The (91,10,1) block diagram is a special symmetrical block diagram . In order to be able to construct it, this combinatorial problem had to be solved: an empty 91 × 91 matrix was filled with ones in such a way that each row of the matrix contains exactly 10 ones and any two rows have exactly 1 one in the same column (not more and not less). That sounds relatively simple, but it is not trivial to solve. There are only certain combinations of parameters (like here v = 91, k = 10, λ = 1) for which such a construction is feasible. The smallest of these (v, k, λ) are listed in this overview .
designation This symmetrical 2- (91,10,1) block plan is called the 9th order projective plane or Desarguean plane .
properties This symmetrical block diagram has the parameters v = 91, k = 10, λ = 1 and thus the following properties:
It consists of 91 blocks and 91 points.
Each block contains exactly 10 points.
Every 2 blocks intersect in exactly 1 point.
Each point lies on exactly 10 blocks.
Each 2 points are connected by exactly 1 block.
Existence and characterization There are exactly four non-isomorphic 2- (91,10,1) block plans. These solutions are:
Solution 1 ( self-duplicated ) with the signature 91 840. It contains 58968 ovals of the 10th order. This is the Desargue's plane
Solution 2 ( self-duplicated ) with the signature 91 840. It contains 9720 10th order ovals . This is the Hughes Plain
Solution 3 ( dual to solution 4) with the signature 91 · 840. It contains 2808 10th order ovals . This is the first reverb level
Solution 4 ( dual to solution 3) with the signature 91 · 840. It contains 2808 10th-order ovals . This is the second reverb level
List of blocks
All the blocks of this block plan are listed here; See this illustration to understand this list
1 2 3 4 5 6 7 8 9 10
1 11 12 13 14 15 16 17 18 19
1 20 21 22 23 24 25 26 27 28
1 29 30 31 32 33 34 35 36 37
1 38 39 40 41 42 43 44 45 46
1 47 48 49 50 51 52 53 54 55
1 56 57 58 59 60 61 62 63 64
1 65 66 67 68 69 70 71 72 73
1 74 75 76 77 78 79 80 81 82
1 83 84 85 86 87 88 89 90 91
2 11 20 29 38 47 56 65 74 83
2 12 21 30 39 48 57 66 75 84
2 13 22 31 40 49 58 67 76 85
2 14 23 32 41 50 59 68 77 86
2 15 24 33 42 51 60 69 78 87
2 16 25 34 43 52 61 70 79 88
2 17 26 35 44 53 62 71 80 89
2 18 27 36 45 54 63 72 81 90
2 19 28 37 46 55 64 73 82 91
3 11 25 35 45 55 57 67 77 87
3 12 20 33 44 49 61 68 82 90
3 13 28 29 43 54 59 71 78 84
3 14 22 30 38 53 64 69 81 88
3 15 26 32 40 47 63 66 79 91
3 16 21 36 42 50 56 73 76 89
3 17 27 31 46 52 60 65 75 86
3 18 24 37 41 48 62 70 74 85
3 19 23 34 39 51 58 72 80 83
4 11 26 36 46 48 58 68 78 88
4 12 24 35 40 52 59 73 81 83
4 13 20 34 45 50 62 69 75 91
4 14 21 29 44 55 60 72 79 85
4 15 23 31 38 54 57 70 82 89
4 16 27 33 41 47 64 67 80 84
4 17 22 37 43 51 56 66 77 90
4 18 28 32 39 53 61 65 76 87
4 19 25 30 42 49 63 71 74 86
5 11 27 37 39 49 59 69 79 89
5 12 26 31 43 50 64 72 74 87
5 13 25 36 41 53 60 66 82 83
5 14 20 35 46 51 63 70 76 84
5 15 22 29 45 48 61 73 80 86
5 16 24 32 38 55 58 71 75 90
5 17 28 34 42 47 57 68 81 85
5 18 23 30 44 52 56 67 78 91
5 19 21 33 40 54 62 65 77 88
6 11 28 30 40 50 60 70 80 90
6 12 22 34 41 55 63 65 78 89
6 13 27 32 44 51 57 73 74 88
6 14 26 37 42 54 61 67 75 83
6 15 20 36 39 52 64 71 77 85
