
The answer to Puzzle 1 is 21 perfect squares.
They are shown in the five diagram on the top illustration.
There is a nice story behind this ingenious old puzzle. It illustrates
a tricky nature of Puzzle 1 very well.
In 1893 professor Louis Hoffmann asked in his famous book Puzzles Old
and New to arrange twenty counters so that they form thirteen
different squares, and in his original solution (he showed a pattern
exactly as our big cross of 20 green dots) stated that there are
seventeen perfect squares.
Several decades later, Henry E. Dudeney, England's greatest puzzle
creator, improved Hoffmann's solution with 17 squares, and did this...
twice  first it was a new solution with 19 squares, and then  21.
Both solutions were published in Dudeney's puzzle books.
The answer to Puzzle 2 which we show on the bottom illustration is exactly as that
from Hoffmann's book  not a single square remains. Moreover, all your
correct solutions fully coincide with this old one! 
