difference i on level j’; and imkl means ‘mixed difference i between levels k and l): I. I. AB,1p1; CD,2p1; EF,3p1; GG,4p1; IM,0m23; JW,4m23; HL,6m23; KN,7m23; PS,1m13; OV,3m13; QT,5m13; RU,6m13. II. HI,1p2; JK,2p2; LM,3p2; NN,4p2; BD,0m12; CV,1m12; EG,2m12; AF,4m12; PR,0m13; TU,2m13; OS,4m13; QW,7m13. III. OP,1p3; QR,2p3; ST,3p3; UU,4p3; CW,3m12; AE,5m12; DG,6m12; BF,7m12; IK,1m23; JV,2m23; LN,3m23;HM,5m23. Partition into 4-cycles: I. (01 , 13 , 52 , 33 ) twice; (01 , 53 , 62 , 63 ) twice; (01 , 11 , 41 , 31 ); (01 , 21 , 41 , 61 ) short; (01 , 21 , 61 , 41 ) short; (01 , 41 , 21 , 61 ) short. II. (01 , 42 , 21 , 43 ) twice; (01 , 12 , 11 , 03 ) twice; (02 , 12 , 42 , 32 ); (02 , 22 , 42 , 62 ) short; (02 , 22 , 62 , 42 ) short; (02 , 42 , 22 , 62 ) short. III. (01 , 32 , 03 , 52 ) twice; (01 , 62 , 03 , 72 ) twice; (03 , 13 , 43 , 33 ); (03 , 23 , 43 , 63 ) short; (03 , 23 , 63 , 43 ) short; (03 , 43 , 23 , 63 ) short. Lemma 5.2. Let n ∈ {48, 84}. Then there exists a TM(Kn ). Proof. For n = 48, consider a TD2 (3, 16) on Z16 × {1, 2, 3}, with groups Z16 × {i}, i = 1, 2, 3. A TM(K16,16,16 ) exists by Lemma 4.3. Put on each of the three groups a copy of TM(K16 ) from Example 4.1, and apply Lemma 4.4. For n = 84, proceed in the same way but use instead a copy of a TM(K28 ) from Example 6.1. 

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Lemma 5.3. A TM(K24n ) exists for all n ≥ 1. Proof. For n = 1, a triple metamorphosis is given in Example 5.1, and for n = 2, it is given in Lemma 5.2. So, we may assume n ≥ 3. Let (X , B ) be a 3-GDD of order 2n, with n groups g1 , . . . , gn of size 2 if n ≡ 0, 1(mod 3), and with one group g1 of size 4 and n − 2 groups g2 , g3 , . . . , gn−1 of size 2 if n ≡ 2(mod 3). If n ≡ 0, 1(mod 3), such a GDD exists for all n ≥ 3, and if n ≡ 2(mod 3), such a GDD exists for all n ≥ 5 [5]. Let V = X × T where T is a set such that |T | = 12 (and so |V | = 24n). If n ≡ 0, 1(mod 3), put on each set gi × T , i = 1, . . . , n, a copy of a TM(K24 ) from Example 5.1, and if n ≡ 2(mod 3), put on g1 × T a copy of a TM(K48 ) from Lemma 5.2, and for each Gi × T , i = 2, 3, . . . , n − 1, a copy of a TM(K24 ) from Example 5.1. For each triple {x, y, z } ∈ B , put a copy of a TM(K12,12,12 ) on {x, y, z } × T such that the groups are {x} × T , {y} × T , {z } × T . By Lemma 4.4, this results in a TM(K24n ).  In view of the absence of a solution for a TM(K12 ), we need a few more small examples in order to be able to handle the case of n ≡ 12(mod 24). Example 5.4. A TM(K36 ). The following is a twofold TS(36, 2)(V , B ) : V = Z12 × {1, 2, 3}; base triples mod (12, 12, 12) are: A. {01 , 11 , 112 }, B. {01 , 11 , 32 }, C. {01 , 21 , 82 }, D. {01 , 21 , 02 }, E. {01 , 31 , 92 }, F. {01 , 31 , 52 }, G. {01 , 41 , 42 }, H. {01 , 41 , 92 }, I. {01 , 51 , 82 }, J. {01 , 51 , 42 }, K. {01 , 61 , 72 }, L. {02 , 12 , 113 }, M. {02 , 12 , 23 }, N. {02 , 22 , 83 }, O. {02 , 22 , 03 }, P. {02 , 32 , 73 }, Q. {02 , 32 , 53 }, R. {02 , 42 , 73 }, S. {02 , 42 , 93 }, T. {02 , 52 , 83 }, U. {02 , 52 , 43 }, V. {02 , 62 , 63 }, X. {01 , 103 , 113 }, Y. {01 , 03 , 13 }, Z. {01 , 63 , 83 }, a. {01 , 03 , 103 }, b. {01 , 63 , 93 }, d. {01 , 23 , 53 }, e. {01 , 33 , 73 }, f. {01 , 53 , 93 }, g. {01 , 33 , 83 }, h. {01 , 43 , 113 }, i. {01 , 13 , 73 }, j. {01 , 12 , 23 }, m. {01 , 72 , 43 }. Matched pairs of orbits in the three respective metamorphoses (for notation, cf. Example 5.1): I. AB,1p1; CD,2p1; EF,3p1; GH,4p1; IJ,5p1; KK,6p1; Mj,1m23, RT,3m23; PU,4m23, QS,5m23; NV,6m23; LO,10m23; Yi,1m13; eg,3m13; hm,4m13; df,5m13; Zb,6m13; Xa,10m13. II. LM,1p2; NO,2p2; PQ,3p2; RS,4p2; TU,5p2; VV,6p2; DG,0m12; BF,2m12; Km,7m12; CI,8m12; EH,9m12; AJ,11m12; Ya,0m13; dj,2m13; ei,7m13; Zg,8m13; bf,9m13;Xh,11m13. III. XY,1p3; Za,2p3; bd,3p3; ef,4p3; gh,5p3; ii,6p3; Kj,1m12; BI,3m12; GJ,4m12; FH,5m12; CE,6m12; AD,10m12; OV,0m23; MQ,2m23; PR,7m23; NT,8m23; Sm,9m23; LU,11m23. Partition into 4-cycles: I. (01 , 11 , 61 , 21 ) twice; (01 , 33 , 02 , 43 ) twice; (01 , 63 , 51 , 103 ) twice, (02 , 63 , 52 , 103 ) twice; (01 , 31 , 91 , 61 ) short, (01 , 31 , 61 , 91 ) short. II. (02 , 12 , 62 , 22 ) twice; (01 , 22 , 21 , 92 ) twice; (01 , 23 , 21 , 93 ) twice; (01 , 112 , 31 , 113 ) twice; (02 , 32 , 92 , 62 ) short; (02 , 32 , 62 , 92 ) short. III. (03 , 13 , 63 , 23 ) twice; (01 , 62 , 51 , 102 ) twice; (02 , 23 , 22 , 113 ) twice; (01 , 32 , 113 , 42 ) twice; (03 , 33 , 93 , 63 ) short; (03 , 33 , 63 , 93 ) short. Lemma 5.5. There exists a TM(K108 ). Proof. Similar to Lemma 5.2, using a TM(K36 ) and a TM(K36,36,36 ).



