I was shown a 4x4 matrix whose entries are distinct positive integers below
32, whose row sums are the same (say S), and whose column products are the same (P).
I believe that there are numerous solutions fitting the above definition, and challenge the solvers to find as many as possible - the values of S and P are up to you.
Rem : It is possible to find a solution or two without a computer, but listing all possible solutions is IMHO a hard nut even for a software-based approach.
Any given such grid can have its rows or its columns permuted at will.
So the columns can be permuted so that the lowest number is in the leftmost column. Then the rows can be permuted so that the leftmost column is in ascending order as one goes down it. Finally, the rightmost 3 columns can be permuted so their top entries are in ascending order, thus making the top row in ascending sequence also, since the lowest number in the grid is at the top left.
The program, when run with maxnum = 31, in accordance with all the numbers being under 32, there are 229 basic solutions as defined in the paragraph above: the lowest number in the upper left corner; the numbers increasing as one goes down the left column; and the numbers increasing as one goes across the top row.
The columns can be permuted in 4! = 24 ways and so can the rows, so each basic solution counts for 24*24 = 576 overall solutions, of which, then, there are 229*576 = 131,904.
The basic solutions are shown below, with the row total shown to the right of the first row and the column product under the grid:
1 4 20 24 49
6 10 12 21
28 9 7 5
30 14 3 2
5040
1 6 14 28 49
7 21 18 3
24 8 2 15
30 5 10 4
5040
1 2 18 24 45
8 15 10 12
20 16 4 5
27 9 6 3
4320
1 2 18 24 45
8 15 10 12
20 16 6 3
27 9 4 5
4320
1 4 16 24 45
8 10 15 12
20 18 2 5
27 6 9 3
4320
1 6 14 28 49
8 24 2 15
21 7 18 3
30 5 10 4
5040
1 9 18 20 48
8 28 5 7
21 10 14 3
30 2 4 12
5040
1 4 21 24 50
8 12 16 14
27 18 2 3
28 7 9 6
6048
1 4 21 24 50
8 12 16 14
27 18 3 2
28 7 6 9
6048
1 4 21 24 50
8 14 16 12
27 18 2 3
28 6 9 7
6048
1 7 18 24 50
8 9 12 21
27 6 14 3
28 16 2 4
6048
1 12 16 21 50
8 4 14 24
27 18 3 2
28 7 9 6
6048
1 2 15 27 45
9 12 16 8
20 18 3 4
24 10 6 5
4320
1 6 12 30 49
9 5 14 21
20 24 3 2
28 7 10 4
5040
1 6 12 30 49
9 5 21 14
20 24 2 3
28 7 10 4
5040
1 6 12 30 49
9 24 2 14
20 5 21 3
28 7 10 4
5040
1 6 12 30 49
9 24 14 2
20 5 3 21
28 7 10 4
5040
1 8 15 21 45
9 18 12 6
20 7 14 4
28 5 2 10
5040
1 8 15 21 45
9 18 14 4
20 7 12 6
28 5 2 10
5040
1 14 18 21 54
9 27 3 15
28 4 20 2
30 5 7 12
7560
1 15 18 20 54
9 14 4 27
28 3 21 2
30 12 5 7
7560
1 2 18 24 45
10 12 8 15
16 20 5 4
27 9 6 3