6 16 23 29 46 49 62 66 81 87
6 17 25 33 38 48 59 72 76 91
6 18 21 35 43 47 58 69 82 86
6 19 24 31 45 53 56 68 79 84
7 11 21 31 41 51 61 71 81 91
7 12 25 32 46 54 56 69 80 85
7 13 23 35 42 48 64 65 79 90
7 14 28 33 45 52 58 66 74 89
7 15 27 30 43 55 62 68 76 83
7 16 20 37 40 53 57 72 78 86
7 17 24 29 39 50 63 67 82 88
7 18 26 34 38 49 60 73 77 84
7 19 22 36 44 47 59 70 75 87
8 11 22 32 42 52 62 72 82 84
8 12 23 37 45 47 60 71 76 88
8 13 26 33 39 55 56 70 81 86
8 14 24 36 43 49 57 65 80 91
8 15 21 34 46 53 59 67 74 90
8 16 28 31 44 48 63 69 77 83
8 17 20 30 41 54 58 73 79 87
8 18 25 29 40 51 64 68 75 89
8 19 27 35 38 50 61 66 78 85
9 11 23 33 43 53 63 73 75 85
9 12 28 36 38 51 62 67 79 86
9 13 24 30 46 47 61 72 77 89
9 14 27 34 40 48 56 71 82 87
9 15 25 37 44 50 58 65 81 84
9 16 22 35 39 54 60 68 74 91
9 17 21 32 45 49 64 70 78 83
9 18 20 31 42 55 59 66 80 88
9 19 26 29 41 52 57 69 76 90
10 11 24 34 44 54 64 66 76 86
10 12 27 29 42 53 58 70 77 91
10 13 21 37 38 52 63 68 80 87
10 14 25 31 39 47 62 73 78 90
10 15 28 35 41 49 56 72 75 88
10 16 26 30 45 51 59 65 82 85
10 17 23 36 40 55 61 69 74 84
10 18 22 33 46 50 57 71 79 83
10 19 20 32 43 48 60 67 81 89
1 2 3 4 5 6 7 8 9 10
1 11 12 13 14 15 16 17 18 19
1 20 21 22 23 24 25 26 27 28
1 29 30 31 32 33 34 35 36 37
1 38 39 40 41 42 43 44 45 46
1 47 48 49 50 51 52 53 54 55
1 56 57 58 59 60 61 62 63 64
1 65 66 67 68 69 70 71 72 73
1 74 75 76 77 78 79 80 81 82
1 83 84 85 86 87 88 89 90 91
2 11 20 29 38 47 56 65 74 83
2 12 21 30 39 48 57 66 75 84
2 13 22 31 40 49 58 67 76 85
2 14 23 32 41 50 59 68 77 86
2 15 24 33 42 51 60 69 78 87
2 16 25 34 43 52 61 70 79 88
2 17 26 35 44 53 62 71 80 89
2 18 27 36 45 54 63 72 81 90
2 19 28 37 46 55 64 73 82 91
3 11 22 30 44 55 63 68 79 87
3 12 20 31 45 53 64 69 77 88
3 13 21 29 46 54 62 70 78 86
3 14 25 33 38 49 57 71 82 90
3 15 23 34 39 47 58 72 80 91
3 16 24 32 40 48 56 73 81 89
3 17 28 36 41 52 60 65 76 84
3 18 26 37 42 50 61 66 74 85
3 19 27 35 43 51 59 67 75 83
4 11 21 31 41 51 61 71 81 91
4 12 22 29 42 52 59 72 82 89
4 13 20 30 43 50 60 73 80 90
4 14 24 34 44 54 64 65 75 85
4 15 25 32 45 55 62 66 76 83
4 16 23 33 46 53 63 67 74 84
4 17 27 37 38 48 58 68 78 88
4 18 28 35 39 49 56 69 79 86
4 19 26 36 40 47 57 70 77 87
5 11 26 32 39 54 60 67 82 88
5 12 27 33 40 55 61 65 80 86
5 13 28 34 38 53 59 66 81 87
5 14 20 35 42 48 63 70 76 91
5 15 21 36 43 49 64 68 74 89
5 16 22 37 41 47 62 69 75 90
5 17 23 29 45 51 57 73 79 85
5 18 24 30 46 52 58 71 77 83
5 19 25 31 44 50 56 72 78 84
6 11 28 33 43 48 62 72 77 85
6 12 26 34 41 49 63 73 78 83
6 13 27 32 42 47 64 71 79 84
6 14 22 36 46 51 56 66 80 88
6 15 20 37 44 52 57 67 81 86
6 16 21 35 45 50 58 65 82 87
6 17 25 30 40 54 59 69 74 91
6 18 23 31 38 55 60 70 75 89
6 19 24 29 39 53 61 68 76 90
7 11 27 34 46 50 57 69 76 89
7 12 28 32 44 51 58 70 74 90
7 13 26 33 45 52 56 68 75 91
7 14 21 37 40 53 60 72 79 83
7 15 22 35 38 54 61 73 77 84
7 16 20 36 39 55 59 71 78 85
7 17 24 31 43 47 63 66 82 86
7 18 25 29 41 48 64 67 80 87
7 19 23 30 42 49 62 65 81 88
8 11 23 35 40 52 64 66 78 90
8 12 24 36 38 50 62 67 79 91
8 13 25 37 39 51 63 65 77 89
8 14 26 29 43 55 58 69 81 84
8 15 27 30 41 53 56 70 82 85
8 16 28 31 42 54 57 68 80 83
8 17 20 32 46 49 61 72 75 87
8 18 21 33 44 47 59 73 76 88
8 19 22 34 45 48 60 71 74 86
