Using 8 of the toothpicks, make a square with two on each side. Using another two, make a smaller square in one corner of the first. Using the remaining two, make a cross in the middle of the second square. One large square on the outside, one medium square inside it and four small squares formed inside that, for a total of six.
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I'm not sure exactly what your question is, but the squares of 4 and 5 do have this property (and are the only perfect squares that do).
There is only one number of this type which is 7744=88^2 where a is 7 and b is 4
A cube is made out of squares ONLY. A Pyramid is out of triangles and sometimes with squares.
Squares have four equal sides and equal angles, rectangles only have two.
A cube has six faces that are all squares so the net can only consist of six squares.