4320
1 2 18 24 45
10 12 8 15
16 20 6 3
27 9 5 4
4320
1 8 12 24 45
10 18 2 15
16 5 20 4
27 6 9 3
4320
1 8 12 24 45
10 18 2 15
16 6 20 3
27 5 9 4
4320
1 9 15 20 45
10 5 24 6
16 8 3 18
27 12 4 2
4320
1 9 15 20 45
10 5 12 18
16 24 3 2
27 4 8 6
4320
1 9 15 20 45
10 5 18 12
16 24 2 3
27 4 8 6
4320
1 9 15 20 45
10 8 24 3
16 5 6 18
27 12 2 4
4320
1 9 15 20 45
10 24 8 3
16 5 18 6
27 4 2 12
4320
1 3 14 30 48
10 12 5 21
18 20 8 2
28 7 9 4
5040
1 3 14 30 48
10 21 5 12
18 20 8 2
28 4 9 7
5040
1 5 12 30 48
10 14 3 21
18 8 20 2
28 9 7 4
5040
1 12 14 21 48
10 3 5 30
18 20 8 2
28 7 9 4
5040
1 12 14 21 48
10 30 5 3
18 2 8 20
28 7 9 4
5040
1 2 18 28 49
10 30 5 4
21 6 7 15
24 14 8 3
5040
1 5 18 21 45
12 9 4 20
15 14 10 6
28 8 7 2
5040
1 9 14 21 45
12 8 5 20
15 10 18 2
28 7 4 6
5040
1 10 14 20 45
12 18 9 6
15 4 5 21
28 7 8 2
5040
1 10 14 20 45
12 21 5 7
15 4 8 18
28 6 9 2
5040
1 4 21 24 50
12 8 16 14
18 27 2 3
28 7 9 6
6048
1 4 21 24 50
12 8 16 14
18 27 3 2
28 7 6 9
6048
1 7 9 28 45
12 4 14 15
20 18 5 2
21 10 8 6
5040
1 7 9 28 45
12 8 10 15
20 5 14 6
21 18 4 2
5040
1 7 9 28 45
12 10 8 15
20 18 5 2
21 4 14 6
5040
1 6 16 27 50
12 7 3 28
21 18 9 2
24 8 14 4
6048
1 7 14 28 50
12 3 8 27
21 16 9 4
24 18 6 2
6048
1 7 14 28 50
12 3 8 27
21 18 9 2
24 16 6 4
6048
1 7 14 28 50
12 8 3 27
21 18 9 2
24 6 16 4
6048
1 7 14 28 50
12 27 8 3
21 2 9 18
24 16 6 4
6048
1 7 14 28 50
12 27 9 2
21 8 3 18
24 4 16 6
6048
1 7 14 28 50
12 27 8 3
21 16 9 4
24 2 6 18
6048
1 8 14 27 50
12 7 3 28
21 18 9 2
24 6 16 4
6048
1 10 20 24 55
12 21 8 14
28 3 9 15
30 16 7 2
10080
1 10 20 24 55
12 21 18 4
28 6 14 7
30 8 2 15
10080
1 10 20 24 55
12 21 7 15
28 16 9 2
30 3 8 14
10080
1 10 20 24 55
12 21 8 14
28 16 9 2
30 3 7 15
10080
1 14 20 21 56
12 18 24 2
28 5 7 16
30 8 3 15
10080
1 8 10 30 49
14 5 28 2
15 7 6 21
24 18 3 4
5040
1 3 18 28 50
14 8 24 4
16 21 7 6
27 12 2 9
6048
1 3 18 28 50
14 21 6 9
16 24 8 2
27 4 7 12
6048
1 3 18 28 50
14 24 8 4
16 7 21 6
27 12 2 9
6048
1 9 12 28 50
14 6 3 27
18 7 21 4
24 16 8 2
6048
1 3 18 28 50
14 12 15 9
20 21 4 5
27 10 7 6
7560
1 5 21 28 55
14 18 8 15
24 7 20 4
30 16 3 6
10080
1 7 20 28 56
14 18 21 3
24 5 12 15
30 16 2 8
10080
1 8 18 28 55
14 21 5 15
24 20 7 4
30 3 16 6
10080
1 8 18 28 55
14 21 16 4
24 20 5 6
30 3 7 15
10080
1 15 18 21 55
14 28 5 8
24 4 7 20
30 6 16 3
10080
1 6 18 30 55
15 8 20 12
24 10 7 14
28 21 4 2
10080
1 6 18 30 55
15 8 20 12
24 10 14 7
28 21 2 4
10080
1 8 16 30 55
15 21 5 14
24 20 7 4
28 3 18 6
10080
1 8 16 30 55
15 21 7 12
24 20 9 2
28 3 10 14
10080
1 10 14 30 55
15 7 12 21
24 9 20 2
28 16 3 8
10080
1 10 14 30 55
15 8 20 12
24 6 18 7
28 21 2 4
10080
1 10 14 30 55
15 16 3 21