9 11 25 36 42 53 58 73 75 86
9 12 23 37 43 54 56 71 76 87
9 13 24 35 41 55 57 72 74 88
9 14 28 30 45 47 61 67 78 89
9 15 26 31 46 48 59 65 79 90
9 16 27 29 44 49 60 66 77 91
9 17 22 33 39 50 64 70 81 83
9 18 20 34 40 51 62 68 82 84
9 19 21 32 38 52 63 69 80 85
10 11 24 37 45 49 59 70 80 84
10 12 25 35 46 47 60 68 81 85
10 13 23 36 44 48 61 69 82 83
10 14 27 31 39 52 62 73 74 87
10 15 28 29 40 50 63 71 75 88
10 16 26 30 38 51 64 72 76 86
10 17 21 34 42 55 56 67 77 90
10 18 22 32 43 53 57 65 78 91
10 19 20 33 41 54 58 66 79 89
1 2 3 4 5 6 7 8 9 10
1 11 12 13 14 15 16 17 18 19
1 20 21 22 23 24 25 26 27 28
1 29 30 31 32 33 34 35 36 37
1 38 39 40 41 42 43 44 45 46
1 47 48 49 50 51 52 53 54 55
1 56 57 58 59 60 61 62 63 64
1 65 66 67 68 69 70 71 72 73
1 74 75 76 77 78 79 80 81 82
1 83 84 85 86 87 88 89 90 91
2 11 20 29 38 47 56 65 74 83
2 12 21 30 39 48 57 66 75 84
2 13 22 31 40 49 58 67 76 85
2 14 23 32 41 50 59 68 77 86
2 15 24 33 42 51 60 69 78 87
2 16 25 34 43 52 61 70 79 88
2 17 26 35 44 53 62 71 80 89
2 18 27 36 45 54 63 72 81 90
2 19 28 37 46 55 64 73 82 91
3 11 22 30 44 55 63 68 79 87
3 12 20 31 45 53 64 69 77 88
3 13 21 29 46 54 62 70 78 86
3 14 25 33 38 49 57 71 82 90
3 15 23 34 39 47 58 72 80 91
3 16 24 32 40 48 56 73 81 89
3 17 28 36 41 52 60 65 76 84
3 18 26 37 42 50 61 66 74 85
3 19 27 35 43 51 59 67 75 83
4 11 21 31 41 51 61 71 81 91
4 12 22 29 42 52 59 72 82 89
4 13 20 30 43 50 60 73 80 90
4 14 24 34 44 54 64 65 75 85
4 15 25 32 45 55 62 66 76 83
4 16 23 33 46 53 63 67 74 84
4 17 27 37 38 48 58 68 78 88
4 18 28 35 39 49 56 69 79 86
4 19 26 36 40 47 57 70 77 87
5 11 26 32 39 52 64 67 78 90
5 12 27 33 40 50 62 65 79 91
5 13 28 34 38 51 63 66 77 89
5 14 20 35 46 48 61 72 76 87
5 15 21 36 44 49 59 73 74 88
5 16 22 37 45 47 60 71 75 86
5 17 23 29 43 55 57 69 81 85
5 18 24 30 41 53 58 70 82 83
5 19 25 31 42 54 56 68 80 84
6 11 28 33 45 48 59 70 80 85
6 12 26 34 46 49 60 68 81 83
6 13 27 32 44 47 61 69 82 84
6 14 22 36 43 53 56 66 78 91
6 15 20 37 41 54 57 67 79 89
6 16 21 35 42 55 58 65 77 90
6 17 25 30 40 51 64 72 74 86
6 18 23 31 38 52 62 73 75 87
6 19 24 29 39 50 63 71 76 88
7 11 27 34 42 53 57 73 76 86
7 12 28 32 43 54 58 71 74 87
7 13 26 33 41 55 56 72 75 88
7 14 21 37 40 52 63 69 80 83
7 15 22 35 38 50 64 70 81 84
7 16 20 36 39 51 62 68 82 85
7 17 24 31 46 47 59 66 79 90
7 18 25 29 44 48 60 67 77 91
7 19 23 30 45 49 61 65 78 89
8 11 23 35 40 54 60 66 82 88
8 12 24 36 38 55 61 67 80 86
8 13 25 37 39 53 59 65 81 87
8 14 26 29 45 51 58 73 79 84
8 15 27 30 46 52 56 71 77 85
8 16 28 31 44 50 57 72 78 83
8 17 20 32 42 49 63 70 75 91
8 18 21 33 43 47 64 68 76 89
8 19 22 34 41 48 62 69 74 90
9 11 25 36 46 50 58 69 75 89
9 12 23 37 44 51 56 70 76 90
9 13 24 35 45 52 57 68 74 91
9 14 28 30 42 47 62 67 81 88
9 15 26 31 43 48 63 65 82 86
9 16 27 29 41 49 64 66 80 87
9 17 22 33 39 54 61 73 77 83
9 18 20 34 40 55 59 71 78 84
9 19 21 32 38 53 60 72 79 85
10 11 24 37 43 49 62 72 77 84
10 12 25 35 41 47 63 73 78 85
10 13 23 36 42 48 64 71 79 83
10 14 27 31 39 55 60 70 74 89
10 15 28 29 40 53 61 68 75 90
10 16 26 30 38 54 59 69 76 91
10 17 21 34 45 50 56 67 82 87
10 18 22 32 46 51 57 65 80 88
10 19 20 33 44 52 58 66 81 86
1 2 3 4 5 6 7 8 9 10
1 11 12 13 14 15 16 17 18 19
1 20 21 22 23 24 25 26 27 28
