THE TRIANGLE INTERSECTION PROBLEM FOR HEXAGON TRIPLE SYSTEMS Except where reference is made to the work of others, the work described in this dissertation is my own or was done in collaboration with my advisory committee. This dissertation does not include proprietary or classi ed information. Carl Stuart Pettis Certi cate of Approval: Dean Ho man Professor Mathematics and Statistics Charles C. Lindner, Chair Distinguished University Professor Mathematics and Statistics Overtoun Jenda Professor Mathematics and Statistics Stephen L. McFarland Dean Graduate School THE TRIANGLE INTERSECTION PROBLEM FOR HEXAGON TRIPLE SYSTEMS Carl Stuart Pettis A Dissertation Submitted to the Graduate Faculty of Auburn University in Partial Ful llment of the Requirements for the Degree of Doctor of Philosophy Auburn, Alabama August 7, 2006 THE TRIANGLE INTERSECTION PROBLEM FOR HEXAGON TRIPLE SYSTEMS Carl Stuart Pettis Permission is granted to Auburn University to make copies of this thesis at its discretion, upon the request of individuals or institutions and at their expense. The author reserves all publication rights. Signature of Author Date Copy sent to: Name Date iii Vita Carl Stuart Pettis was born on June 30, 1979 in Augusta, Georgia, the second of two sons of Charles S. Pettis and Jane H. Pettis. He was an honor graduate of Thomson High School in 1997. He chose Alabama State University in Montgomery, Alabama as his institution of higher learning to pursue his Bachelor of Science degree in Mathematics, which he earned in May, 2001 as a Magna Cum Laude graduate. He would later receive his Master of Science degree in Mathematics from Alabama State University in May, 2003. Auburn University became his home in August, 2003 where he began his pursuit of the Doctorate of Philosophy. iv Dissertation Abstract THE TRIANGLE INTERSECTION PROBLEM FOR HEXAGON TRIPLE SYSTEMS Carl Stuart Pettis Doctor of Philosophy, August 7, 2006 (Master of Science May, 2003) (Bachelor of Science May, 2001) 88 Typed Pages Directed by Charles C. Lindner A hexagon triple is the graph and a hexagon system is an edge disjoint decomposition of 3k n into hexagon triples. Note that a hexagon triple is the union of 3 triangles (= triples). The intersection problem for 3-fold triple systems has been solved for some time now. The purpose of this dissertation is to give a complete solution of the intersection problem for 3-fold triple systems each of which can be organized into hexagon triple systems. v Acknowledgments The author would like to extend his deepest gratitude to all those who aided in this endeavor. To the members of his advisory committee and most especially Dr. Charles C. Lindner who contributed ideas and direction during the course of this research, the author would like to say thank you. The author would like for the faculty and sta of Alabama State University to know that they will always have a special place in his heart. To Mrs. Rosie Torbert, who did a wonderful job with the typing and editing of this dissertation, the author extends his warm thanks. To his family and friends for their undying support and encouragement, the author is forever indebted. vi Style manual or journal used Journal of Approximation Theory (together with the style known as \aums"). Bibliograpy follows van Leunen?s A Handbook for Scholars. Computer software used The document preparation package T E X (speci cally L A T E X) together with the departmental style- le aums1.sty. vii Table of Contents List of Figures ix 1 Introduction 1 1.3 The3-foldConstruction ............................ 2 1.8 The -intersection problem for hexagon triple systems: . . . . . . . . . . 6 2 Preliminaries 7 3 n =7 12 4 n =9 23 5 n =13 47 6 n =15 70 7The6n+1 19 Construction 72 8The6n+3Construction 75 Bibliography 79 viii List of Figures 1.2 Triangle..................................... 2 1.6 Hexagon triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5.1 Hexagon triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 ix Chapter 1 Introduction A Steiner triple system (more simply triple system) of order n is a pair (S;T), where T is a collection of edge disjoint triangles (or triples) which partitions the edge set of K n (= the complete undirected graph on n vertices) with vertex set S. It is straightforward to see thatjTj= n(n?1)=6. In 1846 T. P. Kirkman proved that the spectrum for triple systems (= the set of all n such that a triple system of order n exists) is precisely the set of all n6=1or3(mod6)[2]. Example 1.1 (Two triple systems of order 7) 00110011 01 00110011 0011 00110011 01 00110011 01 00 00 11 11 00 00 11 11 0 0 1 1 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 71 4 61 5 73 5 63 4 13 5272 642 00 00 11 11 0011 00110011 01 00110011 0011 00110011 0011 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 01 00 00 11 11 00 00 11 11 00 00 11 11 0011 637262 1 3 1517 73 4 5 54 642 0011 T 2 =T 1 = Inspection shows that T 1 and T 2 have exactly one triple in common. In [3], C. C. Lindner and A. Rosa gave a complete solution of the intersection problem for triple systems. Let I(n)=f0;1;2;:::;n(n?1)=6=xgnfx?1;x?2;x?3;x?5gand let Int(n)bethesetofallksuch that there exists a pair of triple systems of order n having 1 exactly k triples in common. Lindner and Rosa proved that Int(n)=I(n) for all n 1 or 3 (mod 6), except for n =9. InthiscaseInt(9) = I(9)nf5;8g=f0;1;2;3;4;6;12g. In what follows we will denote by K n the graph on n vertices with each pair of vertices connected by edges. A -fold triple system of order n is a pair (S;T), where T is a collection of triples which partitions the edge set of K n with vertex set S.Itiseasy to show that a necessary condition for the existence of a 3-fold triple system of order n is that n is odd and the number of triples is parenleftbig n 2 . The construction of 3-fold triple systems showing that this condition is su cient is quite easy and whoever did this for the rst time is lost to history. In what follows we will denote the triangle (triple) byfa;b:cgor just abc. 0011 00110011 a bc Figure 1.2: Triangle 1.3 The 3-fold Construction Let (Q; ) be an idempotent commutative quasigroup of order n (n must be odd). Let T =ffa;b;a b = b agja6=b2Qg.Then(Q;T) is a 3-fold triple system. This is quite easy to see. Let a6= b2Q, then the three triplesfa;b;a bg;fa;x;a x= x a = bg, andfb;y;b y = y b = ag2T. 2 Example 1.4 (3-fold triple system of order 7) 74152637 67415263 53741526 46374152 32637415 25263741 11526374 1234567 f1;2;5gf2;4;3gf3;7;5g f1;3;2gf2;5;7gf4;5;1g f1;4;6gf2;6;4gf4;6;5g f1;5;3gf2;7;1gf4;7;2g f1;6;7gf3;4;7gf5;6;2g f1;7;4gf3;5;4gf5;7;6g f2;3;6gf3;6;1gf6;7;3g Now just as we looked at the intersection problem for triple systems, we can look at the intersection problem for 3-fold triple systems. It is not di cult to see (an easy exercise) that if two 3-fold triple systems of order n have k triples in common, k2f0;1;2;:::; parenleftbig n 2 =xgnfx?1;x?2;x?3;x?5g=3I(n). Several people have shown, except for n = 5, that this necessary condition is su cient. (See [1] for example). Example 1.5 (3-fold triple system of order 7) f1;2;5g; f2;4;7g; f3;7;5g f1;3;7g; f2;5;1g; f4;5;3g f1;4;6g; f2;6;4g; f4;6;5g f1;5;2g; f2;7;3g; f4;7;1g f1;6;3g; f3;4;2g; f5;6;7g f1;7;4gf3;5;4g;f5;7;6g f2;3;6g;f3;6;1g;f6;7;2g 3 A cursory check shows that the two 3-fold triple systems in Examples 1.4 and 1.5 have exactly 10 triples in common. They are f1;2;5g;f1;4;6g;f1;4;7g;f2;3;6g;f2;4;6g;f3;4;5g, f1;3;6g;f3;5;7g;f4;5;6g;f5;6;7g. The graph below is called a hexagon triple and a hexagon triple system of order n is a Figure 1.6: Hexagon triple pair (X;H), where H is a collection of edge disjoint hexagon triples which partitions the edge set of 3K n . It is well-known (see [4] for example) that the spectrum for hexagon triple systems is precisely the set of all n 1or3(mod6) 7andthatif (X;H) is a hexagon triple system of order n thatjHj= n(n?1)=6. Note that a hexagon triple consists of 3 triples and so we can think of a hexagon triple system as the piecing together of the triples of a 3-fold triple system into hexagon triples. In the example below the two hexagon triple systems H 1 and H 2 of order 7 have been pieced together from the 3-fold triple systems in Examples 1.4 and 1.5. 