How Many Squares? [solution]

Solution to Puzzle 1
The five diagrams above show all the 21 perfect squares
which can be found in the cross of 20 dots.

Solution to Puzzle 2
The diagram above shows "no-squares" position
with the six dots removed.

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!

 Posted: December 11, 2007