**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 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): <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.