Example 5.6. A TM(K60 ). The following is a twofold TS(60, 2)(V , B ) : V = Z20 × {1, 2, 3}; base triples mod (20, 20, 20) are: A. {01 , 11 , 92 }, B. {01 , 11 , 112 }, C. {01 , 21 , 42 }, D. {01 , 21 , 172 }, E. {01 , 31 , 22 }, F. {01 , 31 , 192 }, G. {01 , 41 , 182 }, H. {01 , 41 , 72 }, I. {01 , 51 , 52 }, J. {01 , 51 , 172 }, K. {01 , 61 , 122 }, L. {01 , 61 , 142 }, M. {01 , 71 , 132 }, N. {01 , 71 , 182 }, O. {01 , 81 , 132 }, P. {01 , 81 , 92 }, Q. {01 , 91 , 162 }, R. {01 , 91 , 42 }, S. {01 , 101 , 102 }, T. {02 , 12 , 93 }, U. {02 , 12 , 113 }, V. {02 , 22 , 43 }, W. {02 , 22 , 173 }, X. {02 , 32 , 23 }, Y. {02 , 32 , 193 }, Z. {02 , 42 , 183 }, a. {02 , 42 , 73 }, b. {02 , 52 , 53 }, d. {02 , 52 , 173 }, e. {02 , 62 , 123 }, f. {02 , 62 , 143 }, g. {02 , 72 , 133 },

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h. {02 , 72 , 183 }, i. {02 , 82 , 133 }, j. {02 , 82 , 93 }, l. {02 , 92 , 163 }, m. {02 , 92 , 43 }, n. {02 , 102 , 103 }, p. {01 , 163 , 173 }, q. {01 , 133 , 143 }, r. {01 , 13 , 33 }, s. {01 , 103 , 123 }, t. {01 , 83 , 113 }, u. {01 , 23 , 53 }, v. {01 , 33 , 73 }, w. {01 , 153 , 193 }, x. {01 , 93 , 143 }, y. {01 , 03 , 153 }, z. {01 , 43 , 183 }, α . {01 , 73 , 133 }, β . {01 , 43 , 113 }, γ . {01 , 93 , 163 }, δ . {01 , 53 , 173 }, ϵ . {01 , 63 , 183 }, λ. {01 , 13 , 123 }, µ. {01 , 83 , 193 }, ρ . {01 , 03 , 103 }, σ . {01 , 12 , 23 }, τ . {01 , 32 , 63 }. Matched pairs of orbits in the three metamorphoses: I. AB,1p1; CD,2p1; EF,3p1; GH,4p1; IJ,5p1; KL,6p1; MN,7p1; OP,8p1; QR,9p1; SS,10p1; jσ ,1m23; Vm,4m23; ib,5m23; eg,6m23; al,7m23; Tf,8m23; Un,10m23; Wd,17m23; Zh,18m23; XY,19m23; rλ,1m13; zβ ,4m13; uδ ,5m13; ϵτ ,6m13; vα ,7m13; tµ,8m13; sρ ,10m13; qx,14m13; wy,15m13; pγ ,16m13. II. TU,1p2; VW,2p2; XY,3p2; Za,4p2; bd,5p2; ef,6p2; gh,7p2; ij,8p2; lm,9p2; nn,10p2; IS,0m12; CE,2m12; Hτ ,3m12; AP,9m12; BN,11m12; JK,12m12; MO,13m12; GL,14m12; DR,15m12; FQ,16m12; yρ ,0m13; uσ ,2m13; rv,3m13; xγ ,9m13; tβ ,11m13; sλ,12m13; qα ,13m13; pδ ,17m13; zϵ ,18m13; wµ,19m13. III. pq,1p3; rs,2p3; tu,3p3; vw,4p3; xy,5p3; zα ,6p3; βγ ,7p3; δϵ ,8p3; λµ,9p3; ρρ ,10p3; Pσ ,1m12; CR,4m12; IO,5m12; KM,6m12; HQ,7m12; AL,8m12; BS,10m12; DJ,17m12; GN,18m12; EF,19m12; bn,0m23; VX,2m23; aτ , 3m23; Tj,9m23; Uh,11m23; de,12m23; gi,13m23; Zf,14m23; Wm,15m23; Yl,16m23. Partition into 4-cycles: I. (01 , 11 , 81 , 21 ) twice; (01 , 31 , 121 , 41 ) twice; (01 , 73 , 11 , 163 ) twice; (02 , 103 , 92 , 173 ) twice; (01 , 103 , 61 , 143 ); (01 , 103 , 91 , 143 ); (61 , 103 , 91 , 143 ); (02 , 63 , 22 , 73 ); (02 , 63 , 82 , 73 ); (22 , 63 , 82 , 73 ); (01 , 51 , 151 , 101 ) short; (01 , 51 , 101 , 151 ) short. II. (02 , 12 , 82 , 22 ) twice; (02 , 32 , 122 , 42 ) twice; (01 , 123 , 101 , 133 ) twice; (01 , 152 , 21 , 162 ) twice; (01 , 93 , 111 , 113 ); (01 , 93 , 121 , 113 ); (111 , 93 , 121 , 113 ); (01 , 32 , 11 , 122 ); (01 , 32 , 31 , 122 ); (11 , 32 , 31 , 122 ); (02 , 52 , 152 , 102 ) short; (02 , 52 , 102 , 152 ) short. III. (03 , 13 , 83 , 23 ) twice; (03 , 33 , 123 , 43 ) twice; (01 , 102 , 91 , 172 ) twice; (02 , 153 , 22 , 163 ) twice; (01 , 62 , 21 , 72 ); (01 , 62 , 81 , 72 ); (21 , 62 , 81 , 72 ); (02 , 33 , 12 , 123 ); (02 , 33 , 32 , 123 ); (12 , 33 , 32 , 123 ); (03 , 53 , 153 , 103 ) short; (03 , 53 , 103 , 153 ) short. Lemma 5.7. There exists a TM(K132 ). Proof. Let (X22 , C ) be a 3-GDD of order 22 with one group g1 of size 4 and three groups g2 , g3 , g4 of size 6 each; such a 3-GDD is known to exist [5]. Let V = X22 × T where T is such that |T | = 6. On g1 × T , put a copy of a TM(K24 ) from Example 5.1; on gi × T , i = 2, 3, 4, put a copy of a TM(K36 ) from Example 5.4. If {x, y, z } is a triple of C , put on {x, y, z } × T a copy of a TM(K6,6,6 ) (Lemma 4.3) so that the groups are {x} × T , {y} × T , {z } × T . Now apply Lemma 4.4.  Lemma 5.8. There exists a TM(K156 ). Proof. Let (X26 , D ) be a 3-GDD of order 26 with one group g1 of size 8 and three groups g2 , g3 , g4 of size 6 each; such a GDD exists by [5]. Let W = X26 × T where |T | = 6. On g1 × T , put a copy of a TM(K48 ) from Lemma 5.2; on gi × T , i = 2, 3, 4, put a copy of a TM(K36 ) from Example 5.4. The rest is as in the proof of Lemma 5.7.  Lemma 5.9. A TM(K24n+12 ) exists for all n ≥ 1. Proof. For n ∈ {1, 2, 3, 4, 5, 6} the statement follows from Examples 5.4 and 5.6, Lemmas 5.2, 5.5, 5.7 and 5.8, respectively. Let now n ≥ 7, and let (X , B ) be a 3-GDD of order 2n + 1 with groups g1 , . . . , gm where the group sizes |gi | ∈ K = {3, 5, 7}. Such a 3-GDD exists for all n ≥ 7 [5]. Let V = X × U where |U | = 12. On each gi × U , i = 1, . . . , m, put a copy of a TM(K36 ) (Example 5.4), TM(K60 ) (Example 5.6), or TM(K84 ) (Lemma 5.2) depending on whether |gi | = 3, 5 or 7. For each triple {x, y, z } ∈ B , put a copy of a TM(K12,12,12 ) from Lemma 4.3 on {x, y, z } × U, with groups {x} × U , {y} × U , {z } × U. It now remains only to invoke Lemma 4.4.  We summarize the results of this section in the following theorem. Theorem 5.10. For all n ≡ 0(mod 12), n ≥ 24, there exists a triple metamorphosis TM(Kn ) of a TS(n, 2) into a 4-cycle system 4C(n, 2). 6. Triple metamorphosis: the case of n ≡ 4(mod 12) We start again with some needed small examples.

