The enormous size of the diagonal Latin squares space makes it unfeasible to enumerate all its objects straightforwardly in reasonable time. So, in order to discover the structure of this space, sophisticated search methods are needed. In RakeSearch project, we implement an application that picks up separate pairs of mutually orthogonal DLSs, which allows to reconstruct full graphs of their orthogonality.

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Processing of the square # 8 completed

Dear participants, yesterday, processing of the square # 8 is fully completed!

1185453085 orthogonal mates and the six place in rating for that square:

0 1 2 3 4 5 6 7 8 9 A B

1 2 0 4 5 3 8 9 A B 6 7

5 3 4 2 0 1 7 8 9 A B 6

B 7 9 8 A 6 5 1 3 2 4 0

A 6 8 7 9 B 0 5 1 3 2 4

6 8 A 9 B 7 1 3 2 4 0 5

2 0 1 5 3 4 A B 6 7 8 9

4 5 3 1 2 0 B 6 7 8 9 A

3 4 5 0 1 2 9 A B 6 7 8

9 B 7 6 8 A 4 0 5 1 3 2

7 9 B A 6 8 3 2 4 0 5 1

8 A 6 B 7 9 2 4 0 5 1 3

With your help, we have strictly established that there are not one or two squares with such large number of orthogonal mates, but at least a set!

Thank you for participation and donation of CPU time!

24 Jan 2022, 12:33:41 UTC
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Processing of the square # 7 completed

Dear participants, another result achieved with the help of your computers!

1586136964 orthogonal mates for this square:

0 1 2 3 4 5 6 7 8 9 A B

1 2 0 4 9 8 B 6 A 3 5 7

5 8 A 6 B 1 4 3 2 7 0 9

B 7 6 8 A 9 2 1 3 5 4 0

7 6 B A 5 3 0 2 4 8 9 1

6 B 7 5 8 4 1 0 9 A 3 2

2 0 1 9 3 A 7 B 5 4 8 6

A 5 8 7 6 0 3 9 1 B 2 4

3 4 9 0 1 6 5 A B 2 7 8

9 3 4 2 0 7 A 8 6 1 B 5

4 9 3 1 2 B 8 5 7 0 6 A

8 A 5 B 7 2 9 4 0 6 1 3

And we have a new item on the third line of "rating"! More correct name for this list - "spectrum", at this moment this square are not placed in spectrum.

Thank you for participation!

3 Jan 2022, 19:32:32 UTC
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Processing of the square # 6 completed

Dear participants, processing of the square # 6 is fully completed. With your help, now we know that square

0 1 2 3 4 5 6 7 8 9 A B

1 2 0 4 5 3 8 9 A B 6 7

6 8 A 9 B 7 1 3 2 4 0 5

B 7 9 8 A 6 5 1 3 2 4 0

A 6 8 7 9 B 0 5 1 3 2 4

5 3 4 2 0 1 7 8 9 A B 6

9 B 7 6 8 A 4 0 5 1 3 2

4 5 3 1 2 0 B 6 7 8 9 A

3 4 5 0 1 2 9 A B 6 7 8

2 0 1 5 3 4 A B 6 7 8 9

7 9 B A 6 8 3 2 4 0 5 1

8 A 6 B 7 9 2 4 0 5 1 3

has 1220317124 orthogonal diagonal Latin squares (ODLS). And it shifted square # 5, processed earlier from the third line of "rating" of of 12th order squares with maximum number of ODLS.

Thank you for project support and donation of CPU time!

31 Dec 2021, 14:36:17 UTC
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Processing of the square # 5 completed

Dear participants!

