Dividing a square into 7 parts of equal area and minimizing the perimeter-to-area ratio. Greetings, Here is something I enjoyed working on. I couldn't find the answer online, so I had to figure it out myself. The challenge was to divide a square into 7 equal parts. Each part had to be as compact as possible (minimum perimeter to area ratio), so I knew that the trivial solution of dividing it into 7 long, thin strips would not work. After drawing several iterations, I came up with this model. A hexagon is centered in the square, rotated at 45 degrees so that two of its vertices align with the diagonal. Each shape has an area of 7, in this 7 x 7 square. If this was a tile fastened to a surface, then the fasteners would be placed at the centroids (stars) to prevent uplift in the most efficient way, supposing that they want to use 7 fasteners and not any other number. Fastener patterns with other numbers are easier to figure out. The next-hardest one was with 5 fasteners, but that's just a simple square in the middle instead of a hexagon.