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Example 6.1. A triple metamorphosis TM(K28 ). The following is a twofold TS(28, 2) : V = Z7 × {1, 2, 3, 4}; base triples mod (7, 7, 7, 7) are: A. {01 , 11 , 23 }, B. {01 , 11 , 63 }, C. {01 , 21 , 32 }, D. {01 , 21 , 62 }, E. {01 , 31 , 44 }, F. {01 , 31 , 64 }, G. {02 , 12 , 24 }, H. {02 , 12 , 64 }, I. {01 , 42 , 62 }, J. {01 , 12 , 32 }, K. {02 , 32 , 43 }, L. {02 , 32 , 63 }, M. {01 , 53 , 63 }, N. {01 , 13 , 23 }, O. {03 , 23 , 34 }, P. {03 , 23 , 64 }, Q. {02 , 33 , 63 }, R. {02 , 13 , 43 }, S. {02 , 54 , 64 }, T. {02 , 14 , 24 }, U. {03 , 44 , 64 }, V. {03 , 14 , 34 }, W. {01 , 34 , 64 }, X. {01 , 14 , 44 }, Y. {01 , 02 , 03 }, Z. {01 , 02 , 04 }, a. {01 , 03 , 04 }, b. {02 , 03 , 04 }, d. {01 , 22 , 43 }, e. {01 , 52 , 24 }, f. {01 , 43 , 24 }, g. {02 , 23 , 44 }, h. {01 , 52 , 33 }, i. {01 , 22 , 54 }, j. {01 , 33 , 54 }, m. {02 , 53 , 34 }. Matched pairs of orbits in the three metamorphoses: I. EF,3p1; KL,3p2; QR,3p3; WX,3p4; di,2m12; eh,5m12; CJ,3m12; DI,4m12; AN,2m13; BM,5m13; GT,2m24; HS,5m24; gj,2m34; fm,5m34; OV,3m34; PU,4m34; Za,0m14; Yb,0m23; II. CJ,1m12; DI,6m12; OV,1m34; PU,6m34; Ya,0m13; AN,1m13; BM,6m13; Zb,0m24; GT,1m24; HS,6m24; EX,1m14, ef,2m14; ij,5m14; FW,6m14; KR,1m23; dg,2m23; hm,5m23; LQ,6m23. III. AB,1p1; CD,2p1; GH,1p2; IJ, 2p2; MN,1p3; OP,2p3; ST,1p4; UV,2p4; YZ,0m12; ab,0m34; hj,3m13; df,4m13; im,3m24; eg,4m24; FW,3m14; EX,4m14; LQ,3m23; KR,4m23. Partition into 4-cycles: I. (01 , 31 , 02 , 32 ) twice; (03 , 33 , 04 , 34 ) twice; (01 , 52 , 04 , 53 ) twice; (01 , 22 , 04 , 23 ); (01 , 22 , 23 , 04 ); (01 , 23 , 22 , 04 ). II. (01 , 62 , 03 , 64 ) twice; (01 , 13 , 62 , 54 ) twice; (01 , 63 , 12 , 24 ) twice; (01 , 12 , 03 , 14 ); (01 , 03 , 12 , 14 ); (01 , 12 , 14 , 03 ). III. (01 , 43 , 02 , 44 ) twice; (01 , 11 , 31 , 21 ); (02 , 12 , 32 , 22 ); (03 , 13 , 33 , 23 ); (04 , 14 , 34 , 24 ); (01 , 33 , 02 , 34 ); (01 , 02 , 33 , 34 ); (01 , 02 , 34 , 33 ); Example 6.2. A triple metamorphosis TM(K52 ). The following is a twofold TS(52, 2)(V , B ) : V = Z13 × {1, 2, 3, 4}; the 68 base triples mod (13, 13, 13, 13) are: A. {01 , 11 , 22 }, B. {01 , 11 , 122 }, C. {01 , 21 , 44 }, D. {01 , 21 , 114 }, E. {01 , 31 .73 }, F. {01 , 31 , 93 }, G. {01 , 41 , 72 }, H. {01 , 41 , 102 }, I. {01 , 51 , 64 }, J. {01 , 51 , 124 }, K. {01 , 61 , 83 }, L. {01 , 61 , 113 }, M. {01 , 12 , 22 }, N. {01 , 112 , 122 }, O. {02 , 22 , 43 }, P. {02 , 22 , 113 }, Q. {02 , 32 , 74 }, R. {02 , 32 , 94 }, S. {01 , 32 , 72 }, T. {01 , 62 , 102 }, U. {02 , 52 , 63 }, V. {02 , 52 , 123 }, W. {02 , 62 , 84 }, X. {02 , 62 , 114 }, Y. {03 , 13 , 24 }, Z. {03 , 13 , 124 }, a. {03 , 23 , 112 }, b. {03 , 23 , 42 }, c. {01 , 43 , 73 }, d. {01 , 63 , 93 }, e. {03 , 43 , 74 }, f. {03 , 43 , 104 }, g. {02 , 13 , 63 }, h. {02 , 73 , 123 }, i. {01 , 23 , 83 }, j. {01 , 53 , 113 }, k. {03 , 14 , 24 }, l. {03 , 114 , 124 }, m. {01 , 24 , 44 }, n. {01 , 94 , 114 }, o. {02 , 44 , 74 }, p. {02 , 64 , 94 }, q. {03 , 34 , 74 }, r. {03 , 64 , 104 }, s. {01 , 14 , 64 }, t. {01 , 74 , 124 }, u. {02 , 24 , 84 }, v. {02 , 54 , 114 }, w. {01 , 02 , 03 }, x. {01 , 02 , 04 }, y. {01 , 03 , 04 }, z. {02 , 03 , 04 }, α . {01 , 52 , 103 }, β . {01 , 82 , 54 }, γ . {01 , 103 , 54 }, δ . {02 , 53 , 104 }, ϵ . {01 , 82 , 33 }, θ . {01 , 52 , 84 }, κ . {01 , 33 , 84 }, λ. {02 , 83 , 34 }, µ. {01 , 42 , 13 }, ξ . {01 , 92 , 104 }, π . {01 , 13 , 104 }, ρ . {02 , 103 , 14 }, σ . {01 , 92 , 123 }, τ . {01 , 42 , 34 }, χ . {01 , 123 , 34 }, ω. {02 , 33 , 124 }. Matched pairs of orbits in the three metamorphoses: I. AB,1p1; GH,4p1; MN,1p2; ST,4p2; YZ,1p3; ef,4p3; kl,1p4; qr,4p4; µπ , 1m13; Fd,6m13; Ec,7m13; σ χ ,12m13; ξ ρ ,1m24, Rp,6m24; Qo,7m24; τ ω,12m24; Cm,2m14; Is,6m14; Jt,7m14; Dn,11m14; Oa,2m23; Ug,6m23; Vh,7m23; Pb,11m23; αθ ,5m12; βϵ ,8m12; Lj,5m13; Ki,8m13; Xv,5m24; Wu,8m24; δκ ,5m34; γ λ,8m34; xy,0m14; wz,0m23. II. EF,3p1; KL,6p1; QR,3p2; WX,6p2; cd,3p3; ij,6p3; op,3p4; uv,6p4; AM,1m12; GS,3m12; HT,6m12; BN,11m12; Yk,1m34; eq,3m34; fr,6m34; Zl,11m34; Is,1m14; βγ ,5m14; θ κ ,8m14; Jt,12m14; Ug,1m23; αδ ,5m23; ϵλ,8m23; Vh,12m23; µτ ,4m12; ξ σ ,9m12; ρχ ,4m34; π ω,9m34; Cm,4m14; Dn,9m14; Oa,4m23; Pb,9m23; wy,0m13; xz,0m24. III. CD,2p1; IJ,5p1; OP,2p2; UV,5p2; ab,2p3; gh,5p3; mn,2p4; st,5p4; AM,2m12; GS,7m12; HT,10m12; BN,12m12; Yk,2m34; eq,7m34; fr,10m34; Zl,12m34; Ki,2m13; Ec,4m13; Fd,9m13; Lj,11m13; Wu,2m24; Qo,4m24; Rp,9m24; Xv,11m24; ϵκ ,3m13; αγ ,10m13; θ λ,3m24; βδ ,10m24; τ χ ,3m14; ξ π ,10m14; σ ω,3m23; µρ ,10m23; wx,0m12; yz,0m34.