24 9 20 2
28 7 12 8
10080
1 10 20 21 52
15 18 3 16
24 8 14 6
28 7 12 5
10080
1 12 18 21 52
15 7 14 16
24 20 5 3
28 6 8 10
10080
1 12 18 21 52
15 20 14 3
24 7 5 16
28 6 8 10
10080
1 7 14 28 50
16 3 27 4
18 24 2 6
21 12 8 9
6048
1 7 14 28 50
16 4 27 3
18 24 2 6
21 9 8 12
6048
1 7 14 28 50
16 4 24 6
18 27 2 3
21 8 9 12
6048
1 8 18 28 55
16 14 20 5
21 6 4 24
30 15 7 3
10080
1 10 15 28 54
16 8 24 6
21 9 4 20
30 14 7 3
10080
1 10 20 24 55
16 28 9 2
21 12 7 15
30 3 8 14
10080
1 10 20 24 55
16 28 9 2
21 12 8 14
30 3 7 15
10080
1 14 15 24 54
16 4 28 6
21 20 3 10
30 9 8 7
10080
1 10 14 30 55
18 6 24 7
20 8 15 12
28 21 2 4
10080
1 10 14 30 55
18 6 24 7
20 21 2 12
28 8 15 4
10080
1 8 24 27 60
18 15 7 20
28 21 9 2
30 6 10 14
15120
1 10 21 27 59
18 24 3 14
28 7 16 8
30 9 15 5
15120
1 15 20 24 60
18 6 27 9
28 8 14 10
30 21 2 7
15120
1 15 20 24 60
18 8 27 7
28 21 2 9
30 6 14 10
15120
1 15 16 27 59
18 24 9 8
28 3 21 7
30 14 5 10
15120
1 8 15 28 52
20 14 6 12
21 5 16 10
24 18 7 3
10080
1 4 24 30 59
20 18 9 12
27 21 5 6
28 10 14 7
15120
1 4 24 30 59
20 18 14 7
27 21 5 6
28 10 9 12
15120
1 7 21 30 59
20 24 6 9
27 18 10 4
28 5 12 14
15120
1 10 18 30 59
20 6 24 9
27 21 7 4
28 12 5 14
15120
1 12 16 30 59
20 6 15 18
27 21 9 2
28 10 7 14
15120
1 12 16 30 59
20 10 15 14
27 21 9 2
28 6 7 18
15120
1 14 21 24 60
20 4 6 30
27 18 12 3
28 15 10 7
15120
1 14 21 24 60
20 6 4 30
27 12 18 3
28 15 10 7
15120
1 14 21 24 60
20 6 4 30
27 18 12 3
28 10 15 7
15120
2 10 20 21 53
6 18 24 5
28 14 3 8
30 4 7 12
10080
2 3 20 28 53
7 16 21 9
24 15 6 8
30 14 4 5
10080
2 10 12 28 52
7 21 15 9
24 16 4 8
30 3 14 5
10080
2 12 18 21 53
7 28 10 8
24 5 4 20
30 6 14 3
10080
2 14 15 21 52
7 5 28 12
24 16 8 4
30 9 3 10
10080
2 15 16 20 53
7 4 14 28
24 21 5 3
30 8 9 6
10080
2 15 16 20 53
7 28 14 4
24 3 5 21
30 8 9 6
10080
2 5 16 30 53
9 21 15 8
20 24 3 6
28 4 14 7
10080
2 5 16 30 53
9 21 15 8
20 24 6 3
28 4 7 14
10080
2 6 15 30 53
9 24 4 16
20 5 21 7
28 14 8 3
10080
2 8 16 24 50
9 12 15 14
20 21 6 3
28 5 7 10
10080
2 8 16 24 50
9 21 6 14
20 5 15 10
28 12 7 3
10080
2 8 16 24 50
9 21 14 6
20 5 15 10
28 12 3 7
10080
2 8 16 24 50
9 21 6 14
20 12 15 3
28 5 7 10
10080
2 7 21 27 57
9 24 10 14
28 6 18 5
30 15 4 8
15120
2 10 18 27 57
9 12 20 16
28 21 3 5
30 6 14 7
15120
2 14 15 20 51
10 9 8 24
18 5 21 7
28 16 4 3
10080
2 14 15 20 51
10 9 24 8
18 5 7 21
28 16 4 3
10080
2 3 18 28 51
10 16 20 5
21 14 7 9
24 15 4 8
10080
2 3 18 28 51
10 16 20 5
21 15 7 8
24 14 4 9
10080
2 3 18 30 53
10 28 8 7
21 6 14 12
24 20 5 4
10080
2 6 14 28 50
10 20 5 15
21 12 9 8
24 7 16 3
10080
2 7 14 28 51
10 8 15 18
21 9 16 5
24 20 3 4
10080
2 7 14 28 51
10 20 3 18
