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edited Mar 7 2010 at 22:35

TonyK
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FG has posted a solution for the 12x1 rectangle, attaining 15/4 181/48 = 3.753.3.7708333...

More results:
11x1 rectangle is 101/27 = 3.740740...
7x3 rectangle is 56/15 = 3.733333...
7x2 rectangle is 67/18 = 3.722222...
8x1 rectangle is 26/7 = 3.714285...
6x1 rectangle is 85/23 = 3.695652...
7x1 rectangle is 11/3 = 3.666666...
5x1 rectangle is 25/7 = 3.571428...

I have removed specific solutions from this answer, as they have been superseded by FG's results.

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edited Mar 3 2010 at 21:49

TonyK
1,453●6●11

Latest figures

FG has posted a solution for the 11x1 12x1 rectangle, attaining 101/27 15/4 = 3.740740..3.75.

I have removed specific solutions from this answer, as they have been superseded by FG's results.

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edited Mar 3 2010 at 14:00

TonyK
1,453●6●11

Latest figures

FG has posted a solution for the 3x7 11x1 rectangle, attaining 56/15 101/27 = 3.733333..3.740740...

More results:
7x3 rectangle is 56/15 = 3.733333...
7x2 rectangle is 67/18 = 3.722222...
8x1 rectangle is 26/7 = 3.714285...
6x1 rectangle is 85/23 = 3.695652...
7x1 rectangle is 11/3 = 3.666666...
5x1 rectangle is 25/7 = 3.571428...

I have removed specific solutions from this answer, as they have been superseded by FG's results.

added 14 characters in body; [made Community Wiki]

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edited Mar 2 2010 at 22:35

TonyK
1,453●6●11

added 85 characters in body

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edited Mar 2 2010 at 21:49

TonyK
1,453●6●11

deleted 2890 characters in body; edited body

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edited Mar 2 2010 at 21:39

TonyK
1,453●6●11

FG has posted a solution for the 8x1 3x7 rectangle, and says that a 7x3 rectangle can attain attaining 56/15 = 3.73333... See comments below3.733333...

More results:
7x2 rectangle is <= 67/18 = 3.72222..., attainable in a 66x61 grid3.722222...
8x1 rectangle is <= 26/7 = 3.714285..., attainable in a 47x40 grid.

These upper bounds are proven, and the grid sizes are big enough to generate consistent lo-hi bounds for all the sub-rectangles. I think this implies that concrete solutions exist. But to generate these solutions is very slow (or perhaps I'm being stupid). The 66x61 grid looks out of reach using my current software; the 47x40 grid will take a day or two.

For a 5x1 rectangle, the maximum sum is 25/7 = 3.571428... This value is attained in the following 29x25 symmetric grid (only the upper left quadrant given here):

2 -3 1 1 -2 2 1 2 0 0 0 1 -3 0 0 -2 0 -1 0 0 -3 2 1 0 0 1 2 1 -1 0 0 2 0 0 0 0 -4 0 -2 0 1 1 2 0 0 0 0 1 0 0 0 0 -2 2 0 3 -3 1 0 0 1 0 0 0 0 1 1 0 -1 0 2 -4 2 -1 -2 0 2 0 2 -1 0 0 0 0 0 2 2 -5 -2 0 2 0 0 -1 1 0 1 0 -2 -2 -19 28 0 -6 0 -4 1 2 2 -5 -6 -2 0 5 4 7 -28 28 -6 0 0 -2 -3 -1 6 4 -2 -2 -3 3 7 -3 -22 19 0 0 0 0 0 0 2 -1 0 -1 -2 6 0 -3 -8 10 0 0 0 0 2 0 1 0 0 -2 3 -7 3 -2 0 -1 0 0 -1 0 2 -2 2 -3 0 8 22 -22 -6 0 2 -2 0 0 -4 -6 6 -4 8 8 -28 -28 28 22 0

The 5x1 rectangle is '28 22 0 22 28', with sum 100. All squares have -28 <= sum <= 283.714285...