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Partition into 4-cycles: I. (01 , 11 , 51 , 41 ); (02 , 12 , 52 , 42 ); (03 , 13 , 53 , 43 ); (04 , 14 , 54 , 44 ); (01 , 63 , 51 , 123 ) twice; (02 , 64 , 52 , 124 ) twice; (01 , 74 , 51 , 114 ) twice; (02 , 73 , 52 , 113 ) twice; (01 , 82 , 04 , 83 ) twice; (01 , 52 , 53 , 04 ); (01 , 53 , 52 , 04 ); (01 , 52 , 04 , 53 ). II. (01 , 31 , 91 , 61 ); (02 , 32 , 92 , 62 ); (03 , 33 , 93 , 63 ); (04 , 34 , 94 , 64 ); (01 , 32 , 51 , 62 ) twice; (03 , 34 , 53 , 64 ) twice; (01 , 84 , 71 , 124 ) twice; (02 , 83 , 72 , 123 ) twice; (01 , 92 , 03 , 94 ) twice; (01 , 42 , 03 , 44 ); (01 , 03 , 42 , 44 ); (01 , 42 , 44 , 03 ). III. (01 , 21 , 71 , 51 ), (02 , 22 , 72 , 52 ); (03 , 23 , 73 , 53 ); (04 , 24 , 74 , 54 ); (01 , 72 , 81 , 102 ) twice; (03 , 74 , 83 , 104 ) twice; (01 , 93 , 71 , 113 ) twice; (02 , 94 , 72 , 114 ) twice; (01 , 103 , 02 , 104 ) twice; (01 , 02 , 33 , 34 ); (01 , 33 , 02 , 34 ); (01 , 02 , 34 , 33 ). Example 6.3. A TM(K76 ). The following is a twofold TS(76, 2)(V , B ) : V = Z19 × {1, 2, 3, 4}; the 100 base triples mod (19, 19, 19, 19) are: A. {01 , 11 , 92 }, B. {01 , 11 , 152 }, C. {01 , 21 , 103 }, D. {01 , 21 , 113 }, E. {01 , 31 , 14 }, F. {01 , 31 , 24 }, G. {01 , 41 , 84 }, H. {01 , 41 , 154 }, I. {01 , 51 , 23 }, J. {01 , 51 , 33 }, K. {01 , 61 , 94 }, L. {01 , 61 , 164 }, M. {01 , 71 , 42 }, N. {01 , 71 , 52 }, O. {01 , 81 , 102 }, P. {01 , 81 , 112 }, Q. {01 , 91 , 133 }, R. {01 , 91 , 153 }, S. {01 , 82 , 92 }, T. {01 , 142 , 152 }, U. {02 , 22 , 104 }, V. {02 , 22 , 114 }, W. {02 , 32 , 13 }, X. {02 , 32 , 23 }, Y. {02 , 42 , 83 }, Z. {02 , 42 , 153 }, a. {02 , 52 , 24 }, b. {02 , 52 , 34 }, d. {02 , 62 , 93 }, e. {02 , 62 , 163 }, f. {01 , 42 , 162 }, g. {01 , 52 , 172 }, h. {01 , 22 , 102 }, i. {01 , 32 , 112 }, j. {02 , 92 , 134 }, l. {02 , 92 , 154 }, m. {03 , 13 , 94 }, n. {03 , 13 , 154 }, p. {01 , 83 , 103 }, q. {01 , 93 , 113 }, r. {02 , 13 , 173 }, s. {02 , 23 , 183 }, t. {02 , 43 , 83 }, u. {02 , 113 , 153 }, w. {01 , 23 , 163 }, y. {01 , 33 , 173 }, α . {02 , 33 , 93 }, β . {02 , 103 , 163 }, γ . {03 , 73 , 44 }, δ . {03 , 73 , 54 }, A. {03 , 83 , 104 }, B. {03 , 83 , 114 }, C. {01 , 43 , 133 }, D. {01 , 63 , 153 }, E. {03 , 84 .94 }, F. {03 , 144 , 154 }, G. {02 , 84 , 104 }, H. {02 , 94 , 114 }, I. {01 , 14 , 174 }, J. {01 , 24 , 184 }, K. {01 , 44 , 84 }, L. {01 , 114 , 154 }, M. {02 , 24 , 164 }, N. {02 , 34 , 174 }, O. {01 , 34 , 94 }, P. {01 , 104 , 164 }, Q. {03 , 44 , 164 }, R. {03 , 54 , 174 }, S. {03 , 24 , 104 }, T. {03 , 34 , 114 }, U. {02 , 44 , 134 }, V. {02 , 64 , 154 }, W. {01 , 02 , 03 }, X. {01 , 02 , 04 }, Y. {01 , 03 , 04 }, Z. {02 , 03 , 04 }, a. {01 , 72 , 143 }, b. {01 , 122 , 74 }, d. {01 , 143 , 74 }, e. {02 , 73 , 144 }, f. {01 , 122 , 53 }, g. {01 , 72 , 124 }, h. {01 , 53 , 124 }, i. {02 , 123 , 54 }, j. {01 , 62 , 13 }, l. {01 , 132 , 144 }, m. {01 , 132 , 183 }, n. {01 , 62 , 54 }, p. {01 , 13 , 144 }, q. {02 , 143 , 14 }, r. {01 , 183 , 54 }, s. {02 , 53 , 184 }, t. {01 , 12 , 73 }, u. {01 , 182 , 64 }, w. {01 , 73 , 64 }, y. {02 , 63 , 74 }, α . {01 , 182 , 123 }, β . {01 , 12 , 134 }, γ . {01 , 123 , 134 }, δ . {02 , 133 , 124 }, Matched pairs of orbits in the three metamorphoses: I. AB,1p1; MN,7p1; OP,8p1; ST,1p2; fg,7p2; hi,8p2; mn,1p3; γ δ ,7p3; AB,8p3; EF,1p4; QR,7p4; ST,8p4; jp,1m13, Iw,2m13; QC,4m13; Dq,9m13; Cp,10m13; RD,15m13; Jy,17m13; mr,18m13; lq,1m24; aM,2m24; jU,4m24; VH,9m24; UG,10m24; lV,15m24; bN,17m24; ns,18m24; FJ,2m14; KO,3m14; GK,4m14; HL,15m14; LP,16m14; EI,17m14; Xs,2m23; dα ,3m23; Yt,4m23; Zu,15m23; eβ ,16m23; Wr,17m23; ag,7m12; bf,12m12; tw,7m13; αγ ,12m13; uy,7m24; βδ ,12m24; eh,7m34; di,12m34; XY,0m14; WZ,0m23. II. EF,3p1; GH,4p1; KL,6p1; WX,3p2; YZ,4p2; de,6p2; rs,3p3; tu,4p3; αβ ,6p3; IJ,3p4; KL,4p4; OP,6p4; Jy,3m13; RD,6m13; Cp,8m13; Dq,11m13; QC,13m13; Iw,16m13; bN,3m24; lV,6m24; UG,8m24; VH,11m24; jU,13m24; aM,16m24; tβ ,1m12; Oh,2m12; Mf,4m12; AS,9m12; Pi,11m12; BT,15m12; Ng,17m12; uα ,18m12; yγ ,1m34; AS,2m34; γ Q,4m34; mE,9m34; BT,11m34; nF,15m34; δ R,17m34; wδ ,18m34; fh,5m13; ad,14m13; nr,5m14; lp,14m14; ms,5m23; jq,14m23; gi,5m24; be,14m24; WX,0m12; YZ,0m34. III. CD,2p1; IJ,5p1; QR,9p1; UV,2p2; ab,5p2; jl,9p2; pq,2p3; wy,5p3; CD,9p3; GH,2p4; MN,5p4; UV,9p4; Pi,3m12, Ng,5m12; AS,8m12; Oh,10m12; BT,14m12; Mf,16m12; BT,3m34; δ R,5m34; mE,8m34; AS,10m34; nF,14m34; γ Q,16m34; EI,1m14; bd,7m14; GK,8m14; KO,9m14; LP,10m14; HL,11m14; gh,12m14; FJ,18m14; Wr,1m23; ae,7m23; Yt,8m23; dα ,9m23; eβ ,10m23; Zu,11m23; fi,12m23; Xs,18m23; jn,6m12; lm,13m12; qr,6m34; ps,13m34; uw,6m14; βγ ,13m14; ty,6m23; αδ ,13m23; WY,0m13; XZ,0m24. Partition into 4-cycles: I. (01 , 81 , 151 , 71 ); (02 , 82 , 152 , 72 ); (03 , 83 , 153 , 73 ); (04 , 84 , 154 , 74 ); (01 , 11 , 34 , 24 ); (01 , 11 , 164 , 154 ); (02 , 12 , 33 , 23 ); (02 , 12 , 163 , 153 ); (01 , 153 , 51 , 93 ) twice; (01 , 13 , 31 , 23 ) twice; (02 , 154 , 52 , 94 ) twice; (02 , 14 , 32 , 24 ) twice; (01 , 34 , 61 , 44 ) twice; (02 , 33 , 62 , 43 ) twice; (01 , 122 , 04 , 123 ) twice; (01 , 72 , 73 , 04 ); (01 , 73 , 72 , 04 ); (01 , 72 , 04 , 73 ).

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II. (01 , 41 , 101 , 61 ); (02 , 42 , 102 , 62 ); (03 , 43 , 103 , 63 ); (04 , 44 , 104 , 64 ); (01 , 31 , 63 , 33 ); (01 , 31 , 93 , 63 ); (02 , 32 , 64 , 34 ); (02 , 32 , 94 , 64 ); (01 , 113 , 31 , 163 ) twice; (02 , 114 , 32 , 164 ) twice; (01 , 22 , 31 , 42 ) twice; (01 , 92 , 131 , 112 ) twice; (03 , 24 , 33 , 44 ) twice; (03 , 94 , 133 , 114 ) twice; (01 , 143 , 02 , 144 ) twice; (01 , 02 , 53 , 54 ); (01 , 53 , 02 , 54 ); (01 , 02 , 54 , 53 ). III. (01 , 91 , 141 , 51 ); (02 , 92 , 142 , 52 ); (03 , 93 , 143 , 53 ); (04 , 94 , 144 , 54 ); (01 , 21 , 52 , 32 ); (01 , 21 , 72 , 52 ); (03 , 23 , 54 , 34 ); (03 , 23 , 74 , 54 ); (01 , 142 , 61 , 162 ) twice; (03 , 144 , 63 , 164 ) twice; (01 , 114 , 41 , 124 ) twice; (01 , 104 , 91 , 184 ) twice; (02 , 113 , 42 , 123 ) twice; (02 , 103 , 92 , 183 ) twice; (01 , 132 , 03 , 134 ) twice; (01 , 62 , 03 , 64 ); (01 , 03 , 62 , 64 ); (01 , 62 , 64 , 03 ). Lemma 6.4. There exists a TM(K100 ). Proof. Let (X50 , B ) be a 3-GDD of order 50, with one group g1 of size 14 and three groups, g2 , g3 , g4 , each of size 12. Such a GDD is known to exist [5]. Consider V = X50 × {1, 2}; put on g1 × {1, 2} a copy of a TM(K28 ) from Example 6.1, and put on gi × {1, 2}, i = 1, 2, a copy of a TM(K24 ) from Example 5.1. If {x, y, z } ∈ B , put on {x, y, z } × {1, 2} a copy of a TM(K2,2,2 ) which exists by Lemma 4.3. Now invoking Lemma 4.4 yields a TM(K100 ) on V .  Lemma 6.5. There exists a TM(K124 ). Proof. The proof is exactly as in the preceding Lemma 6.4 but start instead with a 3-GDD of order 62 having one group of size 14 and four groups of size 12.  Lemma 6.6. There exists a TM(K148 ). Proof. Proceed exactly as in the proof of Lemma 6.4 but start instead with a 3-GDD of order 74 with one group of size 20 and three groups of size 18.  Lemma 6.7. A TM(K24n+4 ) exists for all n ≥ 1. Proof. Let (X , B ) be a 3-GDD of order 6n + 1 with groups g1 , . . . , gm where |gi | ∈ K = {7, 9, 13, 19}. It follows from [5], IV., Lemma 4.2 that such a GDD exists for all n ≥ 7, i.e. 6n + 1 ≥ 43. Now let V = X ×{1, 2, 3, 4}. On each set gi ×{1, 2, 3, 4}, put a copy of a TM(K28 ) (Example 6.1), or TM(K36 ) (Example 5.4), or TM(K52 ) (Example 6.2) or TM(K76 ) (Example 6.3), depending on whether |gi | = 7, 9, 13 or 19. If {x, y, z } ∈ B , put on {x, y, z } × {1, 2, 3, 4} a copy of a TM(K4,4,4 ) (Lemma 4.3) in such a way that the groups are {x} × {1, 2, 3, 4}, {y} × {1, 2, 3, 4}, {z } × {1, 2, 3, 4}. The statement now follows by applying Lemma 4.4.  In addition to an example of a TM(K16 ) given earlier (Example 4.1), we also need an example of a triple metamorphosis for order n = 40. Example 6.8. A TM(K40 ). The following is a twofold TS(40, 2) : V = Z10 × {1, 2, 3, 4}; the base triples mod (10, 10, 10, 10) are: A. {01 , 11 , 12 }, B. {01 , 11 , 03 }, C. {01 , 21 , 62 }, D. {01 , 21 , 53 }, E. {01 , 31 , 34 }, F. {01 , 31 , 84 }, G. {01 , 41 , 92 }, H. {01 , 41 , 13 }, I. {01 , 51 , 74 }, J. {01 , 02 , 12 }, K. {02 , 12 , 04 }, L. {01 , 42 , 62 }, M. {02 , 22 , 54 }, N. {02 , 32 , 03 }, O. {02 , 32 , 53 }, P. {01 , 52 , 92 }, Q. {02 , 42 , 14 }, R. {02 , 52 , 83 }, S. {03 , 13 , 14 }, T. {01 , 03 , 93 }, U. {03 , 23 , 64 }, V. {01 , 33 , 53 }, W. {02 , 03 , 73 }, X. {02 , 23 , 53 }, Y. {03 , 43 , 94 }, Z. {01 , 13 , 73 }, a. {02 , 33 , 83 }, b. {03 , 04 , 14 }, c. {02 , 04 , 94 }, d. {03 , 44 , 64 }, e. {02 , 34 , 54 }, f. {01 , 04 , 34 }, g. {01 , 54 , 84 }, h. {03 , 54 , 94 }, i. {02 , 14 , 74 }, j. {01 , 24 , 74 }, k. {01 , 22 , 83 }, l. {01 , 22 , 64 }, m. {01 , 83 , 64 }, n. {02 , 63 , 44 }, o. {01 , 82 , 23 }, p. {01 , 82 , 44 }, q. {01 , 23 , 44 }, r. {02 , 43 , 64 , }, s. {01 , 32 , 43 }, t. {01 , 32 , 14 }, u. {01 , 43 , 14 }, v. {02 , 13 , 84 }, w. {01 , 72 , 63 }, x. {01 , 72 , 94 }, y. {01 , 63 , 94 }, z. {02 , 93 , 24 }. Matched pairs of orbits in the three metamorphoses: I. AB,1p1; EF,3p1; JK,1p2; NO,3p2; ST,1p3; WX,3p3; bc,1p4; fg,3p4; GP,5m12; DV,3m13; Ij,2m14; Ra,8m23; Qi,7m24; Yh,9m34; CL,6m12; HZ,7m13; Me,5m24; Ud,4m34; oq,2m13; su,4m13; wy,6m13; km,8m13; xz,2m24; ln,4m24; pr,6m24; tv,8m24. II. CD,2p1; GH,4p1; LM,2p2; PQ,4p2; UV, 2p3; YZ,4p3; de,2p4; hi,4p4; II,5p1; RR,5p2; aa,5p3; jj,5p4; Fg,5m14; OX,5m23; AJ,0m12; BT,0m13; Ef,0m14; NW,0m23; Kc,0m24; Sb,0m34; kl,2m12; st,3m12; wx,7m12; op,8m12; qr,2m34; yz,3m34; uv,7m34; mn,8m34. III. CL,4m12; HZ,1m13; Ij,7m14; NW,7m23; Me,3m24; Ud,6m34; GP,9m12; Ef,3m14; Ra,3m23; Sb,1m34; BT,9m13; Fg,8m14, OX,2m23; Qi,1m24; AJ,1m12; DV,5m13; Kc,9m24; Yh,5m34; sv,1m23; or,4m23; kn,6m23; wz,9m23; tu,1m14; pq,4m14; lm,6m14; xy,9m14.

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Partition into 4-cycles: I. (01 , 11 , 41 , 31 ); (02 , 12 , 42 , 32 ); (03 , 13 , 43 , 33 ); (04 , 14 , 44 , 34 ); (01 , 52 , 24 , 33 ); (01 , 52 , 33 , 24 ); (01 , 33 , 52 , 24 ); (01 , 62 , 14 , 73 ) twice; (01 , 63 , 41 , 83 ) twice; (02 , 64 , 42 , 84 ) twice. II. (01 , 21 , 61 , 41 ); (02 , 22 , 62 , 42 ); (03 , 23 , 63 , 43 ); (04 , 24 , 64 , 44 ); (01 , 02 , 04 , 03 ); (01 , 02 , 03 , 04 ); (01 , 03 , 02 , 04 ); (01 , 51 , 04 , 54 ); (02 , 52 , 03 , 53 ); (01 , 72 , 51 , 82 ) twice; (03 , 74 , 53 , 84 ) twice. III. (01 , 42 , 74 , 13 ); (01 , 42 , 13 , 74 ); (01 , 13 , 42 , 74 ); (01 , 92 , 23 , 34 ) twice; (01 , 93 , 72 , 84 ) twice; (01 , 12 , 04 , 53 ) twice; (01 , 64 , 51 , 94 ) twice; (02 , 63 , 52 , 93 ) twice. Lemma 6.9. There exists a TM(K64 ). Proof. Let (X8 , B ) be a 3-GDD of order 8 with four groups gi , i = 1, 2, 3, 4 of size 2. Let W be a set, |W | = 8. Consider V = X8 × W ; for i = 1, 2, 3, 4, put on gi × W a copy of a TM(K16 ) from Example 4.1. For each block {x, y, z } ∈ B , put on {x, y, z } × W a copy of a TM(K8,8,8 ) (Lemma 4.3) so that {x} × W , {y} × W , {z } × W are the groups. Invoking Lemma 4.4 completes the proof.  Lemma 6.10. A TM(K24n+16 ) exists for all n ≥ 0. Proof. For n = 0, 1, 2, the statement follows from Examples 4.1, 6.8, and Lemma 6.9, respectively. So assume n ≥ 3, and let (V , B ) be a 3-GDD of order 6n + 4 where each group is of size 4, 6 or 10; such a GDD is known to exist for all n ≥ 3 [5]. Let X = V × Z where |Z | = 4. For each group g, put on g × Z a copy of a TM(K16 ) from Example 4.1, or a copy of a TM(K24 ) from Example 5.1 or a copy of a TM(K40 ) from Example 6.8, depending on whether |g | = 4, 6 or 10. For each block {x, y, z } ∈ B , put on {x, y, z } × Z a copy of a TM(K4,4,4 ) from Lemma 4.3, and apply Lemma 4.4.  We summarize the results of this section in the following theorem. Theorem 6.11. For all n ≡ 4(mod 12), n ≥ 16, there exists a triple metamorphosis TM(Kn ) of a TS(n, 2) into a 4-cycle system 4C(n, 2). 7. Triple metamorphosis: the case of n ≡ 9(mod 12) First we deal with the case of n ≡ 21(mod 24). As in the preceding sections, we start with necessary small examples. Example 7.1. A TM(K21 ). The following is a simple TS(21, 2)(V , B ) : V = Z7 × {1, 2, 3}; base triples mod (7, 7, 7) are: A. {01 , 11 , 52 }, B. {01 , 11 , 12 }, C. {01 , 21 , 22 }, D. {01 , 21 , 12 }, E. {01 , 31 , 52 }, F. {01 , 31 , 62 }, G. {02 , 12 , 53 }, H. {02 , 12 , 33 }, I. {02 , 22 , 23 }, J. {02 , 22 , 13 }, K. {02 , 32 , 13 }, L. {02 , 32 , 63 }, M. {31 , 03 , 13 }, N. {51 , 03 , 13 }, O. {01 , 03 , 23 }, P. {11 , 03 , 23 }, R. {21 , 03 , 33 }, S. {41 , 03 , 33 }, T. {01 , 32 , 03 }, U. {31 , 02 , 03 }. Matched pairs of orbits in the three respective metamorphoses: I. AB,1p1; CD,2p1; EF,3p1; JK,1m23; HL,3m23; GT,4m23; IU,0m23; PS,6m13; MR,5m13; NO,2m13. II. GH,1p2; IJ,2p2; KL,3p2; BC,0m12; AE,5m12; DF,6m12; OT,0m13; MU,4m13; NS,3m13; PR,1m13. III. MN,1p3; OP,2p3; RS,3p3; AU,4m12; FT,3m12; CE,2m12; BD,1m12; GK,5m23; HI,2m23; JL,6m23. Partition into 4-cycles: I. (02 , 13 , 12 , 43 ) twice; (01 , 31 , 11 , 63 ); (01 , 11 , 31 , 53 ); (01 , 11 , 41 , 63 ). II. (01 , 13 , 11 , 43 ) twice; (01 , 02 , 42 , 52 ); (01 , 02 , 52 , 62 ); (01 , 52 , 32 , 62 ). III. (01 , 22 , 11 , 42 ) twice; (03 , 13 , 33 , 52 ); (03 , 33 , 43 , 52 ); (03 , 33 , 13 , 22 ). Example 7.2. A TM(K45 ). The following is a simple TS(45, 2)(V , B ) : V = Z15 × {1, 2, 3}; base triples mod (15, 15, 15) are: A. {01 , 11 , 02 }, B. {01 , 11 , 12 }, C. {01 , 21 , 32 }, D. {01 , 21 , 92 }, E. {01 , 31 , 52 }, F. {01 , 31 , 132 }; G. {01 , 41 , 62 }, H. {01 , 41 , 142 }, I. {01 , 51 , 132 }, J. {01 , 51 , 92 }, K. {01 , 61 , 112 }, L. {01 , 61 , 122 }, M. {01 , 71 , 32 }, N. {01 , 71 , 42 }, O. {02 , 12 , 03 }, P. {02 , 12 , 23 }, Q. {02 , 22 , 13 }, R. {02 , 22 , 43 }, S. {02 , 32 , 63 }, T. {02 , 32 , 93 }, U. {02 , 42 , 113 }, V. {02 , 42 , 133 }, W. {02 , 52 , 123 }, X. {02 , 52 , 133 }, Y. {02 , 62 , 103 }, Z. {02 , 62 , 113 }, a. {02 , 72 , 103 }, b. {02 , 72 , 123 }, d. {01 , 03 , 13 }, e. {21 , 03 , 13 }, f. {11 , 03 , 23 }, g. {91 , 03 , 23 }, h. {61 , 03 , 33 }, i. {131 , 03 , 33 }, j. {81 , 03 , 43 }, m. {91 , 03 , 43 }, n. {131 , 03 , 53 }, p. {101 , 03 , 53 }, q. {21 , 03 , 63 }, r. {121 , 03 , 63 }, t. {41 , 03 , 73 }, u. {31 , 03 , 73 }, x. {01 , 72 , 03 }, y. {71 , 02 , 03 }.

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Matched pairs of orbits in the three metamorphoses: I. AB,1p1; CD,2p1; EF,3p1; GH,4p1; IJ,5p1; KL,6p1; MN,7p1; Oy,0m23; Xx,8m23; PQ,1m23; RY,4m23; Sa,3m23; TV,9m23; UZ,11m23; Wb,12m23; df,1m13; eq,13m13; gm,6m13; hu,12m13; ip,5m13; jn,7m13; rt,3m13. II. OP,1p2; QR,2p2; ST,3p2; UV,4p2; WX,5p2; YZ,6p2; ab,7p2; dx,0m13; gy,8m13; ef,14m13; hr,9m13; in,2m13; jt,11m13; mp,10m13; qu,4m13; AH,14m12; BC,1m12; DJ,9m12; EG,2m12; FI,13m12; KM,11m12; LN,12m12. III. de,1p3; fg,2p3; hi,3p3; jm,4p3; np,5p3; qr,6p3; tu,7p3; Dx,7m12; Iy,8m12; AB,0m12; CM,3m12; EK,5m12; FH,10m12; GL,6m12; JN,4m12; OQ,14m23; PR,2m23; ST,6m23; UW,7m23; VX,13m23; Ya,10m23; Zb,5m23. Partition into 4-cycles: I. (01 , 51 , 11 , 63 ); (01 , 61 , 11 , 73 ); (01 , 21 , 11 , 133 ); (01 , 31 , 21 , 33 ); (01 , 71 , 131 , 33 ); (01 , 71 , 31 , 13 ); (01 , 21 , 51 , 123 )’; (02 , 13 , 12 , 93 ) twice; (02 , 43 , 12 , 123 ) twice. II. (01 , 12 , 02 , 22 ); (01 , 112 , 92 , 122 ); (01 , 122 , 82 , 132 ); (01 , 132 , 102 , 142 ); (01 , 92 , 22 , 112 ); (01 , 12 , 62 , 142 ); (01 , 22 , 32 , 92 ); (01 , 23 , 21 , 113 ) twice; (01 , 103 , 61 , 143 ) twice. III. (02 , 53 , 43 , 63 ); (02 , 63 , 33 , 73 ); (02 , 133 , 73 , 143 ); (02 , 23 , 83 , 53 ); (02 , 73 , 63 , 103 ); (02 , 103 , 53 , 133 ); (02 , 23 , 43 , 143 ); (01 , 02 , 51 , 82 ) twice; (01 , 52 , 11 , 72 ) twice. Lemma 7.3. A TM(K24n+21 ) exists for all n ≥ 0. Proof. For n = 0, 1, the statement follows from Examples 7.1 and 7.2, respectively, so we may assume n ≥ 2. Let (X , B ) be a 3-GDD of order 6n + 5 with one group g1 of size 5 and 2n groups g2 , . . . , g2n+1 of size 3; such a GDD is known to exist precisely when n ≥ 2 [5]. Let now V = (X × {1, 2, 3, 4}) ∪ {∞}. Put on (g1 × {1, 2, 3, 4}) ∪ {∞} a copy of a TM(K21 ) from Example 7.1; for each i ∈ {2, 3, . . . , 2n}, put on (gi × {1, 2, 3, 4}) ∪ {∞} a copy of a TM(K13 ) which exists by Theorem 3.1. For each block {x, y, z } ∈ B , put on {x, y, z } × {1, 2, 3, 4} a copy of a TM(K4,4,4 ) from Lemma 4.3 in such a way that the groups are {x} × {1, 2, 3, 4}, {y} × {1, 2, 3, 4} and {z } × {1, 2, 3, 4}. The existence of a TM(K24n+21 ) now follows by applying Lemma 4.4.  For the case n = 9, see Section 2 above. There exists no triple metamorphosis TM(K9 ). Example 7.4. A TM(K33 ). The following is a twofold TS(33, 2)(V , B ) : V = Z11 × {1, 2, 3}; base triples mod (11, 11, 11) are: A. {01 , 11 , 02 }, B. {01 , 11 , 22 }, C. {01 , 21 , 62 }, D. {01 , 21 , 102 }, E. {01 , 31 , 72 }, F. {01 , 31 , 32 }, G. {01 , 41 , 22 }, H. {01 , 41 , 92 }, I. {01 , 51 , 82 }, J. {01 , 51 , 12 }, K. {02 , 12 , 03 }, L. {02 , 12 , 23 }, M. {02 , 22 , 13 }, N. {02 , 22 , 53 }, O. {02 , 32 , 73 }, P. {02 , 32 , 93 }, Q. {02 , 42 , 83 }, R. {02 , 42 , 93 }, S. {02 , 52 , 73 }, T. {02 , 52 , 83 }, U. {11 , 03 , 13 }, V. {101 , 03 , 13 }, W. {11 , 03 , 23 }, X. {81 , 03 , 23 }, Y. {71 , 03 , 33 }, Z. {51 , 03 , 33 }, a. {71 , 03 , 43 }, b. {61 , 03 , 43 }, d. {91 , 03 , 53 }, e. {81 , 03 , 53 }, f. {01 , 52 , 03 }, g. {51 , 02 , 03 }. Matched pairs of orbits in the three respective metamorphoses: I. AB,1p1; CD,2p1; EF,3p1; GH,4p1; IJ,5p1; Kg,0m23; Pf,6m23; LM,1m23; NR,5m23; OS,7m23; QT,8m23; UW,10m13; Vd,2m13; Xe,3m13; Ya,4m13; Zb,9m13. II. KL,1p2,MN,2p2; OP,3p2; QR,4p2; ST,5p2; Uf,0m13; Zg,6m13; VW,1m13; Xb,5m13; Yd,7m13; ae,8m13; AD,10m12; BJ,1m12; CE,4m12; FI,3m12; GH, 