4 Example 1.7 (Two hexagon triple systems of order 7 intersecting in 10 triples) 1 5 6 4 2 3 H 2 =H 1 = 4 4 1 5 6 2 3 7 64 5 7 2 1 53 2 6 1 7 2 4 3 5 6 6 1 3 2 4 5 7 7 3 1 4 6 2 7 4 1 5 2 3 3 4 5 5 6 2 4 6 3 7 1 1 2 5 2 6 7 7 1 12 6 3 7 4 7 1 5 2 6 3 5 7 4 5 The object of this thesis is the solution of the following problem: 1.8 The -intersection problem for hexagon triple systems: For each n 1or3(mod6) 7andeachk23I(n)=f0;1;2;3;:::;n(n?1)=2= xgnfx?1;x?2;x?3;x?5gconstruct a pair of 3-fold triple systems of order nintersecting in k triples each of which can be organized into a hexagon triple system. We give a complete solution of this problem, with a few possible exceptions for n = 13. 6 Chapter 2 Preliminaries We collect together in this section some of the ideas and background material nec- essary to obtain the main results. We begin with partial triple systems. A partial triple system of order n is a pair (X;P), whereP is a collection of edge disjoint triples of the edge set of K n . The di erence between a partial triple system and a triple system is that the triples in a partial triple system (X;P) do not necessarily partition the edge set of K n . Wenotethatatriple system is also a partial triple system. Two partial triple systems (X;P 1 )and(X;P 2 ) are said to be balanced if the triples in P 1 and P 2 cover the same edges. Example 2.1 (Balanced partial triple systems of order 6) 2 6 5 1 53 2 43 1 P 1 = 4 P 2 = 6 5 2 43 2 6 4 1 53 1 6 7 The following result is due to Lucia Grionfriddo (unpublished). Lemma 2.2 Let (X;P 1 ) and (X;P 2 ) be partial triple systems that are balanced and disjoint (having no triples in common). Then using each triple in P 2 threetimes,wecan construct a partial hexagon triple system whose inside triples are P 1 . Proof Let fa;b;cg2P 2 and place fa;b;cgon the triples in P 1 containing the edges fa;bg;fa;cg,andfb;cg.SinceP 1 and P 2 are disjoint, the three triples in P 1 are disjoint. Since P 2 is a partial triple system, the three triples in P 2 placed on each triple in P 1 form a hexagon triple. Example 2.3 (Partial hexagon triple system constructed from Example 2.1) 6 2 3 56 2 4 4 1 3 4 2 6 5 1 4 3 2 5 6 5 3 Corollary 2.4 If k2Int(n) for Steiner triple systems, then 3k23Int(n) for hexagon triple systems. Proof Let (S;T 1 )and(S;T 2 ) be a pair of triple systems intersecting in k triples. Luc Teirlinck [7] has shown that every Steiner triple system has a disjoint mate. So let (S;T 1 ) 8 and (S;T 2 ) be triple systems such that T 1 \T 1 = ; and T 2 \T 2 = ;.Ifweplacethe triples of 3T 1 on T 1 and the triples of 3T 2 on T 2 the resulting hexagon triple systems H 1 and H 2 have exactly 3k triples in common. Example 2.5 (Hexagon triple systems of order 7 intersecting in 9 triples) 53 6 73 4 71 T 1 = 5 6 2 71 3 41 5 73 4 52 3 4 2 31 4 62 5 72 5 61 6 1 25 3 65 6 71 4 73 4 51 4 722 1 57 T 1 = 2 6 74 4 51 3 52 6 4 72 4 63 3 71 2 61 T 2 = T 2 = 9 H 1 = 5 6 5 3 4 1 1 5 42 6 1 7 65 3 7 5 6 53 2 4 6 12 7 3 7 4 1 3 2 2 5674 2 1 2 5 4 4 4 1 7 16 6 7 4 2 6 7 3 4 5 1 3 7 2 3 1 6 6 3 4 5 7 2 1 6 2 3 7 54 7 3 H 2 = 3 Finally, we will need both pairwise balanced designs and group divisible designs for the main constructions in this thesis. A pairwise balanced design (PBD) of order n is a pair (X;B), where jXj= n,and Bis a collection of subsets of X called blocks such that each pair of distinct elements of X belongs to exactly one block of B. The blocks in B are not necessarily the same size. In terms of graph theory, we can think of B as a collection of complete graphs (not necessarily of the same size) which partition the edge set of K n with vertex set X.So, for example, a Steiner triple system is a pairwise balanced design in which every block has size 3. A group divisible design (GDD)ofordernis a triple (X;G;B), where (X;G[B)is aPBDof order n and G is a parallel class of blocks, called groups, which partitions X. 10 Example 2.6 (GDD of order 6 with groups of size 2 and blocks of size 3) 8 > > > > < > > > > > : X =f1;2;3;4;5;6g G =ff1;2g;f3;4g;f5;6gg B =ff1;3;5g;f1;4;6g;f2;4;5g;f2;3;6gg Example 2.7 (GDD of order 10 with group sizes 4 and 2 and block size 3) 8 > > > > > > > > < > > > > > > > > : X = f1;2;3;4;5;6;7;8;9;10g G = ff1;2;3;4g;f5;6g;f7;8g;f9;10gg B = ff1;5;9g;f1;6;8g;f1;7;10g;f2;5;10g;f2;6;7g;f2;8;9g; f3;5;8g;f3;7;9g;f3;6;10g;f4;6;9g;f4;5;7g;f4;8;10gg If 2n 0or2(mod6) 6, there is a GDD of order 2n with all groups of size 2 andblocks(=triples)ofsize3. If2n 4(mod6) 10, there is a GDD of order 2n with exactly one group of size 4 and the remaining groups of size 2 and blocks of size 3. (See [6] for example.) 11 Chapter 3 n =7 We give a complete solution of the -intersection problem for n = 7 in this section. In what follows for each x2Int(7) =f0;1;2;:::;21gnf20;19;18;16gwe list the pair of hexagon triple systems having x triples in common. This gives a complete solution. 12 5 2 4 1 7 3 7 23 5 1 3 7 7 4 1 3 5 6 3 4 1 2 7 6 4 1 6 7 4 2 1 5 3 5 4 2 1 6 2 6 2 4 6 7 1 2 5 3 6 7 2 6 37 1 5 61 3 7 4 1 3 5 6 3 6 2 4 1 1 7 4 1 5 7 2 3 6 3 1 3 7 3 5 5 7 5 5 4 1 2 6 3 7 4 6 1 2 6 4 3 2 7 7 6 4 5 2 3 7 1 5 24 6 4 6 2 1 1 5 62 7 3 4 5 3 2 6 7 5 4 6 5 1 3 4 5 1 7 2 4 6 3 7 2 4 5 4 6 7 2 4 3 1 7 5 2 3 14 5 6 5 2 13 7 1 3 4 6 5 2 6 2 5 6 3 1 7 4 15 6 1 5 72 3 4 1 3 4 6 3 2 7 1 4 6 3 7 2 4 5 1 7 6 2 7 5 1 4 4 3 7 5 1 6 1 2 3 7 1 4 3 6 4 7 2 5 6 3 6 2 4 5 4 2 3 5 6 511 31 4 5 6 27 1 7 6 5 3 3 2 1 7 3 7 12 4 6 2 2 6 4 3 7 5 3 7 41 2 5 4 1 6 1 5 2 4 6 7 3 7 6 1 5 2 2 6 37 7 6 1 3 2 2 5 4 3 7 3 2 5 4 7 4 1 5 2 3 3 5 47 6 2 4 76 5 5 14 6 2 3 1 6 4 7 3 5 6 2 6 3 4 1 7 5 1 4 3 5 4 5 3 16 7 6 3 1 2 7 5 1 5 2 3 7 6 5 3 2 1 76 37 1 3 4 6 5 6 2 7 4 1 3 5 6 1 57 6 5 2 4 1 2 45 6 2 2 4 2 2 5 14 7 3 3 7 41 2 6 1 4 2 7 3 5 4 1 5 7 3 6 5 6 4 2 5 7 4 3 2 1 6 7 3 7 1 7 7 3 5 2 4 6 3 5 47 6 1 7 4 2 1 1 3 4 3 6 72 43 4 5 4 7 1 5 4 71 2 5 2 6 1 4 3 5 2 15 1 2 3 1 7 61 2 5 47 3 2 1 72 3 6 6 3 2 1 3 4 3 71 23 2 4 5 7 5 4 1 63 66 2 554 24 2 3 74 4 42 1 1 5 2 3 4 5 3 2 54 433 5 66 7737 511 27 15 2 7 2 1 41 6273 71 37 52 41 3 3 6 46 5 7 1 2 63 45 6 3 3 7 65 74 35 6 24 35 4 7 2 4 71 2 7 6 6 3 12 31 7 5 7 1 6 56 75 4 4 5 7 16 5 5 4 16 7 4 5 6 37 6 1 2 4 7 61 72 3 5 6 35 1 2 1 4 6 7 1 3 3 24 2 4 2 3 4 1 6 6 3352 56 51 7 3 24 7 7 1 2 42 3 5 6 163 3 5 44 5 6 6 7 7 116 3 54 1 731 2 46 1 157 45 54 3 3 5 26 57 1 5 47 64 76 7 6 3 54 74 3 6 5 12 45 4 7 32 6 1 7 2 71 43 5 6 2 5 6 22 7 4 3 3 2 1 5 47 21 6 2 17 1 6 4 7 56 16 4 5 27 63 7 1 63 43 6 4 7 3 1 3 2 5 2 6 6 72 33 13 1 5 5 4 6 4 6 7 1 56 26 5 7 2 72 5 25 2 4 2 6 132 7573 7 5 5 2 1 5 5 4 6 13 7 113 1451 76 7 4 4 5 3 1 4 26 1 7 5 3 4 62 1 4 7 6 2 45 75 3 7 32 52 3 4 31 2 3 2 64 74 3 12 45 7 3 7 13 24 5 7 43 57 61 2 4 18 3 76 4 4 5 6 1 6 4 7 5 24 63 7 1 67 3 3 6 4 1 2 132 6 16 7 14 1 1 4 6 74 6 7 5 2 1 24 7332 3 244 1 5 1 7 1 1 555 27 1 3 5 472 33 1 6 2 6 5 4 1 45 7 2 4 2 5 2 7 6 4 5 2 3 43 16 7 6 7 1 35 3 7 5 7 6 35 3 7 325 3 6 12 7 1 6 2 3 4 7 2 7 25 3 5 41 2 5 4 16 3 5 27 63 54 27 4 4 6 5 19 35 6 5 7 14 1 6 5 4 75 2 7 2 1 2 32 3 4 7 61 4 3 1 7 1 61 537 6 3 7 2 1 7 56 2 4 3 2 4 4 3 2 3 7 3 1 5 6 5 6 3 6 17 6 6 7 7 1 6 2 4 16 4 76 6 5 4 4 3 67 7 2 5 24 6 45 4 3 5 5 2 3 35 7 5 611 2 7 4 3 1 3 66 7 1 3 2 7 4 57 1 1 3 4 2 53 1 3 26 7 4 5 4 5 1 2 4 4 3 1 1 4 2 63 4 1 2 5 74 5 3 2 5 20 7 1 3 45 7 5 6 16 3 5 27 63 41 5 3 4 7 366 3 2 4 3 2 2 1 6 34 1 7 2 1 1 22 5 1 7 2 7 7 12 4 6 7 2 51 2 6 3 7 2 4 1 5 7 1 51 35 1 6 7 1 3 62 1 6 47 6 5 1 36 1 5 2 2 1 5 5 3 5 2 7 2 3 45 47 6 3 17 52 3 2 2 4 1 5 73 4 3 6 2 5 3 1 6 736 3 2 1 6 2 4 7 4 6 7 4 7 463 5 5 13 75 4 7 5 4 6 4 21 The preceding pages give a complete solution to the -intersection problem for hexagon triple systems of order 7. Lemma 3.1 3Int(7) =f0;1;2;:::;21gnf20;19;18;16g. 22 Chapter 4 n =9 We give a complete solution of the -intersection problem for n = 9 in this section. In what follows for each x2Int(9) =f0;1;2;:::;36gnf35;34;33;31gwe list the pair of hexagon triple systems having x triples in common. This gives a complete solution. 