For a
6x1 rectangle , the maximum sum is 85/23 = 3.695652... which is attained in a 36x31 grid. Here is the upper left quadrant:

22 -9 -9 -1 1 -6 7 8 -6 -3 -8 0 1 -7 32 -20 -3 0-15 5 4 -3 0 6 -23 7 8 15 10 -5 0 2 -3 -5 -1 0 -2 0 0 2 0 0 14 -34 19 6 -2 0 -4 -2 -24 28 0 0 1 3 0 -1 0 0 4 6 -11 8 1 0 0 -1 -1 -9 0 0 -1 -3 -6 -1 -1 13 5 0 -9 -28 45 -6 -1 -12 -22 22 5 1 0 -13 -2 0 -10 12 23 -13 -7 -6 -22 26 0 10 1 7 -16 2 6 20 -4 -9 2 -27 -4 17 -7 6 -12 -10 0 12 4 -3 -10 6 -9 4 20 -3 -10 5 4 -11 18 -6 -1 -10 -22 28 45 -45 5 -7-16 12 21 31 -7 3 -38 15 17 17 -13 -8 12 -82 68 24 -7 -19 10 -35 -6 6 40 -16 -14 -28 -10 24 28 -10 3 19 -92 92 -29 15 -1 11 -11 -21 -23 38 32 8 -27 -22 -33 55 -5 42 -52 -40 69 -17 2 0 -2 -4 11 0 18 0 6 -8 -2 -12 30 -36 56 -56 -21 33 0 0 0 5 -6 -16 -1 11 0 3 -7 -23 -9 50 -34 -2 -11 7 0 0 -4 0 -1 0 0 0 14 -6 6 -10 3 9 -8 8 0 -14 2 2 2 -1 6 -12 0 9 -36 40 -4 -1 -11 22 66 -66 -32 18 2 8 0 0 -10 20 -42 -26 60 -74 40 28 16 -88 -92 92 92 -14

The 6x1 rectangle is '92 92 -14 -14 92 92', with sum 340. All squares have -92 <= sum <= 92.

For a
7x1 rectangle , the maximum sum is 11/3 = 3.666666..., which is less than the 6x1 rectangle3.666666...

I will post more results as they come in
5x1 rectangle is 25/7 = 3.571428...

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edited Mar 2 2010 at 17:30

TonyK
1,453●6●11

Latest figures

FG has posted a solution for the 8x1 rectangle, and says that a 7x3 rectangle can attain 56/15 = 3.73333... See comments below.

7x2 rectangle is <= 67/18 = 3.72222..., probably attainable in a 66x61 grid.

8x1 rectangle is <= 26/7 = 3.714285..., probably attainable in a 47x40 grid.

Update FG claims a solution for the 8x1 rectangle.Can you post it, please?

For a 5x1 rectangle, the maximum sum is 25/7 = 3.571428... This value is attained in the following 29x25 symmetric grid (only the upper left quadrant given here):

  2  -3   1   1  -2   2   1   2   0   0   0   1  -3   0   0
 -2   0  -1   0   0  -3   2   1   0   0   1   2   1  -1   0
  0   2   0   0   0   0  -4   0  -2   0   1   1   2   0   0
  0   0   1   0   0   0   0  -2   2   0   3  -3   1   0   0
  1   0   0   0   0   1   1   0  -1   0   2  -4   2  -1  -2
  0   2   0   2  -1   0   0   0   0   0   2   2  -5  -2   0
  2   0   0  -1   1   0   1   0  -2  -2 -19  28   0  -6   0
 -4   1   2   2  -5  -6  -2   0   5   4   7 -28  28  -6   0
  0  -2  -3  -1   6   4  -2  -2  -3   3   7  -3 -22  19   0
  0   0   0   0   0   2  -1   0  -1  -2   6   0  -3  -8  10
  0   0   0   0   2   0   1   0   0  -2   3  -7   3  -2   0
 -1   0   0  -1   0   2  -2   2  -3   0   8  22 -22  -6   0
  2  -2   0   0  -4  -6   6  -4   8   8 -28 -28  28  22   0

The 5x1 rectangle is '28 22 0 22 28', with sum 100. All squares have -28 <= sum <= 28.

For a 6x1 rectangle, the maximum sum is 85/23 = 3.695652... which is attained in a 36x31 grid. Here is the upper left quadrant:

 22  -9  -9  -1   1  -6   7   8  -6  -3  -8   0   1  -7  32 -20  -3   0
-15   5   4  -3   0   6 -23   7   8  15  10  -5   0   2  -3  -5  -1   0
 -2   0   0   2   0   0  14 -34  19   6  -2   0  -4  -2 -24  28   0   0
  1   3   0  -1   0   0   4   6 -11   8   1   0   0  -1  -1  -9   0   0
 -1  -3  -6  -1  -1  13   5   0  -9 -28  45  -6  -1 -12 -22  22   5   1
  0 -13  -2   0 -10  12  23 -13  -7  -6 -22  26   0  10   1   7 -16   2
  6  20  -4  -9   2 -27  -4  17  -7   6 -12 -10   0  12   4  -3 -10   6
 -9   4  20  -3 -10   5   4 -11  18  -6  -1 -10 -22  28  45 -45   5  -7
-16  12  21  31  -7   3 -38  15  17  17 -13  -8  12 -82  68  24  -7 -19
 10 -35  -6   6  40 -16 -14 -28 -10  24  28 -10   3  19 -92  92 -29  15
 -1  11 -11 -21 -23  38  32   8 -27 -22 -33  55  -5  42 -52 -40  69 -17
  2   0  -2  -4  11   0  18   0   6  -8  -2 -12  30 -36  56 -56 -21  33
  0   0   0   5  -6 -16  -1  11   0   3  -7 -23  -9  50 -34  -2 -11   7
  0   0  -4   0  -1   0   0   0  14  -6   6 -10   3   9  -8   8   0 -14
  2   2   2  -1   6 -12   0   9 -36  40  -4  -1 -11  22  66 -66 -32  18
  2   8   0   0 -10  20 -42 -26  60 -74  40  28  16 -88 -92  92  92 -14

The 6x1 rectangle is '92 92 -14 -14 92 92', with sum 340. All squares have -92 <= sum <= 92.

For a 7x1 rectangle, the maximum sum is 11/3 = 3.666666..., which is less than the 6x1 rectangle.

I will post more results as they come in.

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edited Mar 2 2010 at 13:06

TonyK
1,453●6●11

Latest figures

7x2 rectangle is <= 67/18 = 3.72222..., probably attainable in a 66x61 grid.

8x1 rectangle is <= 26/7 = 3.714285..., probably attainable in a 47x40 grid.

Update FG claims a solution for the 8x1 rectangle. Can you post it, please?

For a 5x1 rectangle, the maximum sum is 25/7 = 3.571428... This value is attained in the following 29x25 symmetric grid (only the upper left quadrant given here):

  2  -3   1   1  -2   2   1   2   0   0   0   1  -3   0   0
 -2   0  -1   0   0  -3   2   1   0   0   1   2   1  -1   0
  0   2   0   0   0   0  -4   0  -2   0   1   1   2   0   0
  0   0   1   0   0   0   0  -2   2   0   3  -3   1   0   0
  1   0   0   0   0   1   1   0  -1   0   2  -4   2  -1  -2
  0   2   0   2  -1   0   0   0   0   0   2   2  -5  -2   0
  2   0   0  -1   1   0   1   0  -2  -2 -19  28   0  -6   0
 -4   1   2   2  -5  -6  -2   0   5   4   7 -28  28  -6   0
  0  -2  -3  -1   6   4  -2  -2  -3   3   7  -3 -22  19   0
  0   0   0   0   0   2  -1   0  -1  -2   6   0  -3  -8  10
  0   0   0   0   2   0   1   0   0  -2   3  -7   3  -2   0
 -1   0   0  -1   0   2  -2   2  -3   0   8  22 -22  -6   0
  2  -2   0   0  -4  -6   6  -4   8   8 -28 -28  28  22   0

The 5x1 rectangle is '28 22 0 22 28', with sum 100. All squares have -28 <= sum <= 28.

For a 6x1 rectangle, the maximum sum is 85/23 = 3.695652... which is attained in a 36x31 grid. Here is the upper left quadrant:

 22  -9  -9  -1   1  -6   7   8  -6  -3  -8   0   1  -7  32 -20  -3   0
-15   5   4  -3   0   6 -23   7   8  15  10  -5   0   2  -3  -5  -1   0
 -2   0   0   2   0   0  14 -34  19   6  -2   0  -4  -2 -24  28   0   0
  1   3   0  -1   0   0   4   6 -11   8   1   0   0  -1  -1  -9   0   0
 -1  -3  -6  -1  -1  13   5   0  -9 -28  45  -6  -1 -12 -22  22   5   1
  0 -13  -2   0 -10  12  23 -13  -7  -6 -22  26   0  10   1   7 -16   2
  6  20  -4  -9   2 -27  -4  17  -7   6 -12 -10   0  12   4  -3 -10   6
 -9   4  20  -3 -10   5   4 -11  18  -6  -1 -10 -22  28  45 -45   5  -7
-16  12  21  31  -7   3 -38  15  17  17 -13  -8  12 -82  68  24  -7 -19
 10 -35  -6   6  40 -16 -14 -28 -10  24  28 -10   3  19 -92  92 -29  15
 -1  11 -11 -21 -23  38  32   8 -27 -22 -33  55  -5  42 -52 -40  69 -17
  2   0  -2  -4  11   0  18   0   6  -8  -2 -12  30 -36  56 -56 -21  33
  0   0   0   5  -6 -16  -1  11   0   3  -7 -23  -9  50 -34  -2 -11   7
  0   0  -4   0  -1   0   0   0  14  -6   6 -10   3   9  -8   8   0 -14
  2   2   2  -1   6 -12   0   9 -36  40  -4  -1 -11  22  66 -66 -32  18
  2   8   0   0 -10  20 -42 -26  60 -74  40  28  16 -88 -92  92  92 -14

The 6x1 rectangle is '92 92 -14 -14 92 92', with sum 340. All squares have -92 <= sum <= 92.

