Enhanced Sets of Heptominoes
Since the set is large we will look only at forming squares with an enhanced set. We omit the heptomino with a hole and so need an extra five copies of one of the pieces to form a side 28 square.
For those pieces which form a 2x7 rectangle we can form squares of all sizes. The diagram below shows side 28 and 35 squares.
Since any 7 x even rectangle is possible with these pieces any square can be increased in size by 14. Also these pieces can also form a 21x21 square which will allow squares to be increased in size by 21 thus showing that all squares of side 7k (k>3) can be made.
The remaining pieces can all be used to make squares of side 28. The diagrams are here.
A number of other pieces can also form a side 35 square. These pieces will prbably also form larger squares.
These pieces should also form larger squares in a similar amnner.