21 9 16 5
24 8 15 4
10080
2 7 14 28 51
10 20 16 5
21 9 3 18
24 8 15 4
10080
2 7 14 28 51
10 20 16 5
21 18 3 9
24 4 15 8
10080
2 7 15 28 52
10 30 3 9
21 12 14 5
24 4 16 8
10080
2 8 14 28 52
10 7 5 30
21 12 16 3
24 15 9 4
10080
2 12 16 20 50
10 28 3 9
21 6 15 8
24 5 14 7
10080
2 4 21 30 57
10 14 24 9
27 18 5 7
28 15 6 8
15120
2 4 21 30 57
10 15 24 8
27 18 5 7
28 14 6 9
15120
2 6 20 30 58
10 15 21 12
27 24 4 3
28 7 9 14
15120
2 6 20 30 58
10 24 21 3
27 15 4 12
28 7 9 14
15120
2 12 14 30 58
10 4 20 24
27 15 9 7
28 21 6 3
15120
2 12 20 24 58
10 4 14 30
27 15 9 7
28 21 6 3
15120
2 12 20 24 58
10 14 4 30
27 15 9 7
28 6 21 3
15120
2 12 14 30 58
10 15 9 24
27 4 20 7
28 21 6 3
15120
2 12 14 30 58
10 20 24 4
27 7 3 21
28 9 15 6
15120
2 12 20 24 58
10 30 4 14
27 7 9 15
28 6 21 3
15120
2 12 20 24 58
10 30 4 14
27 7 21 3
28 6 9 15
15120
2 5 21 24 52
12 9 16 15
14 28 3 7
30 8 10 4
10080
2 7 15 28 52
12 16 21 3
14 10 4 24
30 9 8 5
10080
2 8 16 24 50
12 10 21 7
15 9 6 20
28 14 5 3
10080
2 8 16 24 50
12 14 21 3
15 9 6 20
28 10 5 7
10080
2 8 16 24 50
12 14 21 3
15 10 5 20
28 9 6 7
10080
2 8 16 24 50
12 21 14 3
15 6 9 20
28 10 5 7
10080
2 8 16 24 50
12 21 14 3
15 10 5 20
28 6 9 7
10080
2 8 16 24 50
12 28 3 7
20 5 15 10
21 9 14 6
10080
2 8 16 24 50
12 28 7 3
20 5 15 10
21 9 6 14
10080
2 8 16 24 50
12 28 3 7
20 9 15 6
21 5 14 10
10080
2 9 15 24 50
12 28 3 7
20 8 16 6
21 5 14 10
10080
2 10 18 27 57
12 9 20 16
21 28 3 5
30 6 14 7
15120
2 10 12 28 52
14 3 30 5
15 21 7 9
24 16 4 8
10080
2 6 21 28 57
14 7 9 27
18 24 10 5
30 15 8 4
15120
2 7 21 27 57
14 6 9 28
18 24 10 5
30 15 8 4
15120
2 7 20 28 57
14 10 21 12
18 27 9 3
30 8 4 15
15120
2 7 20 28 57
14 16 21 6
18 27 3 9
30 5 12 10
15120
2 5 21 30 58
14 7 9 28
20 24 8 6
27 18 10 3
15120
2 5 21 30 58
14 28 9 7
20 6 8 24
27 18 10 3
15120
2 7 21 28 58
14 5 9 30
20 24 8 6
27 18 10 3
15120
2 7 21 28 58
14 5 30 9
20 24 8 6
27 18 3 10
15120
2 7 21 28 58
14 9 5 30
20 24 8 6
27 10 18 3
15120
2 7 21 28 58
14 9 30 5
20 24 8 6
27 10 3 18
15120
2 7 21 28 58
14 30 5 9
20 24 8 6
27 3 18 10
15120
2 7 21 28 58
14 30 8 6
20 24 5 9
27 3 18 10
15120
2 7 21 28 58
14 30 8 6
20 24 9 5
27 3 10 18
15120
2 7 21 28 58
14 30 9 5
20 24 8 6
27 3 10 18
15120
2 9 18 28 57
14 7 21 15
20 30 4 3
27 8 10 12
15120
2 9 18 28 57
14 10 30 3
20 21 4 12
27 8 7 15
15120
2 10 15 30 57
14 7 28 8
20 12 4 21
27 18 9 3
15120
2 10 15 30 57
14 8 28 7
20 21 12 4
27 9 3 18
15120
2 10 18 28 58
14 9 5 30
20 8 24 6
27 21 7 3
15120
2 10 18 28 58
14 9 5 30
20 24 8 6
27 7 21 3
15120
2 10 15 30 57
14 12 28 3
20 7 9 21
27 18 4 8
15120
2 3 18 28 51
15 24 4 8
16 10 20 5
21 14 7 9
10080
2 3 18 28 51