23 2 7 9 34 68 5 1 5 4 6 2 9 7 5 6 8 7 4 6 58 9 2 7 3 4 9 8 7 6 83 2 1 7 9 58 62 3 7 2 96 45 8 1 24 35 98 1 1 1 3 1 5 8 53 311 2 18 9 6 81 93 15 4 7 48 9 4 8 6 9 2 5 7 22 7 4 2 13 8 21 9 41 2 43 4 2 2 6 14 2 17 47 4 22 5 2 4 6 9 8 76 94 6 5 1 5 3 69 4 5 3 2 7 9 6 2 9 6 1 8 4 3 6 53 15 8 76 9 8 76 84 7 5 2 34 8 2 7 3 8 4 7 3 9 2 3 92 5 3 6 27 86 45 8 9 3 24 1 7 6 3 9 5 9 7 6 1 4 8 8 4 3 8 7 5 4 6 2 9 8 5 17 6 2 1 5 9 3 6 5 4 3 26 8 5 3 27 9 6 2 34 7 8 2 3 8 4 7 5 2 36 8 2 7 9 8 22 1 4 79 54 6 9 62 1 12 1 3 34 14 33 1 1 1 1 113 7 7 3 9 85 6 6 45 8 2 6 95 7 5 6 8 4 6 8 7 1 4 2 9 2 7 3 4 9 8 8 7 6 9 7 9 1 3 6 4 1 5 8 6 2 5 7 9 5 31 7 5 2 4 68 5 1 8 3 17 4 5 3 7 9 7 3 2 5 8 5 2 8 3 1 5 9 3 6 8 1 6 7 3 5 8 1 7 63 9 5 1 6 3 8 4 2 6 7 7 5 4 2 9 6 3 7 5 1 9 25 2 4 5 96 1 8 9 6 41 9 6 5 87 65 96 1 6 53 8 2 6 18 9 9 3 15 7 4 98 7 6 9 5 3 85 4 3 5 9 4 5 1 8 6 9 3 2 1 8 5 4 1 7 6 5 3 8 79 3 1 3 5 4 8 1 8 3 7 2 9 4 9 3 2 1 3 46 8 1 7 6 3 9 9 1 1 5 55 9 2 9 6 7 1 25 9 7 1 3 7 4 7 32 87 6 9 9 2 7 5 4 6 1 5 6 9 1 5 9 8 33 22 22 6 1 4 87 3 15 4 8 54 98 6 9 4 7 2 7 6 95 4 2 8 4 5 6 9 7 8 4 4 3 2 44 22 4 1 52 3 7 84 4 2 7 2 14 26 5 67 5 6 82 6 4 3 9 7 3 41 3 2 8 4 1 6 4 8 42 6 9 4 1 5 76 2 28 7 7 85 9 2 65 8 8 9 12 7 2 6 8 31 7 2 7 9 5 9 4 5 9 5 2 6 3 69 8 5 8 31 1 668 7223 3 344 7 6 9 6 3 31 2 9 77 6 7 2 5 31 6 42 1 8 8 8 3 2 1 2 8 968 113 6 4 9 4 4 5 1 7 3 5 2 7 3 6 2 9 7 5 2 36 4 5 2 4 35 9 3 8 2 5 3 85 1 4 2 9 6 9 7 4 8 2 5 5 4 5 7 7 5 81 6 7 5 81 9 9 7 4 8 1 6 3 7 1 9 8 4 7 1 45 1 68 9 8 7 9 27 7 2 4 3 6 2 5 39 6 96 5 9 8 6 5 2 7 1 6 3 4 7 2 8 9 9 3 3 5 8 9 2 4 6 7 3 19 7 5 2 9 7 4 8 6 5 381 4 7 1 5 28 2 8 5 1 6 1 7 9 1 8 6 8 4 53 4 7 6 12 2 8 9 2 8 4 2 4 9 7 1 5 3 9 2 3 8 1 64 2 4 6 7 5 1 2 3 2 3 4 5 1 7 8 37 6 2 4 3 7 9 2 4 1 3 8 9 3 3 9 2 5 6776 756 93 4 5 9 6 49 5 7 88 2 8 8 5 1 3 5 1 9 4 8 5 5 9 4 6 4 23 6 4 7 4 8 8 5 55 2 6 3 4 97 1 3 96 4 11 21 8 11 28 1 2 8 8 2 3 8 9 4 3 2 7 3 4 5 9 7 9 6 5 1 5 3 7 5 7 7 5 4 96 2 4 1 4 6 5 9 3 2 8 2 1 5 3 76 4 1 7 23 4 8 9 7 4 2 5 1 21 2 9 8 6 3 1211 933 7 192 4 893 5 3 69 4 4 6 5 2 6 3 9 3 5 4 9 3 5 6 8 68 8 4 8 3 1 9 75 4 3 9 9 8 5 6 2 7 5 7 5 8 5 2 1 1 4 5 7 8 3 6 4 3 1 9 7 9 5 4 3 8 7 7 9 4 7 1 5 8 4 6 2 3 5 7 5 81 6 2 3 6 4 1 2 7 9 4 1 8 6 4 7 1 2 6 3 6 9 7 8 9 7 3 4 2 6 5 2 8 1 12 8 6 3 2 8 22 4 29 6 4 8 5 1 65 8 6 9 7 64 87 4 8 7 56 8 73 6 4 9 3 4 69 1 6 4 3 1 4 3 1 66 8 1 5 2 9 6 7 3 2 9 2 8 3 5 2 4 4 85 7 5 5 3 6 2 4 9 9 3 2 1 8 5 7 3 2 1 9 5 9 1 1 3 1 2 7 4 7 1 3 6 7 4 2 3 8 8 4 9 7 5 8 6 3 8 1 5 7 9 4 6 1 3 1 9 7 1 7 3 5 8 2 6 1 2 2 8 9 2 6 9 1 4 3 5 2 8 7 3 6 9 7 8 4 1 2 6 57 4 7 1 8 4 37 95 8 84 2 1 27 5 8 5 3 2 8 5 9 3 9 8 76 2 1 7 4 7 9 6 2 9 4 3 2 7 5 4 7 3 9 2 5 1 9 1 30 1 2 8 5 4 3 9 4 12 9 6 81 96 43 9 5 7 4 9 8 76 9 5 32 7 1 5 78 2 5 2 34 8 9 7 6 3 92 1 5 6 1 2 6 1 4 1 6 8 9 9 3 4 1 8 6 11 2 8 5 4 5 1 1 9 4 8 6 3 2 6 4 4 7 4 13 77 7 44 31 8 5 6 7 5 9 4 8 2 8 44 2 7 8 5 6 6 1 8 1 2 7 2 8 6 7 6 97 8 1 53 8 7 6 4 972 3 6 9 4 6 6 9 6 53 8 92 1 7 6 3 6 2778 7 8 1 4 8 4 3 942 2 33 8 2 44 8 11 5 3 5 6 7 1 6 9 7 3 5 2 75 39 7 9 28 5 9 31 4 8 9 6 85 5 1 78 9 6 3 4 7 454 7 9 8 7 2 7 9 3 6 59 3 5 2 47 7 79 5 6 7 8 6 9 7 4 6 1 2 3 59 9 1 4 1 3 8 1 2 1 5 1 8 8 3 2 4 6 4 4 5 5 1 53 8 3 2 5 1 3 1 1 2 1 8 6 6 7 4 9 4 8 7 2 6 2 6 9 7 9 4 3 1 4 8 3 2 2 9 7 2 1 1 3 6 5 8 22 1 8 5 43 3 28 5 2 9 7 676 5 2 9 8 3 9 5 6 41 93 4 3 969 4 945 7 1 2 6 2 63 2 1 2 9 1 2 1 2 4 2 98 5 6 8 4 7 9 2 6 87 4 22 52 51 7 1 32 76 8 2 7 9 5 32 75 8 3465 51 6 7 82 6 2 7 28 75 31 7 9 28 3 1 8 5 2 3 6 6 8 5 1 1 4 5 5 3 3 5 4 3 8 6 1 5 1 1 4 4 4 2 7 9 28 3 2 7 3 1 4 7 8 3 7 5 8 3 1 5 4 3 8 7 6 7 5 6 4 6 2 3 1 4 2 8 8 5 8 2 1 6 1 7 6 5 3 9 9 7 4 1 6 8 5 9 9 7 3 91 3 6 9 7 8 4 7 5 4 2 6 1 5 9 3 6 8 898 341 7 4 3 9 5 8 3 5 4 2 3 81 2 5 898 76 9 43 17 4 33 5 2 9 1 5 2 2 7 7 9 1 2 5 9 5 8 2 3 6 75 1 4 9 22 2 3 5 2 69 33 5 82 1 7 9 2 8 3 6 2 8 5 1 9 2 5 7 1 6 9 4 1 7 4 6 8 5 8 2 3 6 8 5 1 7 6 2 3 1 6 5 3 8 9 2 7 6 9 7 4 1 6 8 9 7 4 6 8 1 9 7 6 1 4 8 4 5 3 9 4 6 3 9 5 1 3 7 6 2 4 9 5 7 4 2 8 9 4 1 6 97 8 4 7 6 4 7 1 3 9 6 4 1 3 7 9 1 6 8 6 4 3 1 7 9 2 4 8 1 5 9 3 9 5 6 9 3 8 4 3 6 9 7 8 4 7 5 8 2 1 6 4 5 4 6 4 9 5 1 3 6 4 5 1 9 8 7 6 2 4 9 1 7 6 2 2 2 8 5 8 5 1 7 3 1 3 2 9 1 4 7 9 5 4 9 5 4 88 1 9 3 2 5 3 3 2 5 8 7 93 7 428 3 5 6 9 4 34 7 5 8 5 2 5 89 5 7 9 6 4 3 2 7 4 8 1 7 5 6 787 6 6 9 8 3 6 7 1 9 1 72 62 6 8 3 1 7 4 7 8 2 7 4 2 4 4 2 1 2 3 7 8 3 2 4 69 4 3 5 4 1 5 4 1 9 5 4 1 3 6 5 4 9 9 466 3 9 4 7 1 66 1 6 9 7 5 9 9 6 5 2 2 68 3 1 9 1 4 8 1 6 9 2 4 8 5 8 9 5 6 1 31 67 3 8 5 4 2 3 8 7 2 1 7 3 9 8 55 5 7 7 5 5 6 9 8 8 798 7 3 64 8 97 6 4 5 4 9 4 53 6 2 4 1 6 2 1 1 1 33 2 2 6 5 3 8 9 7 9 6 4 2 8 1 7 3 22 8 7 1 3 9 35 1 7 9 3 6 1 3 27 2 9 8 8 2 3 1 9 7 8 4 5 2 2 6 1 4 3 4 6 1 8 5 5 4 2 8 9 5 3 3 2 6 4 3 7 3 5 2 8 7 4 9 5 7 6 3 8 9 2 6 1 1 9 8 6 3 2 8 5 1 7 9 6 5 1 39 5 6 8 2 7 3 5 5 1 9 8 2 4 5 1 9 1 7 6 2 4 9 5 3 8 6 4 9 4 8 5 6 7 1 8 4 6 4 3 1 7 9 2 8 37 9 1 3 6 2 1 3 8 6 2 7 3 9 8 4 2 6 1 4 8 9 6 5 9 5 9 4 7 2 3 6 4 1 5 8 1 7 5 4 9 6 5 6 84 7 3 9 3 4 2 8 4 5 9 3 6 4 3 2 9 7 1 3 7 5 4 2 5 7 2 2 8 6 3 5 9 7 3 1 5 4 2 8 7 6 9 1 5 2 7 8 4 36 3 1 7 3 2 3 9 2 4 4 2 95 1 3 6 2 6 4 7 9 7 3 2 9 7 46 8 1 6 2 9 3 2 1 8 4 98 6 34 7 9 9 1 5 9 6 5 1 3 2 2 8 9 8 1 9 71 2 2 2 5 4 5 3 5 8 7 6 1 2 96 3 2 7 9 5 3 6 4 1 4 7 9 8 5 1 7 5 4 2 6 1 3 6 9 4 8 7 1 2 5 8 7 4 7 5 8 1 6 4 5 2 6 6 4 3 1 7 9 1 5 9 6 8 3 1 7 6 5 3 9 7 5 8 2 415 2 33 3 6 2 8 7 6 28 7 9 1 3 2 7 9 5 32 7 1 2 5 7 6 9 3 4 2 6 6 8 25 8 1 4 2 1 44 9 1 34 8 5 2 34 8 8 31 8 4 9 6 9 3 59 5 7 37 4 3 9 5 2 75 3 7 1 9 867 4 14 2 8 4 2 3 3 58587 8 79 6 8 9 8 7 3 8 6 4 5 6 1 3 8 6 4 1 9 2 5 2 9 3 1 5 1 8 6 5 4 5 1 3 8 3 4 1 8 9 5 9 2 7 6 7 4 5 2 7 2 5 4 2 8 4 1 5 98 4 5 9 3 2 3 4 1 56 46 3 1 9 1 2 8 5 9 3 2 1 8 5 1 7 2 5 4 3 2 6 6 1 5 6 4 9 4 9 7 78 8 1 3 7 4 8 73 9 1 9 2 4 1 8 2 6 51 6 4 1 5 9 5 69 3 7 3 4 4 6 9 1 6 5 7 82 6 1 4 2 8 1 6 9 8 2 1 8 5 7 7 4 87 4 1 77 66 5 9 5 8 4 1 2 6 7 3 5 6 5 38 9 6 4 9 6 43 1 7 6 97 64 9 3 6868 8 7 2 7 5 5 28 7 2 6 37 94 1 1 5 5 2 1 3 8 4 5 1 42 5 8 1 6 4 1 1 4 67 2 4 3 5 9 8 1 21 5 2 93 2 48 4 3 3 9 13 8 2 7 6 8 2 8 3 2 36 3 9 9 44 7 9 7 1 9 2 6 3 5 5 8 3 6 2 1 7 4 7 3 7 5 8 5 7 7 9 6 9 74 4 9 6 36 59 9 7 796 8 5 83 7 4 8 6 7 6 9 8 8 5 95 4 58 6 5 2 8 1 8 1 6 9 2 5 6 4 77 6 9 1 2 4 3 84 2 2 3 3 577 1 6 23 7 2 8 7 6 11 7 4 6 19 7 4 39 8 7 55 3 7 2 6 6 4 3 81 97 8 7 8 6 34 6 8 7 8 6 2 34 3 2 9 5 1 4 1 89 1 3 4 9 1 2 2 9 78 1 2 5 4 7 1 2 4 3 2 6 4 5 9 9 67 3 5 8 5 4 29 3 4 2 7 5 8 4 9 429 9 6 4 2 6 8 9 1 9 6 8 3 1 4 3 4 9 2 5 3 7 1 4 8 1 5 2 2 897 5 2 48 5 4 2 255 9 4 3 6 9 3 7 5 2 6 3 9 6 5 2 4 9 7 9 1 5 7 3 2 1 6 3 3 6 8 2 3 74 7 2 7 7 3 6 74 9 27 1 8 6 1 4 8 1 669 58 9 5 6 54 1 9 7 5 8 3 9 1 5 8 2 3 1 6 4 2 1 6 40 2 3 5 5 2 3 69 2 3 95 1 7 5 3 2 67 9 8 7 676 2 1 7 1 1 2 3 4 3 7 4 8 6 9 7 2 9 4 9 1 8 5 3 2 7 7 6 1 4 8 8 5 2 7 4 1 8 6 88 3 3 2 89 3 9 7 1 63 1 5 5 9 1 7 8 8 69 9 6 8 64 4 6 9 3 76 8 2 1 4 7 3 8 3 2 1 5 4 8 4 2 8 5 1 5 9 5 1 8 6 5 28 6 1 3 5 1 3 7 2 3 1 6 7 8 1 9 3 97 4 3 2 4 5 4 3 3 4 1 7 2 2 6 4 5 9 7 3 5 3 5 2 8 1 6 5 3 8 4 1 6 5 3 8 4 5 8 1 2 7 6 7 7 3 6 9 4 2 8 5 7 3 4 5 8 2 7 6 1 5 7 9 3 9 41 5 3 647 79 8 7 1 9 8 2 6 7 6 2 4 1 6 2 8 5 9 6 44 5 77 68 2 3 2 6 11 9 2 4 6645 1 2 3 4 5 8 1 5 9 5 9 1 3 6 4 2 7 94 7 2 5 3 2 8 6 9 7 1 8 7 4 2 5 9 4 66 73 1 62 1 6 4 5 1 9 3 6 5 1 9 2 49 1 3 5 2 1 97 5 9 4 2 9 4 8 2 6 8 2 7 6 2 1 65 8 9 9 1 3 7 4 1 8 3 3 6 98 3 5 5 4 86 9 4 88 5 7 8 4 3 1 7 9 1 8 3 2 7 4 7 3 9 5 5 4 7 8 6 2 9 3 1 4 8 4 1 8 4 1 8 6 7 2 8 6 9 8 3 2 1 5 5 2 3 6 42 3 3 8 5 97 6 4 4 8 7 6 98 2 9 5 6 95 2 6 25 7 3 9 1 7 2 6 9 7 6 2 8 6 4 9 1 5 6 3 8 1 4 5 6 3 9 3 42 4 5 1 8 7 1 9 4 7 9 1 2 2 1 4 3 5 1 9 1 5 2 6 9 4 1 2 3 2 6 5 3 8 7 3 8 4 4 1 6 6 2 5 1 5 1 9 1 3 7 8 4 2 3 7 4 8 3 1 5 7 3 6 8 2 7 5 4 4 8 8 4 677 46 9 3 7 5 7 1 5 3 5 31 4 7 8 4 9 9 78 1 67 65 78 8 5 9 2 6 7 8 2 48 1 9 