1 29 30 31 32 33 34 35 36 37
1 38 39 40 41 42 43 44 45 46
1 47 48 49 50 51 52 53 54 55
1 56 57 58 59 60 61 62 63 64
1 65 66 67 68 69 70 71 72 73
1 74 75 76 77 78 79 80 81 82
1 83 84 85 86 87 88 89 90 91
2 11 20 29 38 47 56 65 74 83
2 12 21 30 39 48 57 66 75 84
2 13 22 31 40 49 58 67 76 85
2 14 23 32 41 50 59 68 77 86
2 15 24 33 42 51 60 69 78 87
2 16 25 34 43 52 61 70 79 88
2 17 26 35 44 53 62 71 80 89
2 18 27 36 45 54 63 72 81 90
2 19 28 37 46 55 64 73 82 91
3 11 21 31 41 51 61 71 81 91
3 12 22 29 42 52 59 72 82 89
3 13 20 30 43 50 60 73 80 90
3 14 24 34 44 54 64 65 75 85
3 15 25 32 45 55 62 66 76 83
3 16 23 33 46 53 63 67 74 84
3 17 27 37 38 48 58 68 78 88
3 18 28 35 39 49 56 69 79 86
3 19 26 36 40 47 57 70 77 87
4 11 22 30 44 55 63 68 79 87
4 12 20 31 45 53 64 69 77 88
4 13 21 29 46 54 62 70 78 86
4 14 25 33 38 49 57 71 82 90
4 15 23 34 39 47 58 72 80 91
4 16 24 32 40 48 56 73 81 89
4 17 28 36 41 52 60 65 76 84
4 18 26 37 42 50 61 66 74 85
4 19 27 35 43 51 59 67 75 83
5 11 23 35 40 54 60 66 82 88
5 12 24 36 38 55 61 67 80 86
5 13 25 37 39 53 59 65 81 87
5 14 26 29 45 51 58 73 79 84
5 15 27 30 46 52 56 71 77 85
5 16 28 31 44 50 57 72 78 83
5 17 20 32 42 49 63 70 75 91
5 18 21 33 43 47 64 68 76 89
5 19 22 34 41 48 62 69 74 90
6 11 24 37 43 49 62 72 77 84
6 12 25 35 41 47 63 73 78 85
6 13 23 36 42 48 64 71 79 83
6 14 27 31 39 55 60 70 74 89
6 15 28 29 40 53 61 68 75 90
6 16 26 30 38 54 59 69 76 91
6 17 21 34 45 50 56 67 82 87
6 18 22 32 46 51 57 65 80 88
6 19 20 33 44 52 58 66 81 86
7 11 25 36 46 50 58 69 75 89
7 12 23 37 44 51 56 70 76 90
7 13 24 35 45 52 57 68 74 91
7 14 28 30 42 47 62 67 81 88
7 15 26 31 43 48 63 65 82 86
7 16 27 29 41 49 64 66 80 87
7 17 22 33 39 54 61 73 77 83
7 18 20 34 40 55 59 71 78 84
7 19 21 32 38 53 60 72 79 85
8 11 26 32 39 52 64 67 78 90
8 12 27 33 40 50 62 65 79 91
8 13 28 34 38 51 63 66 77 89
8 14 20 35 46 48 61 72 76 87
8 15 21 36 44 49 59 73 74 88
8 16 22 37 45 47 60 71 75 86
8 17 23 29 43 55 57 69 81 85
8 18 24 30 41 53 58 70 82 83
8 19 25 31 42 54 56 68 80 84
9 11 27 34 42 53 57 73 76 86
9 12 28 32 43 54 58 71 74 87
9 13 26 33 41 55 56 72 75 88
9 14 21 37 40 52 63 69 80 83
9 15 22 35 38 50 64 70 81 84
9 16 20 36 39 51 62 68 82 85
9 17 24 31 46 47 59 66 79 90
9 18 25 29 44 48 60 67 77 91
9 19 23 30 45 49 61 65 78 89
10 11 28 33 45 48 59 70 80 85
10 12 26 34 46 49 60 68 81 83
10 13 27 32 44 47 61 69 82 84
10 14 22 36 43 53 56 66 78 91
10 15 20 37 41 54 57 67 79 89
10 16 21 35 42 55 58 65 77 90
10 17 25 30 40 51 64 72 74 86
10 18 23 31 38 52 62 73 75 87
10 19 24 29 39 50 63 71 76 88
Cyclical representation There is a cyclical representation ( Singer cycle ) for solution 1 of this block diagram, it is isomorphic to the above list of blocks. Starting from the block shown, the remaining blocks of the block plan are obtained by cyclic permutation of the points it contains.
1 2 7 11 24 27 35 42 54 56
Orthogonal Latin Squares (MOLS)
This order 9 projective plane is equivalent to these 8 MOLS of order 9:
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[
1
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6th