15 24 4 8
16 20 10 5
21 7 14 9
10080
2 3 18 28 51
15 24 8 4
16 20 5 10
21 7 14 9
10080
2 5 20 24 51
15 8 18 10
16 28 4 3
21 9 7 14
10080
2 7 14 28 51
15 4 8 24
16 20 10 5
21 18 9 3
10080
2 7 14 28 51
15 4 24 8
16 20 10 5
21 18 3 9
10080
2 7 14 28 51
15 8 4 24
16 10 20 5
21 18 9 3
10080
2 7 14 28 51
15 8 4 24
16 20 10 5
21 9 18 3
10080
2 7 14 28 51
15 8 24 4
16 20 10 5
21 9 3 18
10080
2 7 14 28 51
15 24 4 8
16 20 10 5
21 3 18 9
10080
2 7 14 28 51
15 24 8 4
16 20 5 10
21 3 18 9
10080
2 7 14 28 51
15 24 8 4
16 20 10 5
21 3 9 18
10080
2 4 21 30 57
15 10 24 8
18 27 5 7
28 14 6 9
15120
2 4 21 30 57
15 14 20 8
18 27 3 9
28 10 12 7
15120
2 4 21 30 57
15 20 8 14
18 27 9 3
28 7 10 12
15120
2 4 21 30 57
15 20 10 12
18 27 9 3
28 7 8 14
15120
2 4 24 27 57
15 30 7 5
18 21 10 8
28 6 9 14
15120
2 7 21 27 57
15 8 24 10
18 30 5 4
28 9 6 14
15120
2 14 20 21 57
15 30 4 8
18 3 27 9
28 12 7 10
15120
2 14 21 27 64
18 15 24 7
28 16 12 8
30 9 5 20
30240
2 8 24 30 64
20 14 21 9
27 18 12 7
28 15 5 16
30240
2 8 24 30 64
20 21 14 9
27 12 18 7
28 15 5 16
30240
2 7 27 28 64
21 12 16 15
24 18 14 8
30 20 5 9
30240
2 7 27 28 64
21 18 16 9
24 20 5 15
30 12 14 8
30240
2 15 20 27 64
21 9 18 16
24 28 7 5
30 8 12 14
30240
2 16 18 28 64
21 9 7 27
24 15 20 5
30 14 12 8
30240
2 16 18 28 64
21 27 7 9
24 5 20 15
30 14 12 8
30240
3 9 24 27 63
12 21 14 16
28 20 5 10
30 8 18 7
30240
3 16 20 24 63
12 10 27 14
28 9 8 18
30 21 7 5
30240
3 5 27 28 63
16 18 20 9
21 24 8 10
30 14 7 12
30240
3 8 24 28 63
16 20 9 18
21 27 10 5
30 7 14 12
30240
3 8 24 28 63
16 20 18 9
21 27 5 10
30 7 14 12
30240
4 14 16 28 62
12 27 18 5
21 10 7 24
30 8 15 9
30240
5 14 20 21 60
8 12 24 16
27 18 9 6
28 10 7 15
30240
5 14 20 21 60
8 16 12 24
27 9 18 6
28 15 7 10
30240
5 14 20 21 60
8 24 12 16
27 6 18 9
28 15 7 10
30240
5 14 20 21 60
8 24 12 16
27 9 18 6
28 10 7 15
30240
5 12 16 27 60
9 10 21 20
24 14 15 7
28 18 6 8
30240
5 12 16 27 60
9 20 21 10
24 7 15 14
28 18 6 8
30240
5 14 20 21 60
9 18 27 6
24 12 8 16
28 10 7 15
30240
5 14 20 21 60
9 27 18 6
24 8 12 16
28 10 7 15
30240
5 14 20 21 60
12 8 24 16
18 27 9 6
28 10 7 15
30240
5 14 20 21 60
12 16 8 24
18 9 27 6
28 15 7 10
30240
5 14 20 21 60
12 24 8 16
18 6 27 9
28 15 7 10
30240
5 14 20 21 60
12 24 8 16
18 9 27 6
28 10 7 15
30240
5 7 20 28 60
12 27 6 15
21 16 14 9
24 10 18 8
30240
5 7 20 28 60
14 10 21 15
16 24 8 12
27 18 9 6
30240
5 7 20 28 60
14 10 21 15
16 24 12 8
27 18 6 9
30240
5 9 18 28 60
14 21 10 15
16 8 24 12
27 20 7 6
30240
5 7 20 28 60
14 10 21 15
18 27 6 9
24 16 12 8
30240
5 7 20 28 60
14 10 21 15
18 27 9 6
24 16 8 12
30240
|
|
Posted by Charlie
on 2016-02-17 20:16:44 |
computer solution
