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Grid Packing at an Angle

If a rectangular frame is placed over a grid, what is the maximal number of squares it can contain? Usually, lining up the frame with the grid gives an optimal answer.

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 Ed Pegg Jr
"Grid Packing at an Angle"
 http://demonstrations.wolfram.com/GridPackingAtAnAngle/
 Wolfram Demonstrations Project
 Published: September 15, 2010
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