Orthogonal Latin squares of order 10

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  • #644
    Natalia Makarova
    Partecipante

    Dear colleagues!

    I invite you to the Russian distributed computing project

    http://sat.isa.ru/pdsat/index.php

    The project looking for:

    1 pair of diagonal orthogonal LS;
    2. group MOLS of three LS, in which there is incomplete orthogonality.

    The results you can see on the page
    http://sat.isa.ru/pdsat/solutions.php

    Here you can see the results found by other scientists (group MOLS of three LS which incomplete orthogonality):
    http://www.ams.org/journals/mcom/2016-85-298/S0025-5718-2015-03010-5/S0025-5718-2015-03010-5.pdf

    A – square

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

    B – square

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

    C – square

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

    ——

    This is my solution (group MOLS of three LS which incomplete orthogonality)

    A – square

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

    B – square

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

    C – square

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

    See
    http://mathhelpplanet.com/viewtopic.php?p=258012#p258012

    #646
    Natalia Makarova
    Partecipante

    Dear colleagues!

    I invite you to the discussion at the forum in Russia

    http://mathhelpplanet.com/viewtopic.php?f=57&t=46638

    #648
    Natalia Makarova
    Partecipante

    The group MOLS with partial orthogonality (my solution)

    Square A

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

    Square B

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

    Square C

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

    Square D

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

    A – B, A – C, A – D orthogonal pairs. B – C, B – D, C – D do not orthogonal pairs.

    Are there any known similar groups MOLS?

    #649
    Natalia Makarova
    Partecipante

    The group MOLS of five squares with partial orthogonality (my solution)

    Square A

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

    Square B

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

    Square C

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

    Square D

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

    Square E

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

    A – B, A – C, A – D, A – E orthogonal pairs.
    B – C, B – D, B – E, C – D, C – E, D – E do not orthogonal pairs.

    Are there any known similar groups MOLS?

    #651
    Natalia Makarova
    Partecipante

    Dear colleagues!

    I invite you to take part in the Russian project distributed computing
    http://sat.isa.ru/pdsat/index.php

    Please participate in the discussion of problem on the forum
    http://forum.boinc.ru/default.aspx?g=posts&t=1872#post79208

    #652
    Natalia Makarova
    Partecipante

    We consider the normalized Diagonal Latin Square

    The general formula of the Diagonal Latin Square is of the form:

    x1 = 7+2*x90-x83+x79+x80-x73+x70+x68-x63+x60+x57-x53+x50+x46-x43+x40+x35-x33+x30+x24-x23+x20-x8-x9-x6-x7-x4-x5-2*x3 
    x2 = x3+2-x24-x35-x46-x57-x68-x79-x90+x23+x33+x43+x53+x63+x73+x83 
    x10 = 36-x20-x30-x40-x50-x60-x70-x80-x90 
    x11 = -324+2*x89+2*x87+3*x88+2*x86+2*x84+2*x85+2*x82+3*x83+x77+2*x78+x75+x76+2*x73+x74+x69+x67+x68+x65+x66+x63+x64+x62+2*x58+x59+x55+x56+2*x53+x52+2*x48+x49+x47+x44+2*x43+x42+2*x38+x39+x37+x34+x32+2*x33+x29+2*x28+x26+x25+x22+2*x23-x20+2*x8+x6+x7+x4+x5+2*x3 
    x12 = -x3+42+x24+x35+x46+x57+x68+x79+x90-x23-x33-x43-x53-x63-x73-x83-x22-x32-x42-x52-x62-x72-x82 
    x13 = -x3+43-x23-x33-x43-x53-x63-x73-x83 
    x14 = -x4+42-x24-x34-x44-x54-x64-x74-x84 
    x15 = -x5+41-x25-x35-x45-x55-x65-x75-x85 
    x16 = -x6+40-x26-x36-x46-x56-x66-x76-x86 
    x17 = 86-x89-x90-2*x87-2*x88-x86-x84-x85-x82-x83-x77-x78+x72-x67-x68+x63-x58-x57+x54-x48-x47+x45-x38+x36-x37-x28-x8+x9-x7 
    x18 = -x8+38-x28-x38-x48-x58-x68-x78-x88 
    x19 = -x9+37-x29-x39-x49-x59-x69-x79-x89 
    x21 = 92-x89-x90-x87-2*x88-x86-x84-x85-x82-x83-x78+x72-x68+x63-x58+x54-x48+x45-x38+x36-x29-x30-2*x28-x26-x24-x25-x22-x23-x8+x9 
    x27 = -47+x89+x90+x87+2*x88+x86+x84+x85+x82+x83+x78-x72+x68-x63+x58-x54+x48-x45+x38-x36+x28+x8-x9 
    x31 = 45-x32-x33-x34-x35-x36-x37-x38-x39-x40 
    x41 = 45-x42-x43-x44-x45-x46-x47-x48-x49-x50 
    x51 = 45-x52-x53-x54-x55-x56-x57-x58-x59-x60 
    x61 = 45-x62-x63-x64-x65-x66-x67-x68-x69-x70 
    x71 = 45-x72-x73-x74-x75-x76-x77-x78-x79-x80 
    x81 = 45-x82-x83-x84-x85-x86-x87-x88-x89-x90

    We have 70 free variables and 20 dependent variables.

