[superprismatic]

Rearrange the rows of the following

matrix so that each column of the

resulting matrix is an arithmetic

progression modulo 31 reading from

top to bottom (or, equivalently,

from bottom to top).

 30  8 30  5  7 25

 15 10 13  4 21  7

 10 21 28 14  5  1

 11 25 25 12  2 27

 29  4  2  7 10 30

 28  0  5  9 13  4

  5  1 12 24 20 26

  0 12 27  3  4 20

 24 15 17 17 25 24

  7  9  6 20 14 16

 19 26  1 27  9 18

 26 23 11 13 19 14

  8 13  3 18 11 11

  4 28 15 26 23  0

 17 18  7  0 15 28

  2 20 21 30 29 10

 22  7 23 21  0  3

 13  2 19  8 27 17

 21  3 26 23  3  8

  9 17  0 16  8  6

 14  6 16  6 24 12

 25 19 14 15 22 19

 23 11 20 19 28 29

  6  5  9 22 17 21

 27 27  8 11 16  9

 20 30 29 25  6 13

 16 14 10  2 18  2

 18 22  4 29 12 23

  1 16 24  1  1 15

  3 24 18 28 26  5

 12 29 22 10 30 22

[/code]

[bushindo]

Rearrange the rows of the following

matrix so that each column of the

resulting matrix is an arithmetic

progression modulo 31 reading from

top to bottom (or, equivalently,

from bottom to top).

 30  8 30  5  7 25

 15 10 13  4 21  7

 10 21 28 14  5  1

 11 25 25 12  2 27

 29  4  2  7 10 30

 28  0  5  9 13  4

  5  1 12 24 20 26

  0 12 27  3  4 20

 24 15 17 17 25 24

  7  9  6 20 14 16

 19 26  1 27  9 18

 26 23 11 13 19 14

  8 13  3 18 11 11

  4 28 15 26 23  0

 17 18  7  0 15 28

  2 20 21 30 29 10

 22  7 23 21  0  3

 13  2 19  8 27 17

 21  3 26 23  3  8

  9 17  0 16  8  6

 14  6 16  6 24 12

 25 19 14 15 22 19

 23 11 20 19 28 29

  6  5  9 22 17 21

 27 27  8 11 16  9

 20 30 29 25  6 13

 16 14 10  2 18  2

 18 22  4 29 12 23

  1 16 24  1  1 15

  3 24 18 28 26  5

 12 29 22 10 30 22

You can satisfy the OP by reordering the rows so that the first column goes 0,1,2,3,... In general, you can construct a solution by ordering the rows so that the first column looks like the vector (a, 2a, ..., 31*a ) mod 31, for a = 1, 2, ... 30. For instance, for a = 2, the vector is

 

(2  4  6  8 10 12 14 16 18 20 22 24 26 28 30  1  3  5  7  9 11 13 15 17 19 21 23 25 27 29 31)'

For a = 5, the first column should be

(5 10 15 20 25 30  4  9 14 19 24 29  3  8 13 18 23 28  2  7 12 17 22 27  1  6 11 16 21 26 31)'

Edited July 2, 2010 by bushindo

[Guest]

Edited July 2, 2010 by parik

[Guest]

0	4	15	23	28	26

0	3	7	22	21	23

0	4	9	5	13	28

0	6	8	9	16	17

27	3	4	12	0	20

0	7	15	17	18	28

1	5	10	14	21	28

1	24	1	15	1	16

1	9	18	19	26	27

1	20	12	5	24	26

2	2	10	14	16	18

2	10	20	21	29	30

2	8	13	17	19	27

2	11	12	25	25	27

2	4	7	10	29	30

3	5	18	24	26	28

3	3	8	21	26	23

3	8	11	11	13	18

4	7	13	10	15	21

4	12	18	22	23	29

5	6	17	9	21	22

5	7	8	25	30	30

6	6	12	14	16	24

6	13	20	25	29	30

6	7	14	9	16	20

8	9	11	16	27	27

10	12	22	22	29	30

11	13	14	19	23	26

11	19	20	23	28	29

14	15	19	25	19	22

15	17	17	24	24	25


155	279	403	527	651	775

5.00	9.00	13.00	17.00	21.00	25.00

[Guest]

5	7	8	25	30	30

4	7	13	10	15	21

1	5	10	14	21	28

2	11	12	25	25	27

2	4	7	10	29	30

0	4	9	5	13	28

1	20	12	5	24	26

27	3	4	12	0	20

15	17	17	24	24	25

6	7	14	9	16	20

1	9	18	19	26	27

11	13	14	19	23	26

3	8	11	11	13	18

0	4	15	23	28	26

0	7	15	17	18	28

2	10	20	21	29	30

0	3	7	22	21	23

2	8	13	17	19	27

3	3	8	21	26	23

0	6	8	9	16	17

6	6	12	14	16	24

14	15	19	25	19	22

11	19	20	23	28	29

5	6	17	9	21	22

8	9	11	16	27	27

6	13	20	25	29	30

2	2	10	14	16	18

4	12	18	22	23	29

1	24	1	15	1	16

3	5	18	24	26	28

10	12	22	22	29	30


155	279	403	527	651	775

5.00	9.00	13.00	17.00	21.00	25.00

I'm sure there are multiple answers though