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{\ Display style {\ begin {bmatrix} 1 & 2 & 3 & 4 5 & 6 7 & 8 & 9 \\ 2 & 6 & 4 & 9 & 8 & 1 & 5 & 7 & 3 \\ 3 & 4 & 7 & 5 & 2 & 9 & 1 & 6 & 8 \\ 4 & 9 & 5 & 8 & 6 & 3 & 2 1 & 7 \\ 5 & 8 & 2 & 6 & 9 7 & 4 3 & 1 \\ 6 & 1 & 9 & 3 & 7 & 2 & 8 & 5 & 4 \\ 7 & 5 & 1 & 2 & 4 & 8 & 3 & 9 & 6 \\ 8 7 & 6 & 1 & 3 & 5 & 9 & 4 & 2 \\ 9 & 3 & 8 7 & 1 & 4 & 6 & 2 & 5 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 3 & 4 5 & 6 7 & 8 & 9 & 2 \ \ 2 & 4 & 9 & 8 & 1 & 5 & 7 & 3 & 6 \\ 3 & 7 & 5 & 2 & 9 & 1 & 6 & 8 & 4 \\ 4 & 5 & 8 & 6 & 3 2 & 1 & 7 & 9 \\ 5 & 2 & 6 & 9 7 & 4 3 & 1 & 8 \\ 6 & 9 & 3 & 7 & 2 & 8 & 5 4 1 \\ 7 & 1 & 2 & 4 & 8 & 3 & 9 & 6 & 5 \\ 8 & 6 & 1 & 3 & 5 & 9 & 4 2 & 7 \\ 9 & 8 7 & 1 & 4 & 6 & 2 & 5 & 3 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 4 5 & 6 7 & 8 & 9 & 2 & 3 \\ 2 & 9 & 8 & 1 & 5 & 7 & 3 & 6 & 4 \\ 3 & 5 & 2 & 9 & 1 & 6 & 8 & 4 & 7 \\ 4 & 8 & 6 & 3 2 & 1 & 7 & 9 & 5 \\ 5 & 6 & 9 7 & 4 3 & 1 & 8 & 2 \\ 6 & 3 & 7 & 2 & 8 & 5 4 1 9 \\ 7 & 2 & 4 & 8 & 3 & 9 & 6 & 5 & 1 \\ 8 & 1 & 3 & 5 & 9 & 4 2 & 7 & 6 \\ 9 & 7 & 1 & 4 & 6 & 2 & 5 & 3 & 8 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 5 & 6 7 & 8 & 9 & 2 & 3 & 4 \\ 2 & 8 & 1 & 5 & 7 & 3 & 6 & 4 & 9 \\ 3 & 2 & 9 & 1 & 6 & 8 & 4 & 7 & 5 \\ 4 & 6 & 3 2 & 1 & 7 & 9 & 5 & 8 \\ 5 9 7 & 4 3 & 1 & 8 & 2 & 6 \\ 6 & 7 & 2 & 8 & 5 4 1 9 & 3 \\ 7 & 4 & 8 & 3 & 9 & 6 & 5 & 1 & 2 \\ 8 & 3 & 5 & 9 & 4 2 & 7 & 6 & 1 \\ 9 & 1 & 4 & 6 & 2 & 5 & 3 & 8 & 7 \ end {bmatrix}}}
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{\ Display style {\ begin {bmatrix} 1 & 6 7 & 8 & 9 & 2 & 3 & 4 & 5 \\ 2 & 1 & 5 & 7 & 3 & 6 & 4 & 9 & 8 \\ 3 & 9 & 1 & 6 & 8 & 4 & 7 & 5 & 2 \\ 4 3 2 1 7 & 9 & 5 & 8 & 6 \\ 5 & 7 & 4 3 & 1 & 8 & 2 & 6 & 9 \\ 6 & 2 & 8 & 5 4 1 9 & 3 & 7 \\ 7 & 8 & 3 & 9 & 6 & 5 & 1 & 2 & 4 \\ 8 & 5 & 9 & 4 2 & 7 & 6 & 1 & 3 \\ 9 & 4 & 6 & 2 & 5 3 8 7 & 1 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 7 & 8 & 9 & 2 & 3 & 4 5 & 6 \ \ 2 & 5 & 7 & 3 & 6 & 4 & 9 & 8 & 1 \\ 3 & 1 & 6 & 8 & 4 & 7 & 5 & 2 & 9 \\ 4 2 1 7 & 9 & 5 & 8 & 6 & 3 \\ 5 4 3 1 8 & 2 & 6 & 9 & 7 \\ 6 & 8 & 5 4 1 9 & 3 & 7 & 2 \\ 7 & 3 & 9 & 6 & 5 & 1 & 2 & 4 & 8 \\ 8 & 9 & 4 2 & 7 & 6 & 1 & 3 & 5 \\ 9 & 6 & 2 & 5 3 8 7 & 1 & 4 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 8 & 9 & 2 & 3 & 4 5 & 6 & 7 \\ 2 & 7 & 3 & 6 & 4 & 9 & 8 & 1 & 5 \\ 3 & 6 & 8 & 4 & 7 & 5 & 2 & 9 & 1 \\ 4 & 1 & 7 & 9 & 5 & 8 & 6 & 3 & 2 \\ 5 3 1 8 & 2 & 6 & 9 7 & 4 \\ 6 & 5 4 1 9 & 3 & 7 & 2 & 8 \\ 7 & 9 & 6 & 5 & 1 & 2 & 4 & 8 & 3 \\ 8 & 4 2 & 7 & 6 & 1 & 3 & 5 & 9 \\ 9 & 2 & 5 3 8 7 & 1 & 4 & 6 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 9 & 2 & 3 & 4 5 & 6 7 & 8 \\ 2 & 3 & 6 & 4 & 9 & 8 & 1 & 5 & 7 \\ 3 & 8 & 4 & 7 & 5 & 2 & 9 & 1 & 6 \\ 4 & 7 & 9 & 5 & 8 & 6 & 3 2 & 1 \\ 5 & 1 & 8 & 2 & 6 & 9 7 & 4 & 3 \\ 6 & 4 & 1 & 9 & 3 & 7 & 2 & 8 & 5 \\ 7 & 6 & 5 & 1 & 2 & 4 & 8 & 3 & 9 \\ 8 & 2 & 7 & 6 & 1 & 3 & 5 & 9 & 4 \\ 9 & 5 & 3 & 8 & 7 & 1 & 4 & 6 & 2 \ end {bmatrix}}}