    Example

    0 1 2 3 4 5 6 7 8 9 
    9 6 0 1 2 3 7 5 4 8 
    4 5 9 7 6 2 8 1 3 0 
    6 9 3 4 5 7 2 8 0 1 
    5 8 4 0 1 6 9 3 7 2 
    7 2 5 6 0 8 4 9 1 3 
    1 7 6 5 8 9 3 0 2 4 
    3 4 7 8 9 0 1 2 6 5 
    2 3 8 9 7 1 0 4 5 6 
    8 0 1 2 3 4 5 6 9 7
    #664
    Natalia Makarova
    Partecipante

    Groups pairs of Orthogonal Diagonal Latin Squares (ODLS)

    An example of a group of four pairs ODLS we see in the article J. W. Brown and other «Completion of the Spectrum of Orthogonal Diagonal Latin Squares» (1992)

    Square A – the main DLS

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

    This is followed by squares orthogonal for square A

    #1

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

    #2

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

    And the two squares of this group found Oleg Zaikin:

    #3

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

    #4

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

    See
    http://sat.isa.ru/pdsat/additional_solutions.php

    Interesting groups pairs ODLS found my colleague Alex Belyshev.

    Example 1 – a group of two pairs ODLS.

    Square A – the main DLS

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

    This is followed by squares orthogonal for square A

    #1

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

    #2

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

    Example 2 – a group of trhee pairs ODLS.

    Square A – the main DLS

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

    This is followed by squares orthogonal for square A

    #1

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

    #2

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

    #3

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

    To be continued…

    #666
    Natalia Makarova
    Partecipante

    Example 3 – a group of four pairs ODLS.

    Square A – the main DLS

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

    This is followed by squares orthogonal for square A

    #1

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

    #2

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

    #3

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

    #4

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

    Example 4 – a group of six pairs ODLS.

    Square A – the main DLS

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

    This is followed by squares orthogonal for square A

    #1

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

    #2

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

    #3

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

    #4

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

    #5

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

    #6

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

    More examples see here
    https://yadi.sk/d/eXghlY-yqC2Dh

    #667
    Natalia Makarova
    Partecipante

    The general formula of Brown’s DLS

    In the picture you see of Brown’s DLS and its scheme.

    The general formula:

    x1 = 9- x31 
    x10 = 9- x25 
    x11 = - x2- x20+ x25+ x33- x14- x8+ x29+17 
    x12 = 9- x42 
    x13 = 9- x43 
    x15 = 9- x45 
    x17 = 9- x27 
    x18 = 9- x28 
    x19 = 9- x29 
    x23 = 9- x8 
    x26 = 9- x16 
    x3 = 9- x33 
    x30 = 9- x20 
    x32 = 9- x2 
    x4 = 9- x34 
    x41 = x2+ x20- x25- x33+ x14+ x8- x29-8 
    x44 = 9- x14 
    x5 = 9- x35 
    x6 = 9- x21 
    x7 = 9- x22 
    x9 = 9- x24

    We have 19 free variables and 21 dependent variables.

    #668
    Natalia Makarova
    Partecipante

    Groups pairs of Orthogonal Diagonal Latin Squares (ODLS)

    I found a group of eight pairs ODLS.

    Square A – the main DLS

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

    This is followed by squares orthogonal for square A

    #1

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

    #2

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

    #3

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

    #4

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

    #5

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

    #6

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

    #7

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

    #8

    0 1 2 3 4 5 6 7 8 9
    5 7 8 6 9 2 3 0 4 1
    6 9 5 7 0 1 4 8 2 3
    4 3 9 1 8 7 2 5 6 0
    2 4 0 5 6 3 7 9 1 8
    8 2 3 9 1 4 5 6 0 7
    1 6 7 0 5 9 8 2 3 4
    9 8 6 4 2 0 1 3 7 5
    3 5 1 2 7 8 0 4 9 6
    7 0 4 8 3 6 9 1 5 2
    #671
    Natalia Makarova
    Partecipante

    My colleagues found the group MOLS of three LS which incomplete orthogonality with the orthogonal coefficient 85

    Square A from Parker

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

    Square B

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

    Square C

    0 1 2 3 4 5 6 7 8 9
    8 0 4 5 9 3 2 6 7 1
    9 8 3 6 2 7 4 1 0 5
    6 7 1 0 5 9 8 3 2 4
    1 2 5 8 7 4 9 0 3 6
    2 4 9 7 0 1 3 5 6 8
    7 5 0 2 8 6 1 9 4 3
    3 9 6 4 1 2 0 8 5 7
    4 3 7 9 6 8 5 2 1 0
    5 6 8 1 3 0 7 4 9 2
    #679
    Natalia Makarova
    Partecipante

    I and my colleagues found the group MOLS of three LS which incomplete orthogonality with the orthogonal coefficient 86.