3 4 5 5 6 7 2 4 9 1 6 8 3 6 2 8 3 2 9 8 9 5 1 4 9 3 6 88 7 3 2 59 1 6 2 5 4 1 8 5 9 4 43 1 7 94 7 9 5 38 656 4 91 6 8 4 1 6 4 4 23 5 3 6 4 4 3 7 1 6 7 8 2 5 1 8 1 8 2 5 3 5 2 4 5 2 9 1 4 6 4 9 8 7 1 9 3 1 3 1 9 2 9 6 3 5 3 2 7 3 4 9 1 7 6 1 6 7 4 1 5 9 4 9 2 6 7 2 7 1 4 5 13 33 7 3 2 8 2 8 1 28 53 1 8 5 2 1 2 7 49 4 6 71 9 2 8 7 4 89 3 2 9 8 3 4 6 9 4 8 9 2 5 6 9 5 26 97 575 2 79 4 58 6 45 6 2 3 6 5 4 8 2 79 1 3 7 9 5 3 8 51 83 6 9 39 56 95 7 8 9 1 2 3 8 8 1 2 4 5 3 9 6 44 4 79 5 4 7 5 4 38 4 38 6 6 1 4 9 6 8 9 7 4 1 6 8 2 4 7 1 3 8 2 4 5 8 1 6 2 8 7 3 5 1 9 2 7 5 3 4 3 7 6 5 8 2 3 7 6 2 5 8 8 7 4 1 5 1 3 8 7 8 5 6 1 9 9 6 27 8 4 12 979 1 1 6 1 2 6 3 1 7 7 2 4 72 4 3 6 5 99 66 95 7 1 9 2 8 44 1 2 8 9 2 6 7 3 6 5 7 6 2 1 8 11 5 8 6 33 5 2 45 4 8 9 3 3 6 1 6 8 8 2 6 7 2 8 4 9 4 2 5 7 9 2 5 4 7 7 2 5 3 5 3 99 6 3 6 3 1 6 7 3 4 8 2 787 95 9 5 1 7 55 5 8 4 2 1 4 9 2 1 3 8 8 5 4 1 8 The preceding pages give a complete solution of the -intersection problem for hexagon triple systems of order 9. Lemma 4.1 3Int(9) =f0;1;2;:::;36gnf35;34;33;31g. 46 Chapter 5 n =13 n= 13 (A work in progress). To begin with, to keep the pictures from getting out of hand we will denote the hexagon triple by [x 1 ;x 2 ;x 3 ;x 4 ;x 5 ;x 6 ]or[x 1 ;x 6 ;x 5 ;x 4 ;x 3 ;x 2 ] or any cyclic 2-shift. The x 6 x 5 x 4 x 3 x 2 x 1 Figure 5.1: Hexagon triple complete solution of the -intersection problem for hexagon triple systems of order 13 remains elusive; it is a very di cult problem. To date we can show that 60 of the 75 intersection numbers are possible. Hopefully a complete solution can be obtained at a later date. Since f3k j k 2 Int(13)g 3Int(13), we need only look at numbers in 3Int(13)nf3kjk2Int(13)g. We will do this by listing 36 hexagon triple systems and 37 intersections between them. 47 To begin with 3Int(n) f3kjk2Int(n)gfollows from Lemma 2.2. We will list the remaining solutions. This will be done by listing 36 hexagon triple system and then the various intersections between them for a total of 37. [2, 10, 3, 13, 1, 7] [2, 11, 3, 10, 1, 7] [2, 11, 3, 10, 1, 12] [2, 4, 6, 10, 11, 3] [2, 9, 6, 8, 11, 3] [2, 9, 6, 10, 11, 3] [11, 9, 7, 5, 4, 3] [11, 9, 6, 3, 4, 12] [11, 9, 7, 3, 4, 12] [4, 13, 8, 9, 3, 7] [4, 6, 8, 12, 3, 7] [4, 6, 8, 12, 3, 7] [3, 8, 9, 2, 6, 5] [3, 13, 9, 1, 6, 5] [3, 13, 9, 2, 6, 5] [6, 12, 10, 8, 7, 13] [6, 11, 10, 8, 7, 12] [6, 11, 10, 5, 7, 12] [7, 4, 5, 1, 8, 10] [7, 10, 5, 9, 8, 1] [7, 10, 5, 9, 8, 1] [8, 2, 12, 10, 9, 5] [8, 2, 12, 10, 9, 3] [8, 3, 12, 10, 9, 5] [9, 2, 13, 11, 10, 5] [ 9, 2, 13, 4, 10, 5] [9, 3, 13, 4, 10, 12] [10, 4, 1, 11, 5, 7] [10, 4, 1, 11, 5, 7] [10, 3, 1, 11, 5, 7] [5, 11, 2, 1, 12, 13] [ 5, 4, 2, 1, 12, 13] [5, 4, 2, 8, 12, 13] [12, 4, 11, 8, 13, 5] [12, 1, 11, 10, 13, 5] [12, 4, 11, 8, 13, 5] [13, 10, 4, 9, 1, 3] [13, 8, 4, 9, 1, 3] [13, 10, 4, 9, 1, 11] [1, 6, 9, 7, 11, 12] [1, 4, 9, 7, 11, 5] [1, 4, 9, 7, 11, 5] [2, 3, 10, 1, 4, 6] [ 2, 3, 10, 13, 4, 5] [ 2, 1, 10, 13, 4, 5] [3, 4, 11, 2, 5, 6] [3, 4, 11, 2, 5, 6] [3, 2, 11, 8, 5, 6] [4, 9, 12, 7, 6, 8] [4, 11, 12, 10, 6, 2] [4, 11, 12, 7, 6, 1] [3, 9, 13, 6, 7, 12] [3, 9, 13, 6, 7, 12] [3, 9, 13, 2, 7, 4] [6, 9, 1, 5, 8, 11] [6, 13, 1, 5, 8, 4] [6, 13, 1, 7, 8, 4] [7, 1, 2, 13, 9, 11] [7, 13, 2, 6, 9, 11] [7, 13, 2, 6, 9, 11] [8, 6, 11, 13, 10, 2] [8, 13, 11, 6, 10, 2] [8, 13, 11, 6, 10, 2] [9, 12, 4, 2, 5, 10] [9, 12, 4, 7, 5, 8] [9, 8, 4, 2, 5, 1] [10, 1, 3, 8, 12, 6] [10, 1, 3, 8, 12, 9] [10, 8, 3, 1, 12, 9] [5, 3, 6, 1, 13, 12] [5, 3, 6, 1, 13, 1] [5, 3, 6, 1, 13, 12] [12, 3, 7, 8, 1, 11] [12, 6, 7, 8, 1, 2] [12, 6, 7, 8, 1, 2] [13, 4, 8, 12, 1, 7] [13, 11, 8, 10, 2, 7] [13, 6, 8, 10, 2, 7] H 1 H 2 H 3 48 [12, 8, 3, 13, 7, 5] [12, 8, 3, 10, 7, 1] [2, 5, 3, 12, 1, 8] [12, 10, 4, 2, 9, 8] [12, 6, 4, 2, 9, 13] [11, 13, 12, 7, 2, 6] [9, 11, 1, 12, 6, 10] [9, 11, 1, 12, 6, 3] [ 4, 2, 13, 12, 11, 3] [6, 2, 5, 11, 3, 4] [6, 13, 5, 11, 3, 9] [5, 9, 1, 7, 4, 10] [3, 9, 11, 13, 4, 6] [ 3, 5, 11, 13, 4, 8] [6, 11, 2, 3, 5, 12] [4, 12, 10, 5, 1, 13] [4, 12, 10, 13, 1, 5] [7, 5, 11, 2, 6, 10] [1, 4, 5, 9, 8, 6] [1, 10, 5, 9, 8, 6] [8, 6, 4, 1, 7, 3] [5, 6, 2, 12, 11, 10] [5, 12, 2, 6, 11, 10] [9, 1, 5, 13, 8, 11] [11, 12, 13, 1, 10, 8] [11, 12, 13, 9, 10, 8] [ 10, 7, 6, 3, 9, 2] [10, 3, 7, 4, 8, 2] [10, 6, 7, 5, 8, 2] [3, 8, 7, 6, 10, 13] [8, 3, 12, 5, 2, 10] [8, 9, 12, 5, 2, 13] [12, 10, 8, 7, 3, 1] [2, 7, 9, 12, 13, 8] [2, 7, 9, 10, 13, 8] [13, 7, 9, 4, 12, 11] [13, 5, 6, 10, 7, 2] [13, 5, 6, 11, 7, 3] [1, 11, 10, 3, 13, 6] [7, 6, 11, 3, 9, 1] [7, 4, 11, 1, 9, 2] [11, 9, 8, 2, 1, 10] [12, 3, 10, 9, 6, 1] [12, 3, 10, 7, 6, 4] [4, 12, 9, 10, 2, 13] [3, 6, 9, 5, 8, 4] [3, 11, 9, 12, 8, 4] [5, 4, 10, 1, 11, 7] [6, 11, 2, 9, 4, 12] [6, 5, 2, 10, 4, 3] [6, 9, 3, 11, 4, 8] [3, 5, 13, 10, 1, 2] [3, 7, 13, 4, 1, 2] [7, 2, 12, 6, 5, 11] [4, 11, 7, 8, 5, 1] [4, 11, 7, 8, 5, 9] [8, 5, 13, 1, 6, 4] [1, 7, 12, 2, 11, 8] [1, 7, 12, 2, 11, 8] [9, 5, 1, 4, 7, 13] [5, 4, 9, 13, 10, 11] [5, 4, 9, 6, 10, 1] [10, 9, 2, 1, 8, 12] [11, 7, 6, 13, 8, 1] [11, 2, 6, 1, 8, 10] [11, 4, 3, 6, 9, 8] [10, 7, 3, 1, 2, 4] [10, 12, 3, 1, 2, 4] [12, 9, 4, 5, 10, 8] [8, 7, 4, 11, 13, 6] [8, 7, 4, 1, 13, 6] [13, 8, 5, 2, 3, 10] [2, 3, 1, 9, 7, 13] [2, 3, 1, 9, 7, 13] [1, 13, 6, 5, 12, 3] [13, 3, 5, 7, 12, 9] [13, 3, 5, 7, 12, 11] [2, 12, 7, 9, 13, 4] H 4 H 5 H 6 49 [2, 8, 3, 12, 1, 5] [2, 11, 3, 12, 1, 10] [2, 11, 3, 10, 1, 12] [11, 13, 12, 7, 2, 6] [2, 9, 6, 10, 11, 3] [2, 9, 6, 10, 11, 3] [4, 2, 13, 12, 11, 1] [11, 9, 7, 3, 4, 12] [11, 9, 7, 3, 4, 12] [5, 9, 3, 7, 4, 10] [4, 9, 8, 10, 3, 7] [4, 6, 8, 12, 3, 7] [6, 11, 2, 1, 5, 12] [3, 13, 9, 2, 6, 5] [3, 13, 9, 2, 6, 5] [7, 5, 11, 2, 6, 10] [6, 11, 10, 5, 7, 1] [6, 11, 10, 5, 7, 12] [8, 6, 4, 3, 7, 1] [7, 10, 5, 11, 8, 1] [7, 10, 5, 9, 8, 1] [9, 3, 5, 13, 8, 11] [8, 2, 12, 10, 9, 4] [8, 3, 12, 10, 9, 5] [10, 7, 6, 1, 9, 2] [9, 3, 13, 4, 10, 12] [9, 3, 13, 4, 10, 12] [1, 8, 7, 6, 10, 13] [10, 2, 1, 9, 5, 7] [10, 3, 1, 11, 5, 7] [12, 10, 8, 7, 1, 3] [5, 4, 2, 8, 12, 13] [5, 4, 2, 1, 12, 13] [13, 7, 9, 4, 12, 11] [12, 4, 11, 1, 13, 5] [12, 4, 11, 8, 13, 5] [3, 11, 10, 1, 13, 6] [13, 10, 4, 6, 1, 11] [13, 10, 4, 9, 1, 6] [11, 9, 8, 2, 3, 10] [1, 5, 9, 7, 11, 13] [1, 4, 9, 7, 11, 5] [4, 12, 9, 10, 2, 13] [2, 1, 10, 13, 4, 5] [2, 8, 10, 13, 4, 5] [5, 4, 10, 3, 11, 7] [3, 2, 11, 8, 5, 6] [3, 2, 11, 1, 5, 6] [6, 9, 1, 11, 4, 8] [4, 11, 12, 7, 6, 1] [4, 11, 12, 7, 6, 8] [7, 2, 12, 6, 5, 11] [3, 9, 13, 2, 7, 4] [3, 9, 13, 2, 7, 4] [8, 5, 13, 3, 6, 4] [6, 4, 1, 7, 8, 13] [6, 13, 1, 7, 8, 4] [9, 5, 3, 4, 7, 13] [7, 13, 2, 6, 9, 11] [7, 13, 2, 6, 9, 11] [10, 9, 2, 3, 8, 12] [8, 5, 11, 6, 10, 3] [8, 13, 11, 6, 10, 2] [11, 4, 1, 6, 9, 8] [9, 8, 4, 2, 5, 1] [9, 1, 4, 2, 5, 8] [12, 9, 4, 5, 10, 8] [10, 8, 3, 1, 12, 9] [10, 1, 3, 8, 12, 9] [13, 8, 5, 2, 1, 10] [5, 3, 6, 8, 13, 12] [5, 3, 6, 1, 13, 12] [3, 13, 6, 5, 12, 1] [12, 6, 7, 8, 1, 3] [12, 6, 7, 8, 1, 2] [2, 12, 7, 9, 13, 4] [13, 6, 8, 12, 2, 7] [13, 11, 8, 10, 2, 7] H 7 H 8 H 9 50 [12, 6, 3, 2, 1, 8] [1, 2, 3, 8, 12, 6] [1, 2, 3, 10, 12, 7] [7, 13, 2, 11, 12, 5] [12, 11, 2, 13, 7, 5] [7, 10, 6, 4, 12, 1] [9, 12, 13, 2, 7, 3] [7, 2, 13, 12, 9, 1] [9, 1, 11, 4, 7, 2] [6, 4, 1, 11, 9, 10] [9, 11, 3, 4, 6, 10] [1, 10, 5, 4, 9, 11] [5, 7, 12, 3, 6, 2] [6, 1, 12, 7, 5, 2] [6, 12, 4, 13, 1, 8] [7, 6, 11, 10, 5, 12] [5, 10, 11, 6, 7, 12] [11, 8, 10, 7, 6, 2] [8, 5, 9, 1, 11, 3] [11, 3, 9, 5, 8, 1] [5, 3, 11, 10, 8, 7] [4, 1, 6, 13, 8, 7] [8, 13, 6, 3, 4, 7] [4, 10, 2, 12, 5, 9] [10, 11, 5, 3, 4, 12] [4, 1, 5, 11, 10, 12] [10, 9, 13, 1, 4, 2] [3, 8, 11, 5, 10, 13] [10, 5, 11, 8, 1, 13] [8, 4, 3, 12, 10, 11] [2, 10, 8, 11, 3, 1] [1, 11, 8, 10, 2, 3] [2, 5, 12, 9, 8, 13] [13, 11, 4, 9, 2, 7] [2, 9, 4, 11, 13, 7] [13, 3, 7, 9, 2, 8] [1, 7, 10, 3, 13, 5] [13, 1, 10, 7, 3, 5] [3, 6, 9, 10, 13, 7] [7, 4, 8, 12, 1, 10] [3, 12, 8, 4, 7, 10] [7, 11, 4, 8, 3, 13] [4, 2, 9, 13, 12, 10] [12, 13, 9, 2, 4, 10] [9, 13, 10, 3, 12, 8] [6, 9, 10, 1, 7, 11] [7, 3, 10, 9, 6, 11] [8, 5, 7, 12, 1, 6] [5, 4, 3, 7, 9, 8] [9, 7, 1, 4, 5, 