Orthogonal Latin squares of order 10

[Natalia Makarova]

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

• Questo topic è stato modificato 8 anni, 7 mesi fa da Natalia Makarova.

[Natalia Makarova]

Dear colleagues!

I invite you to the discussion at the forum in Russia

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

[Natalia Makarova]

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?

[Natalia Makarova]

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?

[Natalia Makarova]

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

[Natalia Makarova]

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

• Questa risposta è stata modificata 8 anni, 7 mesi fa da Natalia Makarova.

[Natalia Makarova]

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…

• Questa risposta è stata modificata 8 anni, 6 mesi fa da Natalia Makarova.

[Natalia Makarova]

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

[Natalia Makarova]

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.

[Natalia Makarova]

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

• Questa risposta è stata modificata 8 anni, 6 mesi fa da Natalia Makarova.

[Natalia Makarova]

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

[Natalia Makarova]

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

[Natalia Makarova]

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.

[Natalia Makarova]

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.

• Questa risposta è stata modificata 8 anni, 2 mesi fa da Natalia Makarova.
• Questa risposta è stata modificata 8 anni, 2 mesi fa da Natalia Makarova.
• Questa risposta è stata modificata 8 anni, 2 mesi fa da primesmagicgames.

[Natalia Makarova]

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]

[Autore]

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