
Below is a Cayley table of order 25, grouped by the natural cosets of <5>.

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|  0  5 10 15 20 |  1  6 11 16 21 |  2  7 12 17 22 |  3  8 13 18 23 |  4  9 14 19 24 |
|  5 10 15 20  0 |  6 11 16 21  1 |  7 12 17 22  2 |  8 13 18 23  3 |  9 14 19 24  4 |
| 10 15 20  0  5 | 11 16 21  1  6 | 12 17 22  2  7 | 13 18 23  3  8 | 14 19 24  4  9 |
| 15 20  0  5 10 | 16 21  1  6 11 | 17 22  2  7 12 | 18 23  3  8 13 | 19 24  4  9 14 |
| 20  0  5 10 15 | 21  1  6 11 16 | 22  2  7 12 17 | 23  3  8 13 18 | 24  4  9 14 19 |
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|  1  6 11 16 21 |  2  7 12 17 22 |  3  8 13 18 23 |  4  9 14 19 24 |  5 10 15 20  0 |
|  6 11 16 21  1 |  7 12 17 22  2 |  8 13 18 23  3 |  9 14 19 24  4 | 10 15 20  0  5 |
| 11 16 21  1  6 | 12 17 22  2  7 | 13 18 23  3  8 | 14 19 24  4  9 | 15 20  0  5 10 |
| 16 21  1  6 11 | 17 22  2  7 12 | 18 23  3  8 13 | 19 24  4  9 14 | 20  0  5 10 15 |
| 21  1  6 11 16 | 22  2  7 12 17 | 23  3  8 13 18 | 24  4  9 14 19 |  0  5 10 15 20 |
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|  2  7 12 17 22 |  3  8 13 18 23 |  4  9 14 19 24 |  5 10 15 20  0 |  6 11 16 21  1 |
|  7 12 17 22  2 |  8 13 18 23  3 |  9 14 19 24  4 | 10 15 20  0  5 | 11 16 21  1  6 |
| 12 17 22  2  7 | 13 18 23  3  8 | 14 19 24  4  9 | 15 20  0  5 10 | 16 21  1  6 11 |
| 17 22  2  7 12 | 18 23  3  8 13 | 19 24  4  9 14 | 20  0  5 10 15 | 21  1  6 11 16 |
| 22  2  7 12 17 | 23  3  8 13 18 | 24  4  9 14 19 |  0  5 10 15 20 |  1  6 11 16 21 |
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|  3  8 13 18 23 |  4  9 14 19 24 |  5 10 15 20  0 |  6 11 16 21  1 |  7 12 17 22  2 |
|  8 13 18 23  3 |  9 14 19 24  4 | 10 15 20  0  5 | 11 16 21  1  6 | 12 17 22  2  7 |
| 13 18 23  3  8 | 14 19 24  4  9 | 15 20  0  5 10 | 16 21  1  6 11 | 17 22  2  7 12 |
| 18 23  3  8 13 | 19 24  4  9 14 | 20  0  5 10 15 | 21  1  6 11 16 | 22  2  7 12 17 |
| 23  3  8 13 18 | 24  4  9 14 19 |  0  5 10 15 20 |  1  6 11 16 21 |  2  7 12 17 22 |
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|  4  9 14 19 24 |  5 10 15 20  0 |  6 11 16 21  1 |  7 12 17 22  2 |  8 13 18 23  3 |
|  9 14 19 24  4 | 10 15 20  0  5 | 11 16 21  1  6 | 12 17 22  2  7 | 13 18 23  3  8 |
| 14 19 24  4  9 | 15 20  0  5 10 | 16 21  1  6 11 | 17 22  2  7 12 | 18 23  3  8 13 |
| 19 24  4  9 14 | 20  0  5 10 15 | 21  1  6 11 16 | 22  2  7 12 17 | 23  3  8 13 18 |
| 24  4  9 14 19 |  0  5 10 15 20 |  1  6 11 16 21 |  2  7 12 17 22 |  3  8 13 18 23 |
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Below is a random example of finding a pair of disjoint chains.
In this example, we find two disjoint chains whose swaps contain
(7,5,12) and (16,13,4), avoid (0,0,0), and their cells in C(<5>),
along with (0,0,0), can be completed to a transversal of C(<5>).