8] [6, 11, 2, 7, 9, 3] [11, 12, 2, 5, 6, 7] [6, 5, 2, 12, 11, 7] [11, 12, 13, 4, 1, 9] [8, 6, 13, 1, 5, 9] [5, 3, 13, 6, 8, 9] [5, 11, 3, 9, 6, 13] [4, 6, 1, 9, 11, 13] [11, 9, 3, 6, 4, 13] [4, 6, 12, 13, 11, 7] [10, 4, 12, 1, 8, 2] [8, 3, 12, 4, 10, 2] [10, 6, 7, 8, 5, 1] [7, 9, 3, 5, 4, 8] [4, 5, 1, 9, 7, 8] [8, 12, 9, 5, 4, 3] [2, 4, 9, 6, 10, 8] [10, 6, 9, 4, 2, 8] [2, 3, 1, 5, 10, 4] [13, 8, 6, 12, 3, 10] [1, 12, 6, 8, 13, 10] [13, 5, 6, 1, 8, 2] [1, 13, 5, 6, 2, 3] [2, 6, 5, 13, 3, 1] [3, 5, 11, 6, 2, 1] [12, 2, 11, 4, 13, 9] [13, 4, 11, 2, 12, 9] [12, 2, 5, 6, 13, 11] H 10 H 11 H 12 51 [3, 2, 1, 6, 12, 10] [1, 13, 3, 10, 2, 7] [2, 11, 3, 13, 1, 10] [3, 12, 10, 6, 9, 4] [11, 8, 6, 4, 2, 5] [2, 9, 6, 10, 11, 3] [3, 5, 11, 13, 6, 7] [4, 5, 7, 9, 11, 3] [11, 9, 7, 3, 4, 12] [3, 1, 2, 9, 5, 11] [3, 9, 8, 13, 4, 11] [4, 9, 8, 10, 3, 7] [3, 8, 13, 12, 4, 9] [6, 1, 9, 8, 3, 5] [3, 13, 9, 1, 6, 5] [3, 6, 7, 10, 8, 13] [7, 8, 10, 12, 6, 13] [6, 11, 10, 5, 7, 12] [12, 5, 8, 11, 9, 7] [8, 1, 5, 4, 7, 10] [7, 4, 5, 11, 8, 10] [12, 13, 4, 5, 6, 1] [9, 4, 12, 2, 8, 3] [8, 2, 12, 10, 9, 3] [12, 3, 10, 13, 5, 8] [10, 11, 13, 2, 9, 5] [9, 2, 13, 4, 10, 5] [12, 4, 13, 6, 11, 2] [2, 12, 8, 4, 13, 9] [10, 4, 1, 8, 5, 7] [12, 11, 2, 13, 7, 9] [5, 8, 1, 4, 10, 9] [5, 11, 2, 8, 12, 13] [1, 8, 4, 3, 9, 13] [12, 8, 2, 11, 5, 13] [12, 4, 11, 1, 13, 5] [1, 5, 7, 3, 6, 12] [13, 10, 11, 1, 12, 5] [13, 8, 4, 6, 1, 11] [1, 10, 11, 3, 5, 7] [1, 10, 4, 8, 13, 3] [1, 5, 9, 7, 11, 12] [1, 9, 13, 5, 10, 11] [11, 7, 9, 6, 1, 12] [2, 1, 10, 13, 4, 5] [1, 3, 2, 6, 8, 4] [4, 1, 10, 3, 2, 6] [3, 4, 11, 8, 5, 6] [9, 2, 5, 4, 6, 10] [5, 2, 11, 4, 3, 6] [4, 9, 12, 10, 6, 2] [9, 12, 7, 4, 11, 8] [6, 10, 12, 9, 4, 2] [3, 9, 13, 2, 7, 12] [9, 1, 13, 7, 2, 5] [7, 6, 13, 1, 3, 12] [6, 4, 1, 7, 8, 13] [6,8,2,4,10,9] [8,5,1,9,6,11] [7,1,2,6,9,11] [6, 11, 13, 3, 8, 2] [9, 13, 2, 1, 7, 11] [8, 6, 11, 13, 10, 3] [5, 12, 8, 1, 4, 6] [10, 13, 11, 6, 8, 7] [9, 8, 4, 2, 5, 1] [5, 1, 7, 2, 10, 13] [5, 7, 4, 12, 9, 10] [10, 2, 3, 1, 12, 9] [11, 9, 8, 7, 10, 1] [12, 7, 3, 2, 10, 6] [5, 3, 6, 7, 13, 12] [11, 12, 2, 10, 4, 7] [13, 7, 6, 3, 5, 12] [12, 6, 7, 8, 1, 3] [10, 8, 7, 11, 4, 2] [1, 2, 7, 3, 12, 11] [13, 6, 8, 12, 2, 7] H 13 H 14 H 15 52 [6, 11, 3, 12, 1, 10] [2, 11, 3, 12, 1 10] [6, 7, 3, 12, 1, 10] [6,9,2,8,11,5] [2,9,6,8,11,5] [6,9,2,8,7,5] [11, 9, 7, 5, 4, 3] [11, 9, 7, 5, 4, 3] [7, 9, 11, 5, 4, 3] [4, 13, 8, 10, 3, 7] [4, 13, 8, 10, 3, 7] [4, 13, 8, 10, 3, 11] [3, 8, 9, 1, 2, 5] [3, 8, 9, 1, 6, 5] [3, 8, 9, 1, 2, 5] [2, 11, 10, 8, 7, 13] [6, 11, 10, 8, 7, 13] [2, 7, 10, 8, 11, 13] [7, 10, 5, 11, 8, 1] [7, 10, 5, 11, 8, 1] [11, 10, 5, 7, 8, 1] [8, 6, 12, 4, 9, 3] [8, 2, 12, 4, 9, 3] [8, 6, 12, 4, 9, 3] [9, 3, 13, 11, 10, 12] [9, 3, 13, 11, 10, 12] [9, 3, 13, 7, 10, 12] [10, 4, 1, 8, 5, 9] [10, 4, 1, 8, 5, 9] [10, 4, 1, 8, 5, 9] [5, 4, 6, 8, 12, 13] [5, 4, 2, 8, 12, 13] [5, 4, 6, 8, 12, 13] [12, 4, 11, 1, 13, 5] [12, 4, 11, 1, 13, 5] [12, 4, 7, 1, 13, 5] [13, 10, 4, 2, 1, 3] [13, 10, 4, 6, 1, 3] [13, 10, 4, 2, 1, 3] [1, 5, 9, 7, 11, 12] [1, 5, 9, 7, 11, 12] [1, 5, 9, 11, 7, 12] [6, 3, 10, 1, 4, 2] [2, 3, 10, 1, 4, 6] [6, 3, 10, 1, 4, 2] [3, 4, 11, 6, 5, 2] [3, 4, 11, 2, 5, 6] [3, 4, 7, 6, 5, 2] [4, 9, 12, 10, 2, 6] [4, 9, 12, 10, 6, 2] [4, 9, 12, 10, 2, 6] [3, 1, 13, 6, 7, 12] [3, 1, 13, 2, 7, 12] [3, 1, 13, 6, 11, 12] [2,9,1,5,8,13] [6,9,1,5,8,13] [2,9,1,5,8,13] [7, 1, 6, 13, 9, 11] [7, 1, 2, 13, 9, 11] [11, 1, 6, 13, 9, 7] [8, 2, 11, 13, 10, 7] [8, 6, 11, 13, 10, 7] [8, 2, 7, 13, 10, 11] [9, 8, 4, 7, 5, 10] [9, 8, 4, 7, 5, 10] [9, 8, 4, 11, 5, 10] [10, 6, 3, 7, 12, 2] [10, 2, 3, 7, 12, 6] [10, 6, 3, 11, 12, 2] [5, 3, 2, 7, 13, 12] [5, 3, 6, 7, 13, 12] [5, 3, 2, 11, 13, 12] [12, 2, 7, 6, 1, 11] [12, 6, 7, 2, 1, 11] [12, 2, 11, 6, 1, 7] [13, 4, 8, 12, 6, 9] [13, 4, 8, 13, 2, 9] [13, 4, 8, 12, 6, 9] H 16 H 17 H 18 53 [1, 13, 3, 10, 2, 11] [6, 7, 3, 12, 1, 10] [2, 10, 3, 13, 1, 7] [7, 8, 6, 4, 2, 5] [6, 9, 2, 4, 7, 5] [11, 1, 12, 8, 2, 5] [4, 5, 11, 9, 7, 3] [7, 9, 11, 5, 8, 3] [11, 10, 13, 8, 4, 3] [3, 9, 8, 13, 4, 7] [8, 13, 4, 10, 3, 11] [4, 7, 5, 6, 1, 10] [6, 1, 9, 8, 3, 5] [3, 4, 9, 1, 2, 5] [5, 1, 6, 4, 2, 11] [11, 8, 10, 12, 6, 13] [2, 7, 10, 4, 11, 13] [7, 9, 11, 8, 6, 13] [8, 1, 5, 4, 11, 10] [11, 10, 5, 7, 4, 1] [7, 10, 8, 13, 4, 5] [9, 4, 12, 2, 8, 3] [4, 6, 12, 8, 9, 3] [8, 1, 9, 10, 5, 3] [10, 7, 13, 2, 9, 5] [9, 3, 13, 7, 10, 12] [9, 5, 10, 12, 6, 3] [2, 12, 8, 4, 13, 9] [10, 8, 1, 4, 5, 9] [7, 8, 10, 2, 3, 12] [5, 8, 1, 4, 10, 9] [5, 8, 6, 4, 12, 13] [8, 2, 12, 7, 3, 5] [12, 8, 2, 7, 5, 13] [12, 8, 7, 1, 13, 5] [12, 5, 13, 2, 9, 4] [13, 10, 7, 1, 12, 5] [13, 10, 8, 2, 1, 3] [10, 11, 13, 3, 1, 4] [1, 10, 4, 8, 13, 3] [1, 5, 9, 11, 7, 12] [8, 6, 11, 12, 1, 9] [7, 11, 9, 6, 1, 12] [6, 3, 10, 1, 8, 2] [4, 12, 9, 13, 2, 6] [4, 1, 10, 3, 2, 6] [3, 8, 7, 6, 5, 2] [10, 13, 11, 2, 5, 9] [5, 2, 7, 4, 3, 6] [8, 9, 12, 10, 2, 6] [4, 2, 6, 9, 3, 11] [6, 10, 12, 9, 4, 2] [3, 1, 13, 6, 11, 12] [7, 3, 12, 13, 5, 4] [11, 6, 13, 1, 3, 12] [2, 9, 1, 5, 4, 13] [8, 4, 13, 7, 6, 11] [8, 5, 1, 9, 6, 7] [11, 1, 6, 13, 9, 7] [7, 11, 9, 8, 1, 2] [9, 13, 2, 1, 11, 7] [4, 2, 7, 13, 10, 11] [8, 7, 10, 3, 2, 12] [10, 13, 7, 6, 8, 11] [9, 4, 8, 11, 5, 10] [9, 7, 11, 4, 3, 6] [5, 11, 4, 12, 9, 10] [10, 6, 3, 11, 12, 2] [10, 6, 12, 9, 4, 1] [12, 11, 3, 2, 10, 6] [5, 3, 2, 11, 13, 12] [5, 12, 13, 1, 3, 8] [13, 11, 6, 3, 5, 12] [12, 2, 11, 6, 1, 7] [6, 10, 12, 11, 1, 5] [1, 2, 11, 3, 12, 7] [13, 8, 4, 12, 6, 9] [7, 6, 13, 9, 2, 1] H 19 H 20 H 21 54 [2, 11, 3, 12, 1, 10] [2, 10, 3, 13, 1, 7] [2, 10, 3, 13, 1, 7] [6, 10, 11, 3, 2, 9] [11, 1, 12, 7, 2, 6] [11, 1, 12, 8, 2, 5] [7, 9, 11, 12, 4, 3] [11, 12, 13, 8, 4, 3] [11, 10, 13, 8, 4, 3] [4,6,8,10,3,7] [4,7,5,8,1,10] [4,7,5,8,1,10] [6,2,9,13,3,5] [5,3,6,4,2,11] [5,3,6,4,2,11] [7, 5, 10, 11, 6, 12] [7, 9, 11, 8, 6, 10] [7, 9, 11, 8, 6, 13] [7, 1, 8, 9, 5, 10] [7, 3, 8, 13, 4, 5] [7, 10, 8, 13, 4, 5] [9, 10, 12, 2, 8, 4] [8, 3, 9, 10, 5, 1] [8, 3, 9, 10, 5, 1] [10, 4, 13, 3, 9, 12] [9, 2, 10, 12, 6, 1] [9, 5, 10, 12, 6, 1] [5, 7, 10, 2, 1, 11] [7, 8, 10, 13, 3, 12] [7, 8, 10, 2, 3, 12] [5, 13, 12, 1, 2, 4] [8, 2, 12, 1, 3, 9] [8, 2, 12, 7, 3, 5] [12, 5, 13, 1, 11, 4] [12, 5, 13, 2, 9, 4] [12, 5, 13, 2, 9, 4] [4, 10, 13, 6, 1, 9] [10, 11, 13, 6, 1, 4] [10, 11, 13, 3, 1, 4] [9, 7, 11, 13, 1, 5] [8, 9, 11, 10, 1, 2] [8, 6, 11, 12, 1, 9] [4, 13, 10, 8, 2, 5] [4, 12, 9, 13, 2, 6] [4, 12, 9, 13, 2, 6] [5, 8, 11, 2, 3, 6] [10, 13, 11, 2, 5, 9] [10, 13, 11, 2, 5, 9] [6, 7, 12, 11, 4, 1] [4, 8, 6, 5, 3, 11] [4, 2, 6, 9, 3, 11] [7, 2, 13, 9, 3, 4] [7, 3, 12, 6, 5, 11] [7, 3, 12, 13, 5, 4] [6, 13, 8, 7, 1, 4] [8, 5, 13, 7, 6, 11] [8, 4, 13, 7, 6, 11] [7, 11, 9, 6, 2, 13] [7, 13, 9, 5, 1, 4] [7, 11, 9, 6, 1, 2] [10, 6, 11, 5, 8, 3] [8, 7, 10, 3, 2, 12] [8, 7, 10, 3, 2, 12] [5, 1, 9, 8, 4, 2] [9, 7, 11, 4, 3, 6] [9, 7, 11, 4, 3, 8] [10, 9, 12, 8, 3, 1] [10, 8, 12, 9, 4, 5] [10, 6, 12, 9, 4, 1] [6, 8, 13, 12, 5, 3] [5, 12, 13, 1, 3, 2] [5, 12, 13, 1, 3, 6] [7, 6, 12, 3, 1, 8] [6, 10, 12, 11, 1, 9] [6, 10, 12, 11, 1, 5] [8, 11, 13, 7, 2, 12] [7, 6, 13, 4, 2, 1] [7, 6, 13, 9, 2, 1] H 22 H 23 H 24 55 [2, 10, 3, 13, 1, 7] [2, 10, 3, 13, 1, 7] [2, 3, 10, 13, 1, 7] [11, 1, 12, 8, 2, 5] [11, 1, 12, 6, 2, 5] [11, 1, 12, 6, 2, 5] [11, 10, 13, 8, 4, 3] [11, 10, 13, 6, 4, 3] [11, 3, 13, 6, 4, 10] [4,7,5,6,1,10] [4,7,5,8,1,10] [4,7,5,8,1,3] [5,3,6,4,2,11] [5,1,8,4,2,11] [5,1,8,4,2,11] [7, 9, 11, 8, 6, 13] [7, 9, 11, 6, 8, 13] [7, 9, 11, 6, 8, 13] [7, 10, 8, 13, 4, 5] [7, 10, 6, 13, 4, 5] [7, 3, 6, 13, 4, 5] [8, 3, 9, 10, 5, 1] [6, 1, 9, 10, 5, 3] [6, 1, 9, 3, 5, 10] [9, 5, 10, 12, 6, 1] [9, 5, 10, 12, 8, 3] [9, 5, 3, 12, 8, 10] [7, 8, 10, 2, 3, 12] [7, 6, 10, 2, 3, 12] [7, 6, 3, 2, 10, 12] [8, 2, 12, 7, 3, 5] [6, 2, 12, 7, 3, 5] [6, 2, 12, 7, 10, 5] [12, 5, 13, 2, 9, 4] [12, 5, 13, 2, 9, 4] [12, 5, 13, 2, 9, 4] [10, 11, 13, 3, 1, 4] [10, 11, 13, 3, 1, 4] [3, 11, 13, 10, 1, 4] [8, 6, 11, 12, 1, 9] [6, 8, 11, 12, 1, 9] [6, 8, 11, 12, 1, 9] [4, 12, 9, 13, 2, 6] [4, 12, 9, 13, 2, 