9m12. III. UV,1p3; WX,2p3; YZ,3p3; ab,4p3; de,5p3; Cg,6m12; Hf,5m12; AF,0m12; BG,2m12; DI,8m12; EJ,7m12; KM,10m23; LS,2m23; NT,3m23; OQ,4m23; PR,9m23. Partition into 4-cycles: I. (01 , 11 , 61 , 43 ); (01 , 11 , 31 , 23 ); (01 , 31 , 11 , 103 ); (01 , 41 , 11 , 33 ); (01 , 51 , 11 , 43 ); (02 , 13 , 12 , 63 ); (02 , 13 , 12 , 83 ); (02 , 63 , 12 , 83 ). II. (01 , 12 , 22 , 102 ); (01 , 12 , 52 , 32 ); (01 , 32 , 62 , 42 ); (01 , 42 , 102 , 92 ); (01 , 92 , 32 , 102 ); (01 , 13 , 11 , 63 ); (01 , 13 , 11 , 83 ); (01 , 63 , 11 , 83 ). III. (02 , 43 , 103 , 93 ), (02 , 23 , 03 , 103 ); (02 , 93 , 73 , 103 ); (02 , 23 , 63 , 33 ); (02 , 33 , 83 , 43 ); (01 , 22 , 21 , 72 ); (01 , 22 , 21 , 82 ); (01 , 72 , 21 , 82 ). Lemma 7.5. There exists a TM(K57 ). Proof. Let (X , B ) be a 3-GDD of order 28 with one group g1 of size 10 and three groups g2 , g3 , g4 of size 6 (cf. [5]). Let V = (X × {1, 2}) ∪ {∞}, then |V | = 57. On (g1 × {1, 2}) ∪ {∞}, put a copy of TM(K21 ) from Example 7.1, and for i = 2, 3, 4, put on (gi × {1, 2}) ∪ {∞} a copy of a TM(K13 ) given in Theorem 3.1. For a triple {x, y, z } ∈ B , put a copy of a TM(K2,2,2 ) (Lemma 4.3) on {x, y, z } × {1, 2} in such a way that the groups are {x} × {1, 2}, {y} × {1, 2}, {z } × {1, 2}. By Lemma 4.4, this yields a TM(K57 ). 

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Lemma 7.6. There exists a TM(K81 ). Let X be a set with |X | = 20, and let V = (X × {1, 2, 3, 4}) ∪ {∞}. For i ∈ {1, 2, 3, 4}, put on X ∪ {∞} a copy of a TM(K21 ) from Example 7.1. Next consider a 4-GDD of order 80 on X × {1, 2, 3, 4} with groups X × {i}, i = 1, 2, 3, 4 (cf. [5]). Replace each block (of size 4) by a TS(4, 2), and then proceed exactly as in the second part of the construction in Example 4.1 to obtain a TM(K81 ).  Lemma 7.7. There exists a TM(K105 ). Proof. Let (X , B ) be a 3-GDD of order 26 with one group g1 of size 8 and three groups g2 , g3 , g4 of size 6. Put on (g1 ×{1, 2, 3, 4})∪{∞} a copy of a TM(K33 ) from Example 7.4, and for each i = 2, 3, 4, put on (gi ×{1, 2, 3, 4})∪{∞} a copy of a TM(K25 ) from Theorem 3.1. For each triple {x, y, z } ∈ B , put a copy of a TM(K4,4,4 ) from Lemma 4.3 on {x, y, z }×{1, 2, 3, 4} so that the groups are {x} × {1, 2, 3, 4}, {y} × {1, 2, 3, 4}, {z } × {1, 2, 3, 4}. By Lemma 4.4, this results in a TM(K105 ).  Lemma 7.8. There exists a TM(K129 ). Proof. Proceed similarly as in Lemma 7.6 but start instead with a set X such that |X | = 32.



Lemma 7.9. A TM(K24n+9 ) exists for all n ≥ 1. Proof. For n = 1, 2, 3, 4, 5, the statement follows from Example 7.4, Lemmas 7.5–7.8, respectively. So let n ≥ 6, and let (X , B ) be a 3-GDD of order 3n + 1 with group sizes in K = {3, 4, 6, 7, 10}. Such a GDD is known to exist for all n ≥ 6 [5]. Let now V = (X ×Y )∪{∞} where Y is a set with |Y | = 8. For each group g, put on (g ×Y )∪{∞} a copy of a TM(K25 ), or a TM(K33 ), or a TM(K49 ), or a TM(K57 ), or a TM(K81 ), depending on whether |g | = 3, 4, 6, 7 or 10. These triple metamorphoses exist by Theorem 3.1, Example 7.4 and Lemmas 7.5, 7.6. For each block {x, y, z } ∈ B , put a copy of a TM(K8.8.8 ) from Lemma 4.3 on {x, y, z }× Y in such a way that the groups are {x}× Y , {y}× Y and {z }× Y . Finally, apply Lemma 4.4 to obtain a TM(K24n+9 ).  We summarize the results of this section in the following theorem. Theorem 7.10. A triple metamorphosis TM(K12n+9 ) exists for all n ≥ 1 and does not exist for n = 9. 8. Conclusion Our main result is contained in the next theorem. Theorem 8.1. A triple metamorphosis TM(Kn ) exists if and only if n ≡ 0, 1, 4 or 9(mod 12), n > 9, except possibly when n = 12. In view of the enormous number of nonisomorphic simple TS(12, 2)s [12], deciding the existence or nonexistence of a triple metamorphosis TM(K12 ) appears to be a formidable task. Acknowledgments A part of this research was done while the second and the third author visited the first author at Auburn University. They would like to thank him for his wonderful hospitality. The research of the second author was supported by the NCN Grant No. 2011/01/B/ST1/04056. The research of the third author was supported by NSERC Grant No. A7268. The authors thank the referees for their careful reading of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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