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