8] [4, 12, 9, 13, 2, 8] [10, 13, 11, 2, 5, 9] [10, 13, 11, 2, 5, 9] [3, 13, 11, 2, 5, 9] [4,2,6,9,3,11] [4,2,8,9,3,11] [4,2,8,9,10,11] [7, 3, 12, 13, 5, 4] [7, 3, 12, 13, 5, 4] [7, 10, 12, 13, 5, 4] [8, 4, 13, 7, 6, 11] [6, 4, 13, 7, 8, 11] [6, 4, 13, 7, 8, 11] [7, 11, 9, 8, 1, 2] [7, 11, 9, 6, 1, 2] [7, 11, 9, 6, 1, 2] [8, 7, 10, 3, 2, 12] [6, 7, 10, 3, 2, 12] [6, 7, 3, 10, 2, 12] [9, 7, 11, 4, 3, 6] [9, 7, 11, 4, 3, 8] [9, 7, 11, 4, 10, 8] [10, 6, 12, 9, 4, 1] [10, 8, 12, 9, 4, 1] [3, 8, 12, 9, 4, 1] [5, 12, 13, 1, 3, 8] [5, 12, 13, 1, 3, 6] [5, 12, 13, 1, 10, 6] [6, 10, 12, 11, 1, 5] [8, 10, 12, 11, 1, 5] [8, 3, 12, 11, 1, 5] [7, 6, 13, 9, 2, 1] [7, 8, 13, 9, 2, 1] [7, 8, 13, 9, 2, 1] H 25 H 26 H 27 56 [7, 10, 3, 13, 1, 2] [2, 10, 3, 13, 1, 7] [2, 10, 3, 13, 1, 7] [11, 1, 12, 8, 7, 5] [11, 1, 12, 6, 2, 5] [11, 1, 12, 6, 2, 5] [11, 10, 13, 8, 4, 3] [11, 10, 13, 6, 4, 3] [11, 10, 13, 6, 4, 3] [4,2,5,6,1,10] [4,7,5,8,1,10] [4,7,5,6,1,10] [5,1,6,4,7,11] [5,3,8,4,2,11] [5,3,8,4,2,11] [2, 9, 11, 8, 6, 13] [7, 9, 11, 6, 8, 13] [7, 9, 11, 6, 8, 13] [2, 10, 8, 13, 4, 5] [7, 10, 6, 13, 4, 5] [7, 10, 6, 13, 4, 5] [8, 1, 9, 10, 5, 3] [6, 1, 9, 10, 5, 3] [6, 1, 9, 10, 5, 3] [9, 5, 10, 12, 6, 3] [9, 5, 10, 12, 8, 1] [9, 5, 10, 12, 8, 1] [2, 8, 10, 7, 3, 12] [7, 6, 10, 2, 3, 12] [7, 6, 10, 2, 3, 12] [8, 7, 12, 2, 3, 5] [6, 2, 12, 7, 3, 9] [6, 2, 12, 7, 3, 9] [12, 5, 13, 7, 9, 4] [12, 5, 13, 2, 9, 4] [12, 5, 13, 2, 9, 4] [10, 11, 13, 3, 1, 4] [10, 11, 13, 3, 1, 4] [10, 11, 13, 3, 1, 4] [8, 6, 11, 12, 1, 9] [6, 8, 11, 12, 1, 5] [6, 8, 11, 12, 1, 5] [4, 12, 9, 13, 7, 6] [4, 12, 9, 13, 2, 8] [4, 12, 9, 13, 2, 8] [10, 13, 11, 7, 5, 9] [10, 13, 11, 2, 5, 9] [10, 13, 11, 2, 5, 9] [4,7,6,9,3,11] [4,2,8,9,3,11] [4,2,8,9,3,11] [2, 3, 12, 13, 5, 4] [7, 3, 12, 13, 5, 4] [7, 3, 12, 13, 5, 4] [8, 4, 13, 2, 6, 11] [6, 4, 13, 7, 8, 11] [6, 4, 13, 7, 8, 11] [2, 11, 9, 8, 1, 7] [7, 11, 9, 6, 1, 2] [7, 11, 9, 8, 1, 2] [8, 2, 10, 3, 7, 12] [6, 7, 10, 3, 2, 12] [6, 7, 10, 3, 2, 12] [9, 2, 11, 4, 3, 6] [9, 7, 11, 4, 3, 8] [9, 7, 11, 4, 3, 6] [10, 6, 12, 9, 4, 1] [10, 8, 12, 9, 4, 1] [10, 8, 12, 9, 4, 1] [5, 12, 13, 1, 3, 8] [5, 12, 13, 1, 3, 6] [5, 12, 13, 1, 3, 8] [6, 10, 12, 11, 1, 5] [8, 10, 12, 11, 1, 5] [8, 10, 12, 11, 1, 5] [2, 6, 13, 9, 7, 1] [7, 8, 13, 9, 2, 1] [7, 8, 13, 9, 2, 1] H 28 H 29 H 30 57 [2, 10, 3, 4, 1, 7] [1, 13, 3, 10, 6, 11] [2, 12, 8, 4, 13, 9] [11, 1, 12, 8, 2, 5] [7, 8, 2, 4, 6, 5] [1, 13, 3, 10, 2, 7] [11, 3, 13, 8, 4, 10] [4, 5, 11, 9, 7, 8] [11, 8, 6, 4, 2, 5] [4, 7, 5, 6, 1, 3] [3, 9, 8, 13, 4, 7] [4, 5, 7, 9, 11, 3] [5, 1, 6, 4, 2, 11] [2, 1, 9, 8, 3, 5] [3, 9, 8, 13, 4, 11] [7, 9, 11, 8, 6, 13] [11, 8, 10, 12, 2, 13] [6, 1, 9, 8, 3, 5] [7, 10, 8, 13, 4, 5] [8, 1, 5, 4, 11, 10] [7, 3, 10, 12, 6, 13] [8, 1, 9, 10, 5, 3] [9, 4, 12, 6, 8, 3] [8, 1, 5, 4, 7, 10] [9, 4, 12, 10, 6, 3] [10, 7, 13, 6, 9, 5] [9, 4, 12, 2, 8, 3] [7, 8, 10, 2, 3, 12] [6, 12, 8, 4, 13, 9] [10, 11, 13, 2, 9, 5] [8, 2, 12, 7, 3, 5] [5, 8, 1, 4, 10, 9] [5, 8, 1, 4, 10, 9] [10,1, 13, 2, 9, 5] [12, 8, 6, 7, 5, 13] [12, 3, 2, 11, 5, 13] [12, 5, 13, 10, 1, 11] [13, 10, 7, 1, 12, 5] [13, 10, 11, 1, 12, 5] [8, 6, 11, 12, 1, 9] [1, 10, 4, 8, 13, 3] [1, 10, 4, 8, 13, 3] [4, 12, 9, 13, 2, 6] [7, 11, 9, 2, 1, 12] [11, 7, 9, 6, 1, 12] [10, 13, 11, 2, 5, 9] [4, 1, 10, 3, 6, 2] [4, 1, 10, 8, 2, 6] [4, 2, 6, 9, 3, 11] [5, 6, 7, 4, 3, 2] [5, 2, 11, 4, 3, 6] [7, 3, 12, 13, 5, 4] [2, 10, 12, 9, 4, 6] [6, 10, 12, 9, 4, 2] [8, 4, 13, 7, 6, 11] [11, 2, 13, 1, 3, 12] [7, 6, 13, 1, 3, 12] [7, 11, 9, 8, 1, 2] [8, 5, 1, 9, 2, 7] [8, 5, 1, 9, 6, 11] [8, 7, 10, 3, 2, 12] [9, 13, 6, 1, 11, 7] [9, 13, 2, 1, 7, 11] [9, 7, 11, 13, 3, 6] [10, 13, 7, 2, 8, 11] [10, 13, 11, 6, 8, 7] [10, 6, 12, 9, 4, 11] [5, 11, 4, 12, 9, 10] [5, 7, 4, 12, 9, 10] [5, 12, 13, 1, 3, 8] [12, 11, 3, 6, 10, 2] [12, 7, 3, 2, 10, 6] [6, 12, 10, 4, 1, 5] [13, 11, 2, 3, 5, 12] [13, 7, 6, 3, 5, 12] [7, 6, 13, 9, 2, 1] [1, 6, 11, 3, 12, 7] [1, 2, 7, 8, 12, 11] H 31 H 32 H 33 58 [2, 12, 8, 4, 13, 9] [2, 3, 10, 13, 1, 7] [1, 13, 3, 12, 2, 7] [1, 13, 3, 12, 2, 7] [5, 1, 12, 6, 2, 11] [2, 3, 12, 6, 10, 8] [11, 8, 6, 4, 2, 5] [5, 3, 13, 6, 4, 10] [4, 8, 13, 11, 10, 1] [4, 5, 7, 9, 11, 13] [4, 7, 11, 8, 1, 3] [1, 6, 5, 7, 4, 10] [3, 9, 8, 13, 4, 11] [11, 1, 8, 4, 2, 5] [2, 4, 6, 1, 5, 11] [6, 1, 9, 8, 3, 5] [7, 9, 5, 6, 8, 13] [6, 12, 10, 3, 7, 13] [7, 3, 10, 12, 6, 13] [7, 3, 6, 13, 4, 11] [4, 13, 8, 12, 7, 5] [8, 1, 5, 4, 7, 10] [6, 1, 9, 3, 11, 10] [5, 10, 9, 1, 8, 3] [9, 4, 12, 7, 8, 3] [9, 11, 3, 12, 8, 10] [6, 8, 11, 7, 9, 3] [10, 11, 13, 2, 9, 5] [7, 6, 3, 2, 10, 12] [3, 4, 11, 9, 7, 10] [5, 8, 1, 4, 10, 8] [6, 2, 12, 7, 10, 11] [3, 2, 12, 7, 8, 5] [12, 3, 2, 11, 5, 13] [12, 11, 13, 2, 9, 4] [9, 2, 13, 5, 12, 4] [13, 10, 11, 1, 12, 5] [3, 5, 13, 10, 1, 4] [1, 3, 13, 10, 11, 12] [1, 10, 4, 8, 13, 3] [6, 8, 5, 12, 1, 9] [1, 4, 10, 2, 8, 9] [11, 7, 9, 6, 1, 12] [4, 12, 9, 13, 2, 8] [2, 13, 9, 12, 4, 6] [4, 1, 10, 8, 2, 6] [3, 13, 5, 2, 11, 9] [5, 2, 11, 13, 10, 9] [5, 2, 11, 4, 3, 6] [4, 2, 8, 9, 10, 5] [3, 9, 6, 2, 4, 11] [6, 10, 12, 9, 4, 2] [7, 10, 12, 13, 11, 4] [5, 13, 12, 8, 7, 4] [7, 6, 13, 1, 3, 10] [6, 4, 13, 7, 8, 5] [6, 7, 13, 4, 8, 11] [8, 5, 1, 9, 6, 11] [7, 5, 9, 6, 1, 2] [1, 8, 9, 11, 7, 2] [9, 13, 2, 1, 7, 11] [6, 7, 3, 10, 2, 12] [2, 5, 11, 6, 8, 10] [10, 13, 11, 6, 8, 2] [9, 7, 5, 4, 10, 8] [3, 7, 10, 5, 9, 6] [5, 7, 4, 12, 9, 10] [3, 8, 12, 9, 4, 1] [4, 9, 12, 1, 11, 3] [12, 7, 3, 2, 10, 6] [11, 12, 13, 1, 10, 6] [3, 1, 13, 12, 5, 8] [13, 7, 6, 3, 5, 12] [8, 3, 12, 5, 1, 11] [1, 11, 12, 10, 6, 5] [1, 2, 7, 8, 12, 11] [7, 8, 13, 9, 2, 1] [2, 9, 13, 6, 7, 1] H 34 H 35 H 36 59 jH 12 \H 16 j=1; ff2;8;13gg jH 11 \H 16 j=2; ff2;7;13g;f2;7;13gg jH 4 \H 6 j=4; ff2;6;11g;f3;6;9g;f6;7;10g;f11;12;13gg jH 4 \H 7 j=5; ff1;10;13g;f1;10;13g;f6;7;10g;f2;6;11g;f11;12;13gg jH 12 \H 20 j=7; ff8;9;12g;f8;9;12g;f4;6;12g;f4;6;12g;f4;6;12g;f1;7;12g; f1;7;12gg jH 6 \H 18 j=8; ff1;3;12g;f4;9;12g;f4;9;12g;f1;5;9g;f3;4;11g;f2;3;5g; f2;3;5g;f2;3;5gg jH 3 \H 14 j= 10; ff7;9;11g;f7;9;11g;f3;5;6g;f3;5;6g;f5;12;13g;f5;12;13g; f3;5;6g;f2;8;12g;f7;9;11g;f5;12;13gg jH 14 \H 22 j= 11; ff3;5;6g;f3;5;6g;f2;8;12g;f2;8;12g;f7;9;11g;f7;9;11g; f5;12;13g;f5;12;13g;f5;12;13g;f3;5;6g;f7;9;11gg jH 3 \H 19 j= 13; ff2;8;12g;f3;5;6g;f3;5;6g;f3;5;6g;f3;4;7g;f3;4;7g; f3;4;7g;f7;9;11g;f7;9;11g;f7;9;11g;f5;12;13g;f5;12;13g; f5;12;13gg jH 8 \H 20 j= 14; ff1;13;12g;f7;9;11g;f7;9;11g;f7;9;11g;f1;5;9g;f5;8;11g; f5;8;11g;f3;9;13g;f9;10;12g;f2;6;9g;f4;8;9g;f5;12;13g; f5;12;13g;f5;12;13gg jH 8 \H 18 j= 16; ff1;3;12g;f7;9;11g;f7;9;11g;f7;9;11g;f1;5;9g;f4;10;13g; f3;8;10g;f3;9;13g;f3;4;7g;f3;4;7g;f9;10;12g;f2;6;9g; f4;8;9g;f5;12;13g;f5;12;13g;f4;12;13gg 60 jH 18 \H 21 j= 19; ff5;12;13g;f5;12;13g;f4;8;13g;f4;8;13g;f5;9;10g;f5;9;10g; f1;3;13g;f1;3;13g;f4;9;12g;f4;9;12g;f1;4;10g;f1;4;10g; f7;9;11g;f7;9;11g;f2;4;6g;f2;4;6g;f7;9;11g; f3;4;11g;f5;12;13gg jH 9 \H 17 j= 22; ff7;9;11g;f7;9;11g;f3;5;6g;f3;5;6g;f5;12;13g;f5;12;13g; f6;7;12gf7;9;11g;f6;10;11gf4;10;13g;f2;7;13g;f3;9;13g; f3;4;7g;f9;10;12g;f1;7;8g;f2;6;9g;f3;5;6g;f4;11;12g; f5;7;10g;f2;3;11g;f5;12;13g;f2;4;5gg jH 14 \H 18 j= 23; ff4;9;12g;f4;9;12g;f3;4;11g;f1;4;10g;f1;4;10g;f1;5;8g; f1;5;8g;f7;9;11g;f7;9;11g;f3;8;9g;f3;8;9g;f2;4;6g; f2;4;6g;f1;3;13g;f1;3;13g;f5;9;10g;f5;9;10g;f5;9;10g; f4;8;13g;f4;8;13g;f5;12;13g;f5;12;13g;f7;9;11gg jH 23 \H 27 j= 25; ff4;9;12g;f1;5;8g;f1;6;9g;f2;5;11g;f2;9;13g;f7;9;11g; f4;5;7g;f1;11;12g;f6;8;11g;f2;3;10g;f1;2;7g;f5;12;13g; f4;9;12g;f1;5;8g;f1;6;9g;f2;5;11g;f2;9;13g;f7;9;11g; f4;5;7g;f1;11;12g;f6;8;11g;f2;3;10g;f1;2;7g;f5;12;13g; f4;9;12gg jH 2 \H 21 j= 26; ff3;7;12g;f4;9;12g;f3;4;11g;f1;4;10g;f2;5;11g;f10;11;13g; f7;8;10g;f2;8;12g;f2;9;13g;f6;7;13g;f7;9;11g;f6;10;12g; f2;4;6g;f1;3;13g;f4;5;7g;f1;11;12g;f5;9;10g;f6;8;11g; f2;3;10g;f4;8;13g;f1;2;7g;f5;12;13g;f5;12;13g;f5;12;13g; f7;9;11g;f7;9;11gg jH 5 \H 11 j= 28; ff1;2;3g;f1;2;3g;f1;2;3g;f2;11;12g;f4;11;13g;f1;4;5g; f2;5;6g;f6;7;11g;f4;7;8g;f5;8;9g;f6;9;10g;f3;7;10g; f3;8;12g;f9;12;13g;f1;10;13g;f1;8;11g;f2;4;9g;f5;10;11g; f3;4;6g;f5;7;12g;f6;8;13g;f1;7;9g;f2;8;10g;f3;9;11g; f4;10;12g;f3;5;13g;f1;6;12g;f2;7;13gg 