    Square A (DLK)

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

    Square B

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

    Square C

    0 1 2 3 4 5 6 7 8 9
    9 3 5 8 7 0 2 1 6 4
    3 6 9 7 1 4 8 5 2 0
    2 5 1 4 6 8 0 9 3 7
    6 2 4 1 0 9 3 8 7 5
    7 8 6 0 9 3 1 4 5 2
    5 0 7 2 3 1 9 6 4 8
    1 7 8 9 5 2 4 3 0 6
    4 9 0 6 8 7 5 2 1 3
    8 4 3 5 2 6 7 0 9 1
    #680
    Natalia Makarova
    Partecipante

    Systems of four LS of order 10 with partial orthogonality

    Example

    Square A

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

    Square B

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

    Square C

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

    Square D

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

    Orthogonal pairs: A – B, A – C, A – D, partially orthogonal pairs: B – C, B – D, C – D.

    B – C

    00  11  22  33  44  55  66  77  88  99 
    47  83  76  52  01  94  39  60  25  18 
    15  07  43  61  58  80  72  29  96  34 
    73  32  68  20  19  46  91  05  54  87 
    59  76  31  18  82  03  25  94  67  40 
    36  24  85  09  97  61  10  58  43  72 
    98  65  19  87  23  32  54  41  70  06 
    62  49  50  94  76  17  08  83  31  25 
    84  58  97  75  30  29  43  16  02  61 
    21  90  04  46  65  78  87  32  19  53

    00 01 02 03 04 05 06 07 08 09 10 11 15 16 17 18 19 20 21 22 23 24 25 29 30 31 32 33 34 36 39 40 41 43 44 46 47 49 50 52 53 54 55 58 59 60 61 62 65 66 67 68 70 72 73 75 76 77 78 80 82 83 84 85 87 88 90 91 94 96 97 98 99
    The number of unique ordered pairs: U = 73

    B – D

    00  11  22  33  44  55  66  77  88  99 
    48  82  75  59  06  97  34  63  20  11 
    12  08  40  65  53  89  77  24  91  36 
    74  39  67  26  10  41  98  02  55  83 
    51  70  33  17  85  04  29  96  62  48 
    37  23  84  01  99  68  12  50  46  75 
    95  66  18  82  21  30  53  49  74  07 
    63  47  51  90  72  16  05  88  39  24 
    89  54  96  78  37  22  41  15  03  60
    26  95  09  44  68  73  80  31  17  52

    00 01 02 03 04 05 06 07 08 09 10 11 12 15 16 17 18 20 21 22 23 24 26 29 30 31 33 34 36 37 39 40 41 44 46 47 48 49 50 51 52 53 54 55 59 60 62 63 65 66 67 68 70 72 73 74 75 77 78 80 82 83 84 85 88 89 90 91 95 96 97 98 99
    The number of unique ordered pairs: U = 73

    C – D

    00  11  22  33  44  55  66  77  88  99 
    78  32  65  29  16  47  94  03  50  81 
    52  78  30  15  83  09  27  94  61  46 
    34  29  87  06  90  61  18  52  45  73 
    91  60  13  87  25  34  59  46  72  08 
    67  43  54  91  79  18  02  80  36  25 
    85  56  98  72  31  20  43  19  04  67 
    23  97  01  40  62  76  85  38  19  54 
    49  84  76  58  07  92  31  65  23  10 
    16  05  49  64  58  83  70  21  97  32

    00 01 02 03 04 05 06 07 08 09 10 11 13 15 16 18 19 20 21 22 23 25 27 29 30 31 32 33 34 36 38 40 43 44 45 46 47 49 50 52 54 55 56 58 59 60 61 62 64 65 66 67 70 72 73 76 77 78 79 80 81 83 84 85 87 88 90 91 92 94 97 98 99
    The number of unique ordered pairs: U = 73

    The number of unique ordered pairs simultaneously for B – C, B – D, C – D: R=55

    00 47 15 73 59 36 98 62 84 21 11 83 07 65 49 90 22 31 85 50 97 04 33 52 20 18 09 46 44 01 23 30 55 80 03 29 78 66 72 91 10 54 08 77 60 05 16 88 67 70 02 99 34 40 06

    We call R value – characteristic orthogonality of system with partial orthogonality.
    I was unable to find a system of four LS of order 10 with partial orthogonality, for which R> 55.

    #681
    Natalia Makarova
    Partecipante

    I found a new unique group of two pairs orthogonal diagonal latin squares of order 10

    Square A

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

    Square B

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

    Square C

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

    The group found a random generation of DLS with my program.
    Searching orthogonal squares used program S. Belyaev.

    #686
    Natalia Makarova
    Partecipante

    Dear colleagues!

    I bring to your attention an article
    “Systems of N mutually orthogonal diagonal Latin squares of order 10 with a complete orthogonality (N-1) pairs”
    https://yadi.sk/i/S417t0IpwnP6r

    Please send feedback to me at [email protected]

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