Chain 1 (@@): (15, 5,20), (21,16,12), ( 7, 7,14), (23, 8, 6), (24,24,23)
Chain 1 Swap: (15, 8,23), (21,24,20), ( 7, 5,12), (23,16,14), (24, 7, 6)

Chain 2 (**): ( 5,10,15), (16, 6,22), (12,17, 4), ( 3,13,16), ( 9, 4,13)
Chain 2 Swap: ( 5,17,22), (16,13, 4), (12, 4,16), ( 3,10,13), ( 9, 6,15)

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|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  . **  .  . |  .  .  .  .  . |  .  .  . **  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  . @@  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  . @@  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
 ------------------------------------------------------------------------------------
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  . **  .  .  . |  .  .  .  .  . |  .  . **  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  . @@  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  . @@ |
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|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  . @@  .  .  . |  .  .  .  .  . |  . @@  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  . **  . |  .  .  .  .  . | **  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
 ------------------------------------------------------------------------------------
|  .  . **  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  . **  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  . @@  . |  .  .  .  .  . |  . @@  .  .  . |  .  .  .  .  . |
 ------------------------------------------------------------------------------------
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  . **  .  .  . |  .  .  .  .  . |  .  .  .  .  . | **  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  . @@  .  .  . |  .  .  .  .  . |  .  .  .  . @@ |
 ------------------------------------------------------------------------------------

Below is a transversal which may be found from the above disjoint chains.

The completion of ( 0, 0, 0), ( 7, 5,12) and (16,13, 4) to a transversal of C(Z_25):

( 0, 0, 0), ( 1, 1, 2), ( 2,22,24), ( 3,10,13), ( 4,14,18)
( 5,17,22), ( 6,11,17), ( 7, 5,12), ( 8,18, 1), ( 9, 6,15)
(10,20, 5), (11,21, 7), (12, 4,16), (13,23,11), (14,19, 8)
(15, 8,23), (16,13, 4), (17, 2,19), (18, 3,21), (19, 9, 3)
(20,15,10), (21,24,20), (22,12, 9), (23,16,14), (24, 7, 6)

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|  0  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  . 22  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  5 |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  . 23  .  .  . |  .  .  .  .  . |
|  .  .  . 10  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
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|  .  .  .  .  . |  2  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  . 17  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  7 |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  4  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  . 20 |
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|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  . 24 |  .  .  .  .  . |  .  .  .  .  . |
|  . 12  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . | 16  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . | 19  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  9  .  . |  .  .  .  .  . |  .  .  .  .  . |
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|  .  . 13  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  1  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  . 11 |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . | 21  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  . 14  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
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|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  . 18  .  . |
|  .  .  .  .  . |  . 15  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  8  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  .  .  .  . |  .  3  .  .  . |
|  .  .  .  .  . |  .  .  .  .  . |  .  6  .  .  . |  .  .  .  .  . |  .  .  .  .  . |
 ------------------------------------------------------------------------------------

Now the code is looping through all possible partial transversals of
length 3 which include (0,0,0) and each cell belongs to a distinct block
diagonal, and attempting to find a completion to a transversal.

A counter indicating the number of partial transversals which have been
investigated is given. When the exhaustive search concludes, the total
number of such partial transversals the the number of which were
successfully completed to a transversal in C(Z_25) are given.

10000
20000
30000
40000
50000
60000

Total number of partial transversals found: 63540
Total number of successes in completing to a transversal: 63540
Total number of failures in completing to a transversal: 0