For a 7x1 rectangle, the maximum sum is 11/3 = 3.666666..., which is less than the 6x1 rectangle.

I will post more results as they come in.

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edited Mar 2 2010 at 10:33

TonyK
1,453●6●11

Latest figures

7x2 rectangle is <= 67/18 = 3.72222..., probably attainable in a 66x61 grid.

8x1 rectangle is <= 26/7 = 3.714285..., probably attainable in a 47x40 grid.

For a 5x1 rectangle, the maximum sum is 25/7 = 3.571428... This value is attained in the following 29x25 symmetric grid (only the upper left quadrant given here):

  2  -3   1   1  -2   2   1   2   0   0   0   1  -3   0   0
 -2   0  -1   0   0  -3   2   1   0   0   1   2   1  -1   0
  0   2   0   0   0   0  -4   0  -2   0   1   1   2   0   0
  0   0   1   0   0   0   0  -2   2   0   3  -3   1   0   0
  1   0   0   0   0   1   1   0  -1   0   2  -4   2  -1  -2
  0   2   0   2  -1   0   0   0   0   0   2   2  -5  -2   0
  2   0   0  -1   1   0   1   0  -2  -2 -19  28   0  -6   0
 -4   1   2   2  -5  -6  -2   0   5   4   7 -28  28  -6   0
  0  -2  -3  -1   6   4  -2  -2  -3   3   7  -3 -22  19   0
  0   0   0   0   0   2  -1   0  -1  -2   6   0  -3  -8  10
  0   0   0   0   2   0   1   0   0  -2   3  -7   3  -2   0
 -1   0   0  -1   0   2  -2   2  -3   0   8  22 -22  -6   0
  2  -2   0   0  -4  -6   6  -4   8   8 -28 -28  28  22   0

The 5x1 rectangle is '28 22 0 22 28', with sum 100. All squares have -28 <= sum <= 28.

For a 6x1 rectangle, the maximum sum is 85/23 = 3.695652... which is attained in a 36x31 grid. Here is the upper left quadrant:

 22  -9  -9  -1   1  -6   7   8  -6  -3  -8   0   1  -7  32 -20  -3   0
-15   5   4  -3   0   6 -23   7   8  15  10  -5   0   2  -3  -5  -1   0
 -2   0   0   2   0   0  14 -34  19   6  -2   0  -4  -2 -24  28   0   0
  1   3   0  -1   0   0   4   6 -11   8   1   0   0  -1  -1  -9   0   0
 -1  -3  -6  -1  -1  13   5   0  -9 -28  45  -6  -1 -12 -22  22   5   1
  0 -13  -2   0 -10  12  23 -13  -7  -6 -22  26   0  10   1   7 -16   2
  6  20  -4  -9   2 -27  -4  17  -7   6 -12 -10   0  12   4  -3 -10   6
 -9   4  20  -3 -10   5   4 -11  18  -6  -1 -10 -22  28  45 -45   5  -7
-16  12  21  31  -7   3 -38  15  17  17 -13  -8  12 -82  68  24  -7 -19
 10 -35  -6   6  40 -16 -14 -28 -10  24  28 -10   3  19 -92  92 -29  15
 -1  11 -11 -21 -23  38  32   8 -27 -22 -33  55  -5  42 -52 -40  69 -17
  2   0  -2  -4  11   0  18   0   6  -8  -2 -12  30 -36  56 -56 -21  33
  0   0   0   5  -6 -16  -1  11   0   3  -7 -23  -9  50 -34  -2 -11   7
  0   0  -4   0  -1   0   0   0  14  -6   6 -10   3   9  -8   8   0 -14
  2   2   2  -1   6 -12   0   9 -36  40  -4  -1 -11  22  66 -66 -32  18
  2   8   0   0 -10  20 -42 -26  60 -74  40  28  16 -88 -92  92  92 -14

The 6x1 rectangle is '92 92 -14 -14 92 92', with sum 340. All squares have -92 <= sum <= 92.

For a 7x1 rectangle, the maximum sum is 11/3 = 3.666666..., which is less than the 6x1 rectangle.