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{\ Display style {\ begin {bmatrix} 1 & 2 & 3 & 4 5 & 6 7 & 8 & 9 \\ 2 & 3 & 1 & 5 & 6 & 4 & 8 & 9 & 7 \\ 3 & 1 & 2 & 6 & 4 & 5 & 9 & 7 & 8 \\ 4 5 & 6 7 & 8 & 9 & 1 & 2 & 3 \\ 5 & 6 & 4 & 8 & 9 & 7 & 2 & 3 & 1 \\ 6 & 4 & 5 & 9 & 7 & 8 & 3 & 1 & 2 \\ 7 & 8 & 9 & 1 & 2 & 3 & 4 5 & 6 \\ 8 & 9 & 7 & 2 & 3 & 1 & 5 & 6 & 4 \\ 9 & 7 & 8 & 3 & 1 & 2 & 6 & 4 & 5 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 3 2 & 7 & 9 & 8 & 4 & 6 & 5 \ \ 2 & 1 & 3 & 8 7 & 9 & 5 & 4 & 6 \\ 3 2 1 9 & 8 7 & 6 & 5 & 4 \\ 4 & 6 & 5 & 1 & 3 2 & 7 & 9 & 8 \\ 5 4 6 & 2 1 & 3 & 8 7 & 9 \\ 6 & 5 4 3 2 1 9 & 8 & 7 \\ 7 & 9 & 8 & 4 & 6 & 5 & 1 & 3 & 2 \\ 8 & 7 & 9 & 5 4 6 & 2 1 & 3 \\ 9 & 8 7 & 6 & 5 4 3 2 1 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 4 7 & 3 & 6 & 9 & 2 & 5 & 8 \\ 2 & 5 & 8 & 1 & 4 & 7 & 3 & 6 & 9 \\ 3 & 6 & 9 & 2 & 5 & 8 & 1 & 4 & 7 \\ 4 & 7 & 1 & 6 & 9 & 3 & 5 & 8 & 2 \\ 5 & 8 & 2 & 4 & 7 & 1 & 6 & 9 & 3 \\ 6 & 9 & 3 & 5 & 8 & 2 & 4 & 7 & 1 \\ 7 & 1 & 4 & 9 & 3 & 6 & 8 & 2 & 5 \\ 8 & 2 & 5 & 7 & 1 & 4 & 9 & 3 & 6 \\ 9 & 3 & 6 & 8 & 2 & 5 & 7 & 1 & 4 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 5 & 9 & 8 & 3 & 4 & 6 7 & 2 \\ 2 & 6 7 & 9 & 1 & 5 4 8 & 3 \\ 3 & 4 & 8 7 & 2 & 6 & 5 & 9 & 1 \\ 4 & 8 & 3 2 & 6 7 & 9 & 1 & 5 \\ 5 & 9 & 1 & 3 & 4 & 8 7 & 2 & 6 \\ 6 & 7 & 2 & 1 & 5 & 9 & 8 & 3 & 4 \\ 7 & 2 & 6 & 5 & 9 & 1 & 3 & 4 & 8 \\ 8 & 3 & 4 & 6 7 & 2 1 & 5 & 9 \\ 9 & 1 & 5 & 4 & 8 & 3 & 2 & 6 & 7 \ end {bmatrix}}}
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{\ Display style {\ begin {bmatrix} 1 6 & 8 & 5 & 7 & 3 & 9 & 2 & 4 \\ 2 & 4 & 9 & 6 & 8 & 1 & 7 & 3 & 5 \\ 3 5 7 & 4 & 9 & 2 & 8 & 1 & 6 \\ 4 & 9 & 2 & 8 & 1 & 6 & 3 & 5 & 7 \\ 5 & 7 & 3 & 9 & 2 & 4 & 1 & 6 & 8 \\ 6 & 8 & 1 & 7 & 3 & 5 & 2 & 4 & 9 \\ 7 & 3 & 5 & 2 & 4 & 9 & 6 & 8 & 1 \\ 8 & 1 & 6 & 3 & 5 & 7 & 4 & 9 & 2 \\ 9 & 2 & 4 & 1 & 6 & 8 & 5 & 7 & 3 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 7 & 4 2 & 8 & 5 & 3 & 9 & 6 \ \ 2 & 8 & 5 & 3 & 9 & 6 & 1 & 7 & 4 \\ 3 & 9 & 6 & 1 & 7 & 4 2 & 8 & 5 \\ 4 & 1 & 7 & 5 & 2 & 8 & 6 & 3 & 9 \\ 5 & 2 & 8 & 6 & 3 & 9 & 4 & 1 & 7 \\ 6 & 3 & 9 & 4 & 1 & 7 & 5 & 2 & 8 \\ 7 & 4 & 1 & 8 & 5 & 2 & 9 & 6 & 3 \\ 8 & 5 & 2 & 9 & 6 & 3 & 7 & 4 & 1 \\ 9 & 6 & 3 & 7 & 4 & 1 & 8 & 5 & 2 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 8 & 6 & 9 & 4 2 5 3 7 \\ 2 & 9 & 4 & 7 & 5 & 3 & 6 & 1 & 8 \\ 3 & 7 & 5 & 8 & 6 & 1 & 4 2 & 9 \\ 4 & 2 & 9 & 3 & 7 & 5 & 8 & 6 & 1 \\ 5 & 3 & 7 & 1 & 8 & 6 & 9 & 4 & 2 \\ 6 & 1 & 8 & 2 & 9 & 4 & 7 & 5 & 3 \\ 7 & 5 & 3 & 6 & 1 & 8 & 2 & 9 & 4 \\ 8 & 6 & 1 & 4 2 & 9 & 3 & 7 & 5 \\ 9 & 4 2 5 3 7 & 1 & 8 & 6 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 9 & 5 & 6 & 2 & 7 & 8 & 4 3 \\ 2 & 7 & 6 & 4 3 & 8 & 9 & 5 & 1 \\ 3 & 8 & 4 & 5 & 1 & 9 7 & 6 & 2 \\ 4 3 8 & 9 & 5 & 1 & 2 & 7 & 6 \\ 5 & 1 & 9 7 & 6 & 2 & 3 & 8 & 4 \\ 6 & 2 & 7 & 8 & 4 3 & 1 & 9 & 5 \\ 7 & 6 & 2 & 3 & 8 & 4 & 5 & 1 & 9 \\ 8 & 4 3 & 1 & 9 & 5 & 6 & 2 & 7 \\ 9 & 5 & 1 & 2 & 7 & 6 & 4 & 3 & 8 \ end {bmatrix}}}
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{\ Display style {\ begin {bmatrix} 1 & 2 & 3 & 4 5 & 6 7 & 8 & 9 \\ 2 & 3 & 1 & 5 & 6 & 4 & 8 & 9 & 7 \\ 3 & 1 & 2 & 6 & 4 & 5 & 9 & 7 & 8 \\ 4 5 & 6 7 & 8 & 9 & 1 & 2 & 3 \\ 5 & 6 & 4 & 8 & 9 & 7 & 2 & 3 & 1 \\ 6 & 4 & 5 & 9 & 7 & 8 & 3 & 1 & 2 \\ 7 & 8 & 9 & 1 & 2 & 3 & 4 5 & 6 \\ 8 & 9 & 7 & 2 & 3 & 1 & 5 & 6 & 4 \\ 9 & 7 & 8 & 3 & 1 & 2 & 6 & 4 & 5 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 3 2 & 7 & 9 & 8 & 4 & 6 & 5 \ \ 2 & 1 & 3 & 8 7 & 9 & 5 & 4 & 6 \\ 3 2 1 9 & 8 7 & 6 & 5 & 4 \\ 4 & 6 & 5 & 1 & 3 2 & 7 & 9 & 8 \\ 5 4 6 & 2 1 & 3 & 8 7 & 9 \\ 6 & 5 4 