61 jH 23 \H 28 j= 29; ff4;9;12g;f3;4;11g;f1;4;10g;f10;11;13g;f6;10;12g;f1;3;13g; f1;11;12g;f5;9;10g;f6;8;11g;f4;8;13g;f1;2;7g;f5;12;13g; f4;9;12g;f3;4;11g;f1;4;10g;f10;11;13g;f6;10;12g;f1;3;13g; f1;11;12g;f5;9;10g;f6;8;11g;f4;8;13g;f1;2;7g;f5;12;13g; f7;9;13g;f3;6;9g;f5;7;11g;f4;9;12g;f3;4;11gg jH 17 \H 18 j= 32; ff4;9;12g;f4;9;12g;f3;4;7g;f1;4;10g;f1;4;10g;f1;5;8g; f1;5;8g;f7;9;11g;f7;9;11g;f3;8;9g;f3;8;9g;f2;4;6g; f2;4;6g;f1;3;13g;f1;3;13g;f5;9;10g;f5;9;10g;f4;8;13g; f4;8;13g;f5;12;13g;f5;12;13g;f1;13;12g;f7;9;11g;f1;5;9g; f4;10;13g;f3;8;10g;f3;9;13g;f3;4;11g;f9;10;12g;f2;6;9g; f4;8;9g;f5;12;13gg jH 14 \H 15 j= 34; ff3;7;12g;f4;9;12g;f1;6;9g;f3;5;6g;f3;4;11g;f1;4;10g; f1;5;8g;f2;5;11g;f10;11;13g;f7;8;10g;f2;8;12g;f2;9;13g; f6;7;13g;f7;9;11g;f6;10;12g;f3;8;9g;f2;4;6g;f1;3;13g; f4;5;7g;f1;11;13g;f5;9;10g;f6;8;11g;f2;3;10g;f4;8;13g; f1;2;7g;f5;12;13g;f3;5;6g;f3;5;6g;f2;8;12g;f2;8;12g; f7;9;11g;f7;9;11g;f5;12;13g;f5;12;13gg jH 15 \H 16 j= 35; ff3;7;12g;f4;9;12g;f3;4;11g;f1;4;10g;f1;5;8g;f10;11;13g; f7;8;10g;f2;7;13g;f2;7;13g;f7;9;11g;f7;9;11g;f3;8;9g; f2;4;6g;f1;3;13g;f4;5;7g;f1;11;12g;f5;9;10g;f4;8;13g; f5;12;13g;f5;12;13g;f1;3;12g;f7;9;11g;f1;5;9g;f3;8;10g; f6;7;13g;f3;9;13g;f3;4;7g;f9;10;12g;f1;7;8g; f2;6;9g;f1;11;13g;f4;11;12g;f5;7;10g;f4;8;9g;f5;12;13gg 62 jH 23 \H 29 j= 40; ff3;7;12g;f3;7;12g;f4;9;12g;f4;9;12g;f3;8;9g;f1;5;8g; f3;4;11g;f3;4;11g;f1;4;10g;f1;4;10g;f1;6;9g;f2;5;11g; f2;5;11g;f10;11;13g;f10;11;13g;f2;9;13g;f2;9;13g;f7;9;11g; f7;9;11g;f3;5;6g;f1;3;13g;f1;3;13g;f4;5;7g;f4;5;7g; f1;11;12g;f1;11;12g;f5;9;10g;f5;9;10g;f6;8;11g;f6;8;11g; f2;3;10g;f2;3;10g;f1;2;7g;f1;2;7g;f5;12;13g;f5;12;13g; f4;9;12g;f6;7;10g;f3;4;1g;f8;10;12gg jH 17 \H 26 j= 43; ff3;7;12g;f4;9;12g;f3;8;9g;f1;5;8g;f3;4;11g;f1;4;10g; f1;6;9g;f2;5;11g;f10;11;13g;f2;9;13g;f7;9;11g;f3;5;6g; f1;3;13g;f4;5;7g;f1;11;12g;f5;9;10g;f6;8;11g;f2;3;10g; f1;2;7g;f5;12;13g;f3;7;12g;f4;9;12g;f3;8;9g;f1;5;8g; f3;4;11g;f1;4;10g;f1;6;9g;f2;5;11g;f10;11;13g; f2;9;13g;f7;9;11g;f3;5;6g;f1;3;13g;f4;5;7g;f1;11;12g; f5;9;10g;f6;8;11g;f2;3;10g;f1;2;7g;f5;12;13g;f7;9;11g; f3;5;6g;f5;12;13gg jH 23 \H 26 j= 44; ff3;7;12g;f4;9;12g;f3;8;9g;f1;5;8g;f3;4;11g;f1;4;10g; f1;6;9g;f2;5;11g;f10;11;13g;f2;9;13g;f7;9;11g;f3;5;6g; f1;3;13g;f4;5;7g;f1;11;12g;f5;9;10g;f6;8;11g;f2;3;10g; f1;2;7g;f5;12;13g;f3;7;12g;f4;9;12g;f3;8;9g;f1;5;8g; f3;4;11g;f1;4;10g;f1;6;9g;f2;5;11g;f10;11;13g;f2;9;13g; f7;9;11g;f3;5;6g;f1;3;13g;f4;5;7g;f1;11;12g;f5;9;10g; f6;8;11g;f2;3;10g;f1;2;7g;f5;12;13g;f4;9;12g;f6;7;10g; f3;4;11g;f8;10;12gg 63 jH 1 \H 15 j= 46; ff3;7;12g;f4;9;12g;f1;6;9g;f3;5;6g;f3;4;11g;f1;4;10g; f1;5;8g;f2;5;11g;f10;11;13g;f7;8;10g;f2;8;12g;f2;9;13g; f6;7;13gf7;9;11g;f6;10;12g;f3;8;9g;f2;4;6g;f1;3;13g; f4;5;7g;f1;11;12g;f5;9;10g;f6;8;11g;f2;3;10g;f4;8;13g; f1;2;7g;f5;12;13g;f3;5;6g;f2;8;12g;f7;9;11g;f5;12;13g; f6;7;12g;f7;9;11g;f6;10;11g;f4;10;13g;f2;7;13g;f3;9;13g; f3;4;7g;f9;10;12g;f1;7;8g;f2;6;9g;f3;5;6g;f4;11;12g; f5;7;10g;f2;3;11g;f5;12;13g;f2;4;5gg jH 17 \H 21 j= 47; ff3;7;12g;f4;9;12g;f3;4;11g;f1;4;10g;f2;5;11g;f10;11;13g; f7;8;10g;f2;8;12g;f2;9;13g;f6;7;13g;f7;9;11g;f6;10;12g; f2;4;6g;f1;3;13g;f4;5;7g;f1;11;12g;f5;9;10g;f6;8;11g; f2;3;10g;f4;8;13g;f1;2;7g;f5;12;13g;f3;7;12g;f4;9;12g; f3;4;11g;f1;4;10g;f2;5;11g;f10;11;13g;f7;8;10g;f2;8;12g; f2;9;13g;f6;7;13g;f7;9;11g;f6;10;12g;f2;4;6g;f1;3;13g; f4;5;7g;f1;11;12g;f5;9;10g;f6;8;11g;f2;3;10g;f4;8;13g; f1;2;7g;f5;12;13g;f5;12;13g;f7;9;11g;f8;8;12gg 64 jH 16 \H 18 j= 50; ff4;9;12g;f4;9;12g;f1;2;9g;f2;3;5g;f2;3;5g;f1;2;9g;f3;4;7g; f1;9;10g;f1;4;10g;f1;5;8g;f1;5;8g;f6;8;12g;f6;8;12g; f6;9;13g;f6;9;13g;f7;9;11g;f7;9;11g;f2;10;12g;f2;10;12g; f3;8;;9g;f3;8;9g;f2;4;6g;f2;4;6g;f1;3;13g;f1;3;13g; f5;9;10g;f5;9;10g;f3;6;10g;f3;6;10g;f4;8;13g;f4;8;13g; f5;12;13g;f5;12;13g;f1;3;12g;f7;9;11g;f1;5;9g;f1;2;4g; f4;5;6g;f4;10;13g;f3;8;10g;f6;8;12g;f3;9;13g;f3;4;11g; f9;10;12g;f2;6;9g;f2;3;5g;f4;8;9g;f16;10g;f2;8;13g; f5;12;13gg jH 14 \H 30 j= 52; ff3;7;12g;f3;7;12g;f3;7;12g;f4;9;12g;f4;9;12g;f4;9;12g; f3;8;9g;f1;5;8g;f3;4;11g;f3;4;11g;f3;4;11g;f1;4;10g; f1;4;10g;f1;4;10g;f1;6;9g;f2;5;11g;f2;5;11g;f2;5;11g; f10;11;13g;f10;11;13g;f2;9;13g;f2;9;13g;f2;9;13g;f7;9;11g; f7;9;11g;f7;9;11g;f3;5;6g;f1;3;13g;f1;3;13g;f1;3;13g; f4;5;7g;f4;5;7g;f4;5;7g;f1;11;12g;f1;11;12g;f1;11;12g; f5;9;10g;f5;9;10g;f5;9;10g;f6;8;11g;f6;8;11g;f6;8;11g; f2;3;10g;f2;3;10g;f2;3;10g;f1;2;7g;f1;2;7g;f1;2;7g; f5;12;13g;f5;12;13g;f5;12;13g;f10;11;13gg 65 jH 14 \H 17 j= 56; ff3;7;12g;f4;9;12g;f4;9;12g;f1;6;9g;f1;6;9g;f3;5;6g;f3;5;6g; f3;4;11g;f3;4;11g;f1;4;10g;f1;4;10g;f1;5;8g;f1;5;8g; f2;5;11g;f2;5;11g;f10;11;13g;f10;11;13g;f7;8;10g;f7;8;10g; f2;8;12g;f2;8;12g;f2;9;13g;f2;9;13g:f6;7;13g;f6;7;13g; f7;9;11g;f7;9;11g;f6;10;12g;f6;10;12g;f3;8;9g;f3;8;9g; f2;4;6g;f2;4;6g;f1;3;13g;f1;3;13g;f4;5;7g;f4;5;7g; f1;11;12g;f1;11;12g;f5;9;10g;f5;9;10g;f6;8;11g;f6;8;11g; f2;3;10g;f2;3;10g;f4;8;13g;f4;8;13g;f1;2;7g;f1;2;7g; f5;12;13g;f5;12;13g;f3;5;6g;f2;8;12g;f7;9;11g;f5;12;13g; f3;7;12gg jH 2 \H 15 j= 58; ff3;7;12g;f4;9;12g;f1;6;9g;f3;5;6g;f3;4;11g;f1;4;10g; f1;5;8g;f2;5;11g;f10;11;13g;f7;8;10g;f2;8;12g;f2;9;13g; f6;7;13g;f7;9;11g;f6;10;12g;f3;8;9g;f2;4;6g;f1;3;13g; f4;5;7g;f1;11;12g;f5;9;10g;f6;8;11g;f2;3;10g;f4;8;13g; f1;2;7g;f5;12;13g;f6;7;12g;f6;7;12g;f7;9;11g;f7;9;11g; f6;10;11g;f6;10;11g;f4;10;13g;f4;10;13g;f2;7;13g;f2;7;13g; f3;9;13g;f3;4;7g;f3;4;7g;f3;9;13g;f9;10;12g;f9;10;12g; f1;7;8g;f1;7;8g;f2;6;9g;f2;6;9g;f3;5;6g;f3;5;6g; f4;11;12g;f4;11;12g;f5;7;10g;f5;7;10g;f2;3;11g;f2;3;11g; f5;12;13g;f5;12;13g;f2;4;5g;f2;4;5gg 66 jH 14 \H 25 j= 70; ff3;7;12g;f3;7;12g;f3;7;12g;f4;9;12g;f4;9;12g;f4;9;12g; f1;6;9g;f3;5;6g;f3;4;11g;f3;4;11g;f3;4;11g;f1;4;10g; f1;4;10g;f1;4;10g;f1;5;8g;f2;5;11g;f2;5;11g;f2;5;11g; f10;11;13g;f10;11;13g;f10;11;13g;f7;8;10g;f7;8;10g;f7;8;10g; f2;8;12g;f2;8;12g;f2;8;12g;f2;9;13g;f2;9;13g;f2;9;13g; f6;7;13g;f6;7;13g;f6;7;13g;f7;9;11g;f7;9;11g;f7;9;11g; f6;10;12g;f6;10;12g;f6;10;12g;f3;8;9g;f2;4;6g;f2;4;6g; f2;4;6g;f1;3;13g;f1;3;13g;f1;3;13g;f4;5;7g;f4;5;7g; f4;5;7g;f1;11;12g;f1;1;12g;f1;11;12g;f5;9;10g;f5;9;10g; f5;9;10g;f6;8;11g;f6;8;11g;f2;3;10g;f2;3;10g;f2;3;10g; f4;8;13g;f4;8;13g;f4;8;13g;f1;2;7g;f1;2;7g;f1;2;7g; f5;12;13g;f5;12;13g;f5;12;13g;f6;8;11gg jH 14 \H 24 j= 74; ff3;7;12g;f3;7;12g;f3;7;12g;f4;9;12g;f4;9;12g;f4;9;12g; f1;6;9g;f1;6;9g;f3;5;6g;f3;5;6g;f3;4;11g;f3;4;11g; f3;4;11g;f1;4;10g;f1;4;10g;f1;4;10g;f1;5;8g;f1;5;8g; f2;5;11g;f2;5;11g;f2;5;11g;f10;11;13g;f10;11;13g;f10;11;13g; f7;8;10g;f7;8;10g;f7;8;10g;f2;8;12g;f2;8;12g;f2;8;12g; f2;9;13g;f2;9;13g;f2;9;13g;f6;7;13g;f6;7;13g;f6;7;13g; f7;9;11g;f7;9;11g;f7;9;11g;f6;10;12g;f6;10;12g;f6;10;12g; f3;8;9g;f3;8;9g;f2;4;6g;f2;4;6g;f2;4;6g;f1;3;13g; f1;3;13g;f1;3;13g;f4;5;7g;f4;5;7g;f4;5;7g;f1;11;12g; f1;11;12g;f1;11;12g;f5;9;10g;f5;9;10g;f5;9;10g;f6;8;11g; f6;8;11g;f6;8;11g;f2;3;10g;f2;3;10g;f2;3;10g;f4;8;13g; f4;8;13g;f4;8;13g;f1;2;7g;f1;2;7g;f1;2;7g; f5;12;13g;f5;12;13g;f5;12;13gg: 67 jH 16 \H 34 j= 31; ff4;9;12g;f4;9;12g;f3;4;11g;f3;4;11g;f1;4;10g;f1;5;8g f1;5;8g;f10;11;13g;f10;11;13g;f6;7;13g;f7;9;11g;f7;9;11g; f1;4;10g;f7;9;11g;f3;8;9g;f3;8;9g;f2;4;6g;f2;4;6g; f1;3;13g;f1;3;13gf4;5;7g;f4;5;7g;f1;11;12g;f1;11;12g; f5;9;10g;f5;9;10g;f4;8;13g;f4;8;13g;f5;12;13g;f5;12;13g; f5;12;13gg jH 33 \H 35 j= 17; ff4;9;12g;f4;9;12g;f4;9;12g;f1;6;9g;f1;6;9g;f1;6;9g; f2;5;11g;f2;5;11g;f2;5;11g;f2;9;13g;f2;9;13g;f2;9;13g; f1;2;7g;f1;2;7g;f1;2;7g;f2;3;10g;f2;3;10gg: jH 17 \H 36 j= 38; ff4;9;12g;f4;9;12g;f3;4;11g;f3;4;11g;f1;4;10g;f1;4;10g; f2;5;11g;f2;5;11g;f10;11;13g;f10;11;13g;f2;9;13g;f2;9;13g; f6;7;13g;f6;7;13g;f7;9;11g;f7;9;11g;f6;10;12g;f6;10;12g; f2;4;6g;f2;4;6g;f1;3;13g;f1;3;13g;f4;5;7g;f4;5;7g; f1;11;12g;f1;11;12g;f5;9;10;g;f5;9;10g;f6;8;11g;f6;8;11g; f4;8;13g;f4;8;13g;f1;2;7g;f1;2;7g;f5;12;13g;f5;12;13g; f5;12;13g;f7;9;11gg: jH 31 \H 32 j= 20; ff4;9;12g;f4;9;12g;f4;9;12g;f7;9;11g;f7;9;11g;f7;9;11g; f2;4;6g;f2;4;6g;f2;4;6g;f5;9;10g;f5;9;10g;f4;8;13g; f4;8;13g;f4;8;13g;f5;12;13g;f5;12;13g;f5;12;13g;f1;4;10g; f1;3;13g;f5;9;10gg 68 jH 21 \H 33 j= 