I will post more results as they come in.

added 520 characters in body; added 9 characters in body

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edited Mar 2 2010 at 10:01

TonyK
1,453●6●11

Latest figures

7x2 rectangle is <= 67/18 = 3.72222..., probably attainable in a 66x61 grid.

8x1 rectangle is <= 26/7 = 3.714285..., probably attainable in a 47x40 grid.

These upper bounds are proven, and the grid sizes are big enough to generate consistent lo-hi bounds for all the sub-rectangles. I think this implies that solutions exist. But to generate these solutions is very slow (or perhaps I'm being stupid). The 66x61 grid looks out of reach using my current software; the 47x40 grid will take a day or two.

For a 5 x 1 5x1 rectangle, the maximum sum is 25/7 = 3.571428... This value is attained in the following 29 x 25 29x25 symmetric grid (only the upper left quadrant given here):

  2  -3   1   1  -2   2   1   2   0   0   0   1  -3   0   0
 -2   0  -1   0   0  -3   2   1   0   0   1   2   1  -1   0
  0   2   0   0   0   0  -4   0  -2   0   1   1   2   0   0
  0   0   1   0   0   0   0  -2   2   0   3  -3   1   0   0
  1   0   0   0   0   1   1   0  -1   0   2  -4   2  -1  -2
  0   2   0   2  -1   0   0   0   0   0   2   2  -5  -2   0
  2   0   0  -1   1   0   1   0  -2  -2 -19  28   0  -6   0
 -4   1   2   2  -5  -6  -2   0   5   4   7 -28  28  -6   0
  0  -2  -3  -1   6   4  -2  -2  -3   3   7  -3 -22  19   0
  0   0   0   0   0   2  -1   0  -1  -2   6   0  -3  -8  10
  0   0   0   0   2   0   1   0   0  -2   3  -7   3  -2   0
 -1   0   0  -1   0   2  -2   2  -3   0   8  22 -22  -6   0
  2  -2   0   0  -4  -6   6  -4   8   8 -28 -28  28  22   0

The 5x1 rectangle is '28 22 0 22 28', with sum 100. All squares have -28 <= sum <= 28.

For a 6x1 rectangle, the maximum sum is 85/23 = 3.695652... which is attained in a 36x31 grid. Here is the upper left quadrant:

 22  -9  -9  -1   1  -6   7   8  -6  -3  -8   0   1  -7  32 -20  -3   0
-15   5   4  -3   0   6 -23   7   8  15  10  -5   0   2  -3  -5  -1   0
 -2   0   0   2   0   0  14 -34  19   6  -2   0  -4  -2 -24  28   0   0
  1   3   0  -1   0   0   4   6 -11   8   1   0   0  -1  -1  -9   0   0
 -1  -3  -6  -1  -1  13   5   0  -9 -28  45  -6  -1 -12 -22  22   5   1
  0 -13  -2   0 -10  12  23 -13  -7  -6 -22  26   0  10   1   7 -16   2
  6  20  -4  -9   2 -27  -4  17  -7   6 -12 -10   0  12   4  -3 -10   6
 -9   4  20  -3 -10   5   4 -11  18  -6  -1 -10 -22  28  45 -45   5  -7
-16  12  21  31  -7   3 -38  15  17  17 -13  -8  12 -82  68  24  -7 -19
 10 -35  -6   6  40 -16 -14 -28 -10  24  28 -10   3  19 -92  92 -29  15
 -1  11 -11 -21 -23  38  32   8 -27 -22 -33  55  -5  42 -52 -40  69 -17
  2   0  -2  -4  11   0  18   0   6  -8  -2 -12  30 -36  56 -56 -21  33
  0   0   0   5  -6 -16  -1  11   0   3  -7 -23  -9  50 -34  -2 -11   7
  0   0  -4   0  -1   0   0   0  14  -6   6 -10   3   9  -8   8   0 -14
  2   2   2  -1   6 -12   0   9 -36  40  -4  -1 -11  22  66 -66 -32  18
  2   8   0   0 -10  20 -42 -26  60 -74  40  28  16 -88 -92  92  92 -14

The 6x1 rectangle is '92 92 -14 -14 92 92', with sum 340. All squares have -92 <= sum <= 92.

For a 7x1 rectangle, the maximum sum is 11/3 = 3.666666..., which is less than the 6x1 rectangle.

I will post more results as they come in.

added 1245 characters in body

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edited Mar 2 2010 at 7:51

TonyK
1,453●6●11

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answered Mar 1 2010 at 23:10

TonyK
1,453●6●11