Problem 59 by Dutchman

In the interwar period (1918-1939) Poland had two important centres of mathematical science – Warsaw and Lviv (today Ukraine). In this second city working over maths became a kind of fun.

Centered aroung mathematical genius prof. Stefan Banach, mathematicians used to have informal meetings in the Scottish Café, where they initially used a marble table for substitution of a blackboard. Unhappily some very valuable solutions were lost when a janitor wiped them out of the table. In 1935 mathematicians used to write their problems down in a special excersise book brought to the scientists by Łucja Banach, the wife of the famous profesor.

In this excersise book – the legendary Scottish Book, which is presently in family hands – mathematicians used to place some problems of a different importance. Fundamental questions are mixed there with few problems on a lighter note. Some of the problems were solved immediately, some had to wait for many years for a result. In over twenty cases lucky scientists who managed to solve some problems were granted awards by authors of those problems.

One of the problem of a lighter note was a problem number 59, writen in the book by Stanisław Ruziewicz (end of 1935 or begin of 1936). The question was:

Is it possible to divide a square into a number of smaller squares that all have different sizes?

That problem was solved by three men in three different ways. The first of them was Zbigniew Morón, who solved this problem for a recktangle of 32 x 33. There are 9 squares on his drawing.

In 1940 the problem was solved by Roland Sprague. He divided the squire in 55 squares all with different sizes. So the question arose “what is the minimum number of squares that is needed to form the big square”. In the same year R.L. Brooks presented a solution with 26 squares.

In 1948 T.H. Willcocks found a solution with 24 small squares.

And finally it was a Dutchman, A.J.W. Duijvesteijn, who in 1962 presented a theoretical mathematical prove that at least 21 little squares are needed. On  March 27, 1978 he presented the drawing.

What is interesting – following Duijvesteijn solution Bob Mackay constructed a cupboard faithfully reflecting the shape of that solution!

Read also: Maths in Scottish Café


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