62; ff3;7;12g;f3;7;12g;f4;9;12g;f4;9;12g;f4;9;12g;f3;4;11g; f3;4;11g;f3;4;11g;f1;4;10g;f1;4;10g;f1;4;10g;f2;5;11g; f2;5;11g;f2;5;11g;f10;11;13g;f10;11;13g;f10;11;13g;f7;8;10g; f7;8;10g;f2;8;12g;f2;8;12g;f2;9;13g;f2;9;13g;f2;9;13g; f7;9;11g;f7;9;11g;f7;9;11g;f6;10;12g;f6;10;12g;f6;10;12g; f2;4;6g;f2;4;6g;f2;4;6g;f1;3;13g;f1;3;13g;f1;3;13g; f4;5;7g;f4;5;7g;f4;5;7g;f1;11;12g;f1;11;12g;f1;11;12g; f5;9;10g;f5;9;10g;f5;9;10g;f6;8;11g;f6;8;11g;f6;8;11g; f2;3;10g;f2;3;10g;f4;8;13g;f4;8;13g;f4;8;13g;f1;2;7g; f1;2;7g;f1;2;7g;f5;12;13g;f5;12;13g;f5;12;13g;f6;7;13g; f6;7;13g;f6;7;13gg: Combining all of the above gives the following lemma. Lemma 5.2 3Int(13) f0;1;2;3;4;5;6;7;8;9;10;11;12;13;14;15;16;17;18;19, 20;21;22;23;24;25;26;27;28;29;30;31;32;33;34;35;36;38;39;40;42;43;44;45, 46;47;48;50;51;52;54;56;57;58;60;62;66;70;74;78g. 69 Chapter 6 n =15 Let (X;F)and(X;G) be two 1-factorizations ofK 2n ,whereF=fF 1 ;F 2 ;:::;F 2n?1 g and G = fG 1 ;G 2 ;:::G 2n?1 g.Wesaythat(X;F)and(X;G)havekedges in common provided P 2n?1 i=1 jF i \G i j = k. The intersection problem for 1-factorization of K 2n was solved in 1982 by C. C. Lindner and W. D. Wallis [5]. In particular, for 2n =8the intersection numbers aref0;1;2;:::;28gnf27;26;25;23g. We will need the following construction; the 2n + 1 Construction specialized to 2n+ 1 = 15. Let (S;T) be a 3-fold triple system of order 7 where S = f1;2;3;4;5;6;7g.Let (X;F) be 1-factorization of K 8 where S\X = ; and F = fF 1 ;F 2 ;F 3 ;F 4 ;F 5 ;F 6 ;F 7 g. De ne a collection of triples T as follows: (1) T T ,and (2) for each edge fx;yg2F i , place three copies of fi;x;ygin T . Then (S[X;T ) is a 3-fold triple system. The following lemma is immediate. Lemma 6.1 Let (X;F) and (X;G) have k triples in common. Then the type (2) triples in (2) have 3k-triples in common. Lemma 6.2 If (S;T) can be organized into hexagon triples, then (S[X;T ) can be organized into hexagon triples. Proof It is only necessary to organize the triples of type (2) into hexagon triples. This is quite easy. Let (X;G) be a 1-factorization of K 8 where G\F = ;.Thenby Corollary 2.4 we can place the triples of type (2) on the triples fi;x;yg2G i 2Gto obtain a collection of hexagon triples. 70 Corollary 6.3 If x2 3Int(7) and k2f0;1;2;:::;28gnf27;26;25;23g,thenx+3k2 3Int(15). Proof Let (S;T 1 )and(S;T 2 ) be 3-fold triple system havingk triples in common which can be organized into hexagon triples and let (X;F)and(X;G) be 1-factorizations of K 8 having k edges in common. Then the 3-fold triple systems (S[X;T 1 )and(S[X;T 2 ) constructed using the 2n+ 1 Construction can be organized into hexagon triple systems having x +3ktriples in common. Lemma 6.4 3Int(15) =f0;1;2;:::;78gnf77;76;75;73g. Proof Each n23Int(15) can be written in the form n = x +3k,wherex23Int(7) and k2f0;1;2;:::;28gnf27;26;25;23g. 71 Chapter 7 The 6n+1 19 Construction Since we have a solution for 7;9;13 (modulo a few exceptions), and 15 we need consider only the cases 6n+1 19. Before giving the 6n+ 1 Construction, we need the following example. 72 Example 7.1 (Two decompositions of 3K 3;3;3 into hexagon triples having no triples in common.) 2 54 54 6 54 5 6 9 3 6 1 8 3 7 1 9 3 8 6 14 7 65 1 26 5 5 3 1 9 1 7 7 8 3 8 9 3 7 8 3 8 52 9 2 9 3 2 5 4 3 4 43 6 6 6 4 4 3K 3;3 =H 2 =H 1 9 8 7 6 5 41 4 7 6 2 4 9 3 8 8 1 5 2 7 1 9 2 8 1 7 6 2 9 3 7 1 4 8 2 9 1 9 7 5 1 7 2 8 3 7 2 6 9 5 2 8 Corollary 7.2 There exists a pair of 3K 3;3;3 hexagon triple systems having 0 or 27 triples in common. 73 With this example in hand, we can proceed to the 6n +1 19 Construction. The 6n + 1 19 Construction Let (X;G;T) be a group divisible design (GDD)oforder2n 6withatmostone group of size 4 and the remaining groups of size 2. If 2n 0 or 2 (mod 6) all groups are of size 2 and if 2n 4 (mod 6) exactly one group is of size 4, the others of size 2. (See [6].) Let S =f1g[(X f1;2g) and de ne a collection of hexagon triples H as follows: (1) For each g2G,let(f1g[(g f1;2;3g);H(g)) be a hexagon triple system of order 7 or 13, as the case may be, and place the hexagon triples of H(g)inH. (2) For each block t = fa;b;cg2T,let(3K 3;3;3 ;T(t)) be a decomposition of 3K 3;3;3 with parts fag f1;2;3g;fbg f1;2;3g,andfcg f1;2;3ginto hexagon triples and place these hexagon triples in H. Then (S;H) is a hexagon triple system. Lemma 7.3 There exists a pair of hexagon triple systems of order 6n+1 (mod 6) 19 having x triples in common for all x23Int(n). Proof Let x 2 3Int(n). If 2n 0 or 2 (mod 6), we can write x = P n i=1 a i + P jTj i=1 f0;27g where the a i ?s belong to 3Int(7). If 2n 4 (mod 6), we can write x = a 1 + P n?2 i=1 a i + P jTj i=1 f0;27g,wherea 1 2f0;78g(see Lemma 6.4) and a 2 ;a 3 ;:::;a n?2 2 3Int(7). Combining the results in Chapters 3, 4, 5, 6, and 7 giving the following theorem. Theorem 7.4 3Int(n)=3I(n)for all n 1(mod 6), with possibly a few exceptions for n =13(see Section 5). 74 Chapter 8 The 6n+3Construction Before giving the 6n+ 3 Construction we will need the following lemma. Lemma 8.1 There exist a pair of partial hexagon triple systems of order 9 which are disjoint, balanced, and cover the edges of 3K 9 n3K 3 . Proof Let (S;T 1 )and(S;T 2 ) be a pair of triple systems of order 9 having exactly the one triple t in common. Then (S;T 1 nt )and(S;T 2 nt ) are disjoint and balanced. Putting the triples of T 1 nt on the triples of T 2 nt and vice versa gives the desired pair of partial hexagon triple systems. 75 Example 8.2 (A pair of partial hexagon triple systems of order 9.) 9 3 1 647 6 3 6 9 4 8 5 7 3 8 81 64 1 976 7 2 4 69 2 61 813 1 1 552 7 6 8 1 2 4 1 5 55 2 5 452 31 2 8 2 25 8 54679 111 485 7 3 8 3 7 T 1 nt = T 2 nt = 596 8 33 45894 7 3 42 67695 11 t =f1;23g T 2 =T 1 = 47596 8 3 3 9 85 4 3 8 7 1 9496 9 3 76 7 856 2 2 4 475 96 8 76 4 1 5 7 14 6 5 2 6 6 9 1 7 51 4 8 4 7 8 3 24 9 2 5 7H 2 =H 1 =6 7 5 2 3 9 6 8 5 8 5 7 86 2 5 3 7 5 99 4 1 9 5 9 2 9 4 1 87 9 8 3 3 7 97 85 8 4 9 4 8 3 1 6 2 3 6 4 6 78 2 2 68 4 34 7 3 5 6 8 3 46 3 1 6 9 3 8 78 4 2 8 6 9 2 71 5 4 7 5 2 1 5 6 4 15 9 1 We can now give the 6n + 3 Construction. Let (X;G;B)beaGDD of order 2n with at most one group of size 4 and the remaining groups of size 2. Set S =f1 1 ;1 2 ;1 3 g[(X f1;2;3g) and de ne a collection of hexagon triples H as follows: Let G =fg 1 ;g 2 ;g 3 ;:::;g 2n gbe the groups of G withjg 1 j=4ifGcontains a group of size 4. (1) Place a hexagon triple system on f1 1 ;1 2 ;1 3 g[(g 1 f1;2;3g) and place these hexagon triples in H. 77 (2) For each g i 2G, i 2, place a partial hexagon triple system on f1 1 ;1 2 ;1 3 g[ fg i f1;2;3g) as in Lemma 8.1 (3) For each triple fa;b;cg2B, place a hexagon triple system on 3K 3;3;3 with parts fag f1;2;3g;fbg f1;2;3g,andfcg f1;2;3g. Then (S;H) is a hexagon triple system of order 6n+3. Theorem 8.3 The intersection numbers for hexagon triple systems of order n 3(mod 6) are precisely f0;1;2;:::; parenleftbig n 2 =xgnfx?1;x?2;x?3;x?5gg. Proof 9 and 15 are taken care of in Sections 4 and 6. So we need only concern ourselves here with 6n+3 21. In case jg 1 j= 4, any number in f0;1;2;::: parenleftbig n 2 =xgnfx?1;x?2;x?3;x?5gcan be written as a sum a+ P b i + P f0;27g,wherea23Int(15) and b i 2f0;33g.Ifjg i j=2 then any number inf0;1;2;:::; parenleftbig n 2 =xgnfx?1;x?2;x?3;x=5gcan be written as asuma+ P b i + P f0;27gwhere a23Int(9). 78 Bibliography [1] S. Ajodani-Namini and G. B. Khosrovshahi, On a conjecture of A. Hartman, in Combinatorics Advances (Ed. C. J. Colbourn and E. Mahmoodian), Kluwer Aca- demic, Dordrecht, (1995), 1-12. [2] T. P. Kirkman, On a problem in combinations, Cambridge and Dublin Math. J. 2, 1847, 191-204. [3] C. C. Lindner and A. Rosa, Construction of Steiner triple systems having a pre- scribed number of triples in common, Canad. J. Math., XXVII (1975), 1166-1175. [4] C. C. Lindner and W. D. Wallis, A note on one-factorizations having a prescribed number of edges in common, Annals of Discrete Mathematics 12 (1982), 203-209. [5] C. C. Lindner and S. 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