# Professor Arie Duijvestijn and the Scottish Book

### Professor Arie Duijvestijn and the Scottish Book

In the 1930s and 1940s, a group of mathematicians used to meet regularly in the Scottish Café in what was then the Polish city of Lwów. During their meetings they scribbled down 193 mathematical problems in a notebook that later became famous as the Scottish Book. The man who solved problem 59 from that book was none other than University of Twente professor Arie Duijvestijn.

When Poland gained independence after the First World War, the government of the newly-established country did everything it could to promote the development of the hard sciences. And their efforts were crowned with success: in the early 1930s, Lwów – the modern-day Ukrainian city of Lviv – was home to a collective of leading Polish mathematicians who were referred to as the Lwów mathematical school. The mathematicians met regularly in the Szkocka Café (Scottish Café) and initially wrote their mathematical problems on the café’s marble tables, but because those tables were removed at the end of the day, one of the group, Stefan Banach, bought a notebook that was kept at the café.

On 17 July 1935, Banach wrote the first problem in that notebook, which would later come to be known as the Scottish Book. It contains a total of 193 problems, and the notebook promises a reward to the person who solves some of them: for instance five glasses of beer, a cup of coffee or even a live goose.

While the members of the mathematical school were busy filling their notebook in the Scottish Café, Arie Duijvestijn (1927–1998) was still at primary school. Decades later, in 1962, he gained his doctorate from Eindhoven Technical College (now Eindhoven University of Technology) in the field of perfect squares; his PhD thesis resolved one of the 193 problems described in the Scottish Book. To be precise: problem 59, written down by Stanislaw Ruziewicz in late 1935 or early 1936. The problem read: ‘Can a square be divided into squares that all have different dimensions?’ Today the problem is known as ‘squaring the square’.

In 1925, Polish mathematician Zbigniew Morón divided a 32 x 33 rectangle into nine squares that each had different dimensions. It was long believed that this would be impossible in a square, until 1936 when a group of students at Trinity College in Cambridge became fascinated with the problem. In 1940, they produced a square made up of 69 different squares; one of them, R.L. Brooks, would later manage to get that number down to 39.

In 1962, Arie Duijvestijn proved that it would be impossible to form a square out of fewer than 21 different squares, but he had not yet worked out whether that goal of 21 squares could be achieved. After achieving his doctorate in Eindhoven, Duijvestijn became professor by special appointment at the new Technische Hogeschool Twente (‘Technical College Twente’) in Enschede. In 1965, he was appointed professor in the Electrical Engineering department, with a further appointment two years later in the Mathematics subdivision. He worked on Problem 59 for many years with the help of a computer, until in March 1978 he discovered his famous 21-square square.

Alongside his work as a professor, Duijvestijn also advised Philips and IBM on their computers, in which role he was one of the driving forces behind the development of information technology in the Netherlands and a joint founder of the Nederlands Genootschap voor Informatica (Dutch Information Technology Society, NGI). And, of course, he was known as the man who solved Problem 59. His academic masterpiece was published on the cover of the Journal of Combinational Theory and forms the logo of the Trinity Mathematical Society. He may not have received five beers or a live goose, but at least he achieved worldwide fame.

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