3 2 1 9 & 8 & 7 \\ 7 & 9 & 8 & 4 & 6 & 5 & 1 & 3 & 2 \\ 8 & 7 & 9 & 5 4 6 & 2 1 & 3 \\ 9 & 8 7 & 6 & 5 4 3 2 1 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 4 7 & 3 & 6 & 9 & 2 & 5 & 8 \\ 2 & 5 & 8 & 1 & 4 & 7 & 3 & 6 & 9 \\ 3 & 6 & 9 & 2 & 5 & 8 & 1 & 4 & 7 \\ 4 & 7 & 1 & 8 & 2 & 5 & 9 & 3 & 6 \\ 5 & 8 & 2 & 9 & 3 & 6 & 7 & 1 & 4 \\ 6 & 9 & 3 & 7 & 1 & 4 & 8 & 2 & 5 \\ 7 & 1 & 4 & 5 & 8 & 2 & 6 & 9 & 3 \\ 8 & 2 & 5 & 6 & 9 & 3 & 4 & 7 & 1 \\ 9 & 3 & 6 & 4 & 7 & 1 & 5 & 8 & 2 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 5 & 9 & 8 & 3 & 4 & 6 7 & 2 \\ 2 & 6 7 & 9 & 1 & 5 4 8 & 3 \\ 3 & 4 & 8 7 & 2 & 6 & 5 & 9 & 1 \\ 4 & 8 & 3 & 5 & 9 & 1 & 2 & 6 & 7 \\ 5 & 9 & 1 & 6 7 & 2 & 3 & 4 & 8 \\ 6 & 7 & 2 & 4 & 8 & 3 & 1 & 5 & 9 \\ 7 & 2 & 6 & 3 & 4 & 8 & 9 & 1 & 5 \\ 8 & 3 & 4 & 1 & 5 & 9 & 7 & 2 & 6 \\ 9 & 1 & 5 & 2 & 6 & 7 & 8 & 3 & 4 \ end {bmatrix}}}
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{\ Display style {\ begin {bmatrix} 1 6 & 8 & 5 & 7 & 3 & 9 & 2 & 4 \\ 2 & 4 & 9 & 6 & 8 & 1 & 7 & 3 & 5 \\ 3 5 7 & 4 & 9 & 2 & 8 & 1 & 6 \\ 4 & 9 & 2 & 3 & 5 & 7 & 6 & 8 & 1 \\ 5 & 7 & 3 & 1 & 6 & 8 & 4 & 9 & 2 \\ 6 & 8 & 1 & 2 & 4 & 9 & 5 & 7 & 3 \\ 7 & 3 & 5 & 8 & 1 & 6 & 2 & 4 & 9 \\ 8 & 1 & 6 & 9 & 2 & 4 3 & 5 & 7 \\ 9 & 2 & 4 & 7 & 3 & 5 & 1 & 6 & 8 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 7 & 4 2 & 8 & 5 & 3 & 9 & 6 \ \ 2 & 8 & 5 & 3 & 9 & 6 & 1 & 7 & 4 \\ 3 & 9 & 6 & 1 & 7 & 4 2 & 8 & 5 \\ 4 & 1 & 7 & 9 & 6 & 3 & 8 & 5 & 2 \\ 5 & 2 & 8 7 & 4 & 1 & 9 & 6 & 3 \\ 6 & 3 & 9 & 8 & 5 & 2 & 7 & 4 & 1 \\ 7 & 4 & 1 & 6 & 3 & 9 & 5 & 2 & 8 \\ 8 & 5 & 2 & 4 & 1 & 7 & 6 & 3 & 9 \\ 9 & 6 & 3 & 5 & 2 & 8 & 4 & 1 & 7 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 8 & 6 & 9 & 4 2 5 3 7 \\ 2 & 9 & 4 & 7 & 5 & 3 & 6 & 1 & 8 \\ 3 & 7 & 5 & 8 & 6 & 1 & 4 2 & 9 \\ 4 & 2 & 9 & 6 & 1 & 8 & 3 & 7 & 5 \\ 5 3 7 & 4 & 2 & 9 & 1 & 8 & 6 \\ 6 & 1 & 8 & 5 & 3 & 7 & 2 & 9 & 4 \\ 7 & 5 & 3 2 & 9 & 4 & 8 & 6 & 1 \\ 8 & 6 & 1 & 3 & 7 & 5 & 9 & 4 & 2 \\ 9 & 4 2 & 1 & 8 & 6 & 7 & 5 & 3 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 9 & 5 & 6 & 2 & 7 & 8 & 4 3 \\ 2 & 7 & 6 & 4 3 & 8 & 9 & 5 & 1 \\ 3 & 8 & 4 & 5 & 1 & 9 7 & 6 & 2 \\ 4 & 3 & 8 & 2 & 7 & 6 & 5 & 1 & 9 \\ 5 & 1 & 9 & 3 & 8 & 4 & 6 & 2 & 7 \\ 6 & 2 & 7 & 1 & 9 & 5 4 3 8 \\ 7 & 6 & 2 & 9 & 5 & 1 & 3 & 8 & 4 \\ 8 & 4 3 & 7 & 6 & 2 & 1 & 9 & 5 \\ 9 & 5 & 1 & 8 & 4 & 3 & 2 & 7 & 6 \ end {bmatrix}}}
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{\ Display style {\ begin {bmatrix} 1 & 3 2 & 7 & 9 & 8 & 4 & 6 & 5 \\ 2 & 1 & 3 & 8 7 & 9 & 5 & 4 & 6 \\ 3 2 1 9 & 8 7 & 6 & 5 & 4 \\ 4 & 6 & 5 & 1 & 3 2 & 7 & 9 & 8 \\ 5 4 6 & 2 1 & 3 & 8 7 & 9 \\ 6 & 5 4 3 2 1 9 & 8 & 7 \\ 7 & 9 & 8 & 4 & 6 & 5 & 1 & 3 & 2 \\ 8 & 7 & 9 & 5 4 6 & 2 1 & 3 \\ 9 & 8 7 & 6 & 5 4 3 2 1 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 2 & 3 & 4 5 & 6 7 & 8 & 9 \ \ 2 & 3 & 1 & 5 & 6 & 4 & 8 & 9 & 7 \\ 3 & 1 & 2 & 6 & 4 & 5 & 9 & 7 & 8 \\ 4 5 & 6 7 & 8 & 9 & 1 & 2 & 3 \\ 5 & 6 & 4 & 8 & 9 & 7 & 2 & 3 & 1 \\ 6 & 4 & 5 & 9 & 7 & 8 & 3 & 1 & 2 \\ 7 & 8 & 9 & 1 & 2 & 3 & 4 5 & 6 \\ 8 & 9 & 7 & 2 & 3 & 1 & 5 & 6 & 4 \\ 9 & 7 & 8 & 3 & 1 & 2 & 6 & 4 & 5 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 7 & 4 2 & 8 & 5 & 3 & 9 & 6 \\ 2 & 8 & 5 & 3 & 9 & 6 & 1 & 7 & 4 \\ 3 & 9 & 6 & 1 & 7 & 4 2 & 8 & 5 \\ 4 & 1 & 7 & 9 & 6 & 3 & 8 & 5 & 2 \\ 5 & 2 & 8 7 & 4 & 1 & 9 & 6 & 3 \\ 6 & 3 & 9 & 8 & 5 & 2 & 7 & 4 & 1 \\ 7 & 4 & 1 & 6 & 3 & 9 & 5 & 2 & 8 \\ 8 & 5 & 2 & 4 & 1 & 7 & 6 & 3 & 9 \\ 9 & 6 & 3 & 5 & 2 & 8 & 4 & 1 & 7 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 9 & 5 & 6 & 2 & 7 & 8 & 4 3 \\ 2 & 7 & 6 & 4 3 & 8 & 9 & 5 & 1 \\ 3 & 8 & 4 & 5 & 1 & 9 7 & 6 & 2 \\ 4 & 3 & 8 & 2 & 7 & 6 & 5 & 1 & 9 \\ 5 & 1 & 9 & 3 & 8 & 4 & 6 & 2 & 7 \\ 6 & 2 & 7 & 1 & 9 & 5 4 3 8 \\ 7 & 6 & 2 & 9 & 5 & 1 & 3 & 8 & 4 \\ 8 & 4 3 & 7 & 6 & 2 & 1 & 9 & 5 \\ 9 & 5 & 1 & 8 & 4 & 3 & 2 & 7 & 6 \ end {bmatrix}}}
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{\ Display style {\ begin {bmatrix} 1 & 8 & 6 & 9 & 4 2 5 3 7 \\ 2 & 9 & 4 & 7 & 5 & 3 & 6 & 1 & 8 \\ 3 & 7 & 5 & 8 & 6 & 1 & 4 2 & 9 \\ 4 & 2 & 9 & 6 & 1 & 8 & 3 & 7 & 5 \\ 5 3 7 & 4 & 2 & 9 & 1 & 8 & 6 \\ 6 & 1 & 8 & 5 & 3 & 7 & 2 & 9 & 4 \\ 7 & 5 & 3 2 & 9 & 4 & 8 & 6 & 1 \\ 8 & 6 & 1 & 3 & 7 & 5 & 9 & 4 & 2 \\ 9 & 4 2 & 1 & 8 & 6 & 7 & 5 & 3 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 4 7 & 3 & 6 & 9 & 2 & 5 & 8 \ \ 2 & 5 & 8 & 1 & 4 & 7 & 3 & 6 & 9 \\ 3 & 6 & 9 & 2 & 5 & 8 & 1 & 4 & 7 \\ 4 & 7 & 1 & 8 & 2 & 5 & 9 & 3 & 6 \\ 5 & 8 & 2 & 9 & 3 & 6 & 7 & 1 & 4 \\ 6 & 9 & 3 & 7 & 1 & 4 & 8 & 2 & 5 \\ 7 & 1 & 4 & 5 & 8 & 2 & 6 & 9 & 3 \\ 8 & 2 & 5 & 6 & 9 & 3 & 4 & 7 & 1 \\ 9 & 3 & 6 & 4 & 7 & 1 & 5 & 8 & 2 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 6 & 8 & 5 & 7 & 3 & 9 & 2 & 4 \\ 2 & 4 & 9 & 6 & 8 & 1 & 7 & 3 & 5 \\ 3 5 7 & 4 & 9 & 2 & 8 & 1 & 6 \\ 4 & 9 & 2 & 3 & 5 & 7 & 6 & 8 & 1 \\ 5 & 7 & 3 & 1 & 6 & 8 & 4 & 9 & 2 \\ 6 & 8 & 1 & 2 & 4 & 9 & 5 & 7 & 3 \\ 7 & 3 & 5 & 8 & 1 & 6 & 2 & 4 & 9 \\ 8 & 1 & 6 & 9 & 2 & 4 3 & 5 & 7 \\ 9 & 2 & 4 & 7 & 3 & 5 & 1 & 6 & 8 \ end {bmatrix}} \ quad {\ begin {bmatrix} 1 & 5 & 9 & 8 & 3 & 4 & 6 7 & 2 \\ 2 & 6 7 & 9 & 1 & 5 4 8 & 3 \\ 3 & 4 & 8 7 & 2 & 6 & 5 & 9 & 1 \\ 4 & 8 & 3 & 5 & 9 & 1 & 2 & 6 & 7 \\ 5 & 9 & 1 & 6 7 & 2 & 3 & 4 & 8 \\ 6 & 7 & 2 & 4 & 8 & 3 & 1 & 5 & 9 \\ 7 & 2 & 6 & 3 & 4 & 8 & 9 & 1 & 5 \\ 8 & 3 & 4 & 1 & 5 & 9 & 7 & 2 & 6 \\ 9 & 1 & 5 & 2 & 6 & 7 & 8 & 3 & 4 \ end {bmatrix}}}
oval An oval of the block plan is a set of its points, no three of which are on a block. Here is an example of a maximum order oval for each solution to this block diagram:
1 2 11 21 32 55 63 69 76 88
1 2 11 21 32 49 62 73 79 90
1 2 11 21 42 50 64 72 80 88
1 2 11 21 42 50 64 72 80 88
literature Individual evidence
↑ Zvonimir Janko , Tran van Trung : The Classification of Projective Planes of Order 9 Which Possess an Involution. In: Journal of Combinatorial Theory, Series A. Vol. 33, No. 1, 1982, pp. 65-75, doi : 10.1016 / 0097-3165 (82) 90079-6
^ Rudolf Mathon, Alexander Rosa : 2- (ν, κ, λ) Designs of Small Order. In: Charles J. Colbourn , Jeffrey H. Dinitz (Eds.): Handbook of Combinatorial Designs. 2nd edition. Chapman and Hall / CRC, Boca Raton FL et al. 2007, ISBN 978-1-4200-1054-1 , pp. 25-57.
^ Clement WH Lam , Galina Kolesova, Larry Thiel: A computer search for finite projective planes of order 9. In: Discrete Mathematics. Vol. 92, No. 1/3, 1991, pp. 187-195, doi : 10.1016 / 0012-365X (91) 90280-F
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