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Smaller symmetric example
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F_G
F_G
  • 837
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  • 7

Update : using Leonid's remark on imposing symmetry w.l.o.g., here is a smaller and symmetric 13x16 example with a 7/2 rectangle (scale by 1/4):

 0  0  0  0  1 -1  0  0  0  0 -1  1  0  0  0  0
 1  0  0 -2 -2  4 -1  0  0 -1  4 -2 -2  0  0  1
-2  0  0  2  0 -4  4 -1 -1  4 -4  0  2  0  0 -2
 1  0  0  0  0  0 -4  3  3 -4  0  0  0  0  0  1
 0  0  0  0  0  0  0 -1 -1  0  0  0  0  0  0  0
 0  0  0  0  2  2 -2 -1 -1 -2  2  2  0  0  0  0
 0  0  0  0 -2 -4  4  3  3  4 -4 -2  0  0  0  0
 0  0  0  0  2  2 -2 -1 -1 -2  2  2  0  0  0  0
 0  0  0  0  0  0  0 -1 -1  0  0  0  0  0  0  0
 1  0  0  0  0  0 -4  3  3 -4  0  0  0  0  0  1
-2  0  0  2  0 -4  4 -1 -1  4 -4  0  2  0  0 -2
 1  0  0 -2 -2  4 -1  0  0 -1  4 -2 -2  0  0  1
 0  0  0  0  1 -1  0  0  0  0 -1  1  0  0  0  0

I also tried to find an example with a ratio better than 7/2, keeping the symmetry and a 1x4 rectangle in the middle, but could not find any (the largest size I am able to check is 33x38).

Update : using Leonid's remark on imposing symmetry w.l.o.g., here is a smaller and symmetric 13x16 example with a 7/2 rectangle (scale by 1/4):

 0  0  0  0  1 -1  0  0  0  0 -1  1  0  0  0  0
 1  0  0 -2 -2  4 -1  0  0 -1  4 -2 -2  0  0  1
-2  0  0  2  0 -4  4 -1 -1  4 -4  0  2  0  0 -2
 1  0  0  0  0  0 -4  3  3 -4  0  0  0  0  0  1
 0  0  0  0  0  0  0 -1 -1  0  0  0  0  0  0  0
 0  0  0  0  2  2 -2 -1 -1 -2  2  2  0  0  0  0
 0  0  0  0 -2 -4  4  3  3  4 -4 -2  0  0  0  0
 0  0  0  0  2  2 -2 -1 -1 -2  2  2  0  0  0  0
 0  0  0  0  0  0  0 -1 -1  0  0  0  0  0  0  0
 1  0  0  0  0  0 -4  3  3 -4  0  0  0  0  0  1
-2  0  0  2  0 -4  4 -1 -1  4 -4  0  2  0  0 -2
 1  0  0 -2 -2  4 -1  0  0 -1  4 -2 -2  0  0  1
 0  0  0  0  1 -1  0  0  0  0 -1  1  0  0  0  0

I also tried to find an example with a ratio better than 7/2, keeping the symmetry and a 1x4 rectangle in the middle, but could not find any (the largest size I am able to check is 33x38).

Source Link
F_G
F_G
  • 837
  • 5
  • 7

A computer search led to the following 16x16 example with a 7/2 rectangle in the middle (scale everything by 1/12):

 0  -3   3   0   0   0   0   0   0   0   0  -6   6   0   0   0
-1   1  -3   0   3   3  -3   0   0  -3   0   6   0  -3   0   0
 4   2   0   0 -12   9  -3   0   0   0   9 -12   0  -3   6   0
-6   0   0   0   6 -12  12  -6   0  12 -12   3   3   2  -2  -6
 6   0   0   0   6   0 -12   6  12 -12   0   3   2  -1  -4  12
-3   0   0   0  -3   2  -2   0  -6   6  -6  12  -5  -7   0   0
 0   0   0   0   3   4  -4  -6   0 -12  12  -6   0  12   0  -6
 0   0   0   0  -6 -12  12   6  12  12 -12  -6   0   0   0   0
 0   9  -6   0   6   2  -2  -4  -2 -10  10   0   5  -9  -2   0
 3  -3   0   0   0   4  -4  -2  -4   4  -4   6   0   4   2  -6
 0  -6   6   0   0   0 -12   6  12 -12   0  -3  -2   1   4   6
-6   0   0   0   6 -12  12   2  -8  12 -12   3   3   0   0  -6
 8   0  -6   4 -12   9   3  -5  -1   0   9  -6  -4   4   0   3
 0   0   0  -4   4   0   0   3  -7   0   3   1  -2  -1  -4   3
-2   0   3  -1   0   3   0  -1   2   0  -6  -1   7   0   0   0
-3   0   6   1   0  -4   0   1  -1  -3   3   0   0   0   0   0