Recently we publish the results of processing the first bunch or tasks (square # 4). Processing of the second tasks set (square # 5) was completed today and now we can announce that the square

0 1 2 3 4 5 6 7 8 9 A B

1 2 3 4 5 0 8 6 A B 9 7

5 0 1 2 3 4 7 B 6 A 8 9

8 A 9 B 7 6 5 4 0 2 1 3

A 9 B 7 6 8 0 5 1 3 2 4

6 8 A 9 B 7 4 3 5 1 0 2

2 3 4 5 0 1 A 8 9 7 B 6

4 5 0 1 2 3 B 9 7 8 6 A

B 7 6 8 A 9 2 1 3 5 4 0

9 B 7 6 8 A 1 0 2 4 3 5

7 6 8 A 9 B 3 2 4 0 5 1

3 4 5 0 1 2 9 A B 6 7 8

has 1212560768 orthogonal diagonal Latin squares (ODLS) and it took the third place in the "rating" of 12th order squares with maximum number of ODLS.

Thank you for project support and donation of CPU time!

21 Dec 2021, 20:31:54 UTC
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The Square 12/30192/3855983322 finally confirmed!

Dear participants!

As previously announced (Ru), on August 18, 2021, as part of a separate computational experiment (which did not require calculations in the desktop grid, but is a separate interesting task) Eduard Vatutin found a very interesting diagonal Latin square of order 12 with 30192 diagonal transverses:

0 1 2 3 4 5 6 7 8 9 A B

1 2 3 4 9 8 B 5 A 0 6 7

5 8 A 6 B 4 1 3 9 7 0 2

B 7 5 8 A 2 9 1 3 6 4 0

7 5 8 A 6 3 0 2 4 B 9 1

9 0 1 2 3 7 A B 5 4 8 6

6 B 7 5 8 1 4 0 2 A 3 9

A 6 B 7 5 0 3 9 1 8 2 4

3 4 9 0 1 6 5 A B 2 7 8

2 3 4 9 0 A 7 8 6 1 B 5

4 9 0 1 2 B 8 6 7 3 5 A

8 A 6 B 7 9 2 4 0 5 1 3

This number of transverses was a record (and is so at the moment), which suggested that this square may be accompanied by many orthogonal diagonal Latin squares (ODLS). Search were launched in the Gerasim@Home project, which ended on October 24, 2021 (Ru), issued a record value of the ODLS, but the results obtained were questioned by the authors and declared preliminary, since during processing 7 out of 1214514 results corresponding to one of the 30192 transverses suffered due to a failure in one of the memory banks (which was then replaced). Recomputing of 480 workunits from the "broken" 1421 transversal (Ru) was executed very quickly and by October 27, a new, slightly larger value was already known.

But the following questions remain:

1) How reliable are the calculations performed in grids from a PC? (It is raised at every conference dedicated to supercomputing and grid technologies);

2) The fact that 7 corrupted results were found did not mean that there were only 7 of them;

3) When we talk about calculating this characteristic (as well as many other integer ones), it is important for us to get an absolutely accurate value, even +/- 1 - it will already be incorrect;

4) The project Gerasim@Home is a independent development by SerVal, that compatible with BOINC infrastructure through support of BOINC protocol, but still is not "original BOINC server".

Taking into account the above, the decision was made to perform a complete recomputing for this square in the project Gerasim@Home and two weeks later, on November 5 in RakeSearch project.

On November 11, a sudden help came - information appeared (Ru) that one of the active participants of distributed computing from the Russia Team - CoolAtchOk, using programs based on the algorithm of Alexey Belyshev (whitefox), processed the square of Eduard, obtained a value that coincides with that obtained as a result of recalculation on October 24 - 27.

And today, on the 15th of December, 2021, after conducting a full check in RakeSearch and Gerasim@Home projects, we can confidently say that:

1. The number of ODLS for this record (at the moment) square is 3855983322, which coincides with the values obtained after recalculation of the 1421st transversal and the CoolAtchOk participant;

2. Calculations in PC-based grids are really reliable due to the quorum and we have another "hard argument" to such a question.

What's next? Now the last hundreds of tasks for another square are being counted (square # 5), and the result may be close to a record, also the processing of another one (square # 6) is in full thrust!

Thank you for participation and donation of CPU time!

15 Dec 2021, 22:25:54 UTC
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©2022 The searchers team, Karelian Research Center of the Russian Academy of Sciences