Cut-the-Knot

In my previous post titled Four Fours Representation of 32, I made reference to a site called Cut-the-Knot. In this post, I'll look at the site more closely and I feel it certainly deserves closer attention. Here is what one reviewer wrote about the site.

The Web may contain almost every possible problem, puzzle, and article imaginable, but it’s decentralized nature makes it’s hard to locate good content in a sea of endless tutorials, amusing pictures, and commercial promotions. If you’re trying to find extracurricular mathematical materials you need to know where to look, but more importantly, what to look for. Knowing an erudite guide makes life much easier. Alexander Bogomolny, a professional mathematician and curator of mathematical recreations and other topics, is that guide. His site Cut-the-Knot is an enormous collection of fascinating articles, illustrations, and animations covering a wide range of mostly non-advanced mathematics. One of the defining features of his articles are the interactive Java applets that illustrate a problem or principle. The site has been continuously updated since 1997, which makes it among the most comprehensive such repositories online. Unfortunately, because it was created more than fifteen years ago, its age shows in the design and technology used (Java applets are no longer the preferred delivery mechanism for interactive media). Although Cut-the-Knot has garnered over twenty awards, including one from Scientific American, it is not as well known as it should be. If you’re looking for a source of enrichment for regular math classes this is one of the best places to start.

In this post, I'll just look at the first problem that occurs under the subject of Arithmetic and this is 100 Grasshoppers on a Triangular Board. Here is the problem statement:

A triangular board has been cut into 100 small triangular cells by the lines parallel to its sides. Two cells that share a side are said to be neighbours. In each cell there is a grasshopper. All at once, the grasshoppers hop from their cells to neighboring cells. This happens 9 times. Prove that at least 10 cells are now empty.

The solution is as follows:

The board is naturally coloured into two colours so that the neighbouring cells are coloured differently. Let the colours be red and brown and label the grasshoppers by the colour of their cells. Convince yourself that there are 55 red and 45 brown cells. In one hop, the brown grasshoppers move to the red cells thus emptying 55 cells. On the same hop, the brown grasshoppers move to the red cells thus filling at most 45 red cells. Inevitably, at least 10 red cells will remain empty.

The next time the grasshoppers move, they can move into their original cells, thus filling all of them. The situation will return to the original one, while the argument will apply to every odd hop. Perhaps there may happen a greater mix-up of the red and brown hoppers, but the best that may be claimed is that after every odd hop, there are at least 10 empty cells.

In general, there are \(n(n+1)/2\) red cells and \(n(n-1)/2\) brown cells. The difference being \(n(n+1)/2 - n(n-1)/2 = n\), there are always at least \(n\) empty cells after an odd number of hops.

The specific number of hops (9) is nothing but a red herring.

So, an interesting little problem, and of course this site is replete with many more, 244 in fact under the subject heading of Arithmetic. The subject headings are:

• Arithmetic with 244 articles
• Algebra with 1059 articles
• Geometry with 1824 articles
• Probability with 204 articles
• Trigonometry with 36 (unnumbered) articles
• Visual Illusions with 51 (unnumbered) articles

There is a glossary with more articles and links to many more. Unfortunately, the author of the site, Alex Bogomolny, died in 2018 at the age of 70 but he has left behind a treasure trove of mathematical information. 

Dr. Alexander Bogomolny, May 2017
(In Costa Rica, holding a sloth)

Here is what his Wikipedia entry had to say:

Alexander Bogomolny (January 4, 1948  – July 7, 2018) was a Soviet-born Israeli American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Mathematics, senior instructor at Hebrew University and software consultant at Ben Gurion University. He wrote extensively about arithmetic, probability, algebra, geometry, trigonometry and mathematical games.

He was known for his contribution to heuristics and mathematics education, creating and maintaining the mathematically themed educational website Cut-the-Knot for the Mathematical Association of America (MAA) Online. He was a pioneer in mathematical education on the internet, having started Cut-the-Knot in October 1996.

Bogomolny attended Moscow school No. 444, for gifted children, then entered Moscow State University, where he graduated with a master's degree in mathematics in 1971.[3] From 1971 to 1974 he was a junior research fellow at the Moscow Institute of Electronics and Mathematics. He emigrated to Israel and became a senior programmer at Lake Kinneret Research Laboratory in Tiberias, Israel (1974 – 1977) and a software consultant at Ben Gurion University in Negev, Be’er Sheva, Israel (1976 – 1977). From 1976 to 1983 he was a Senior Instructor and researcher at Hebrew University in Jerusalem. He received his Ph.D. in mathematics at Hebrew University in 1981. His dissertation is titled, A New Numerical Solution for the Stamp Problem and his thesis advisor was Gregory I. Eskin. From 1981 to 1982 he was also a Visiting Professor at Ohio State University where he taught mathematics.

From 1982 to 1987 he was Professor of Mathematics at the University of Iowa. From August 1987 to August 1991 he was Vice President of Software Development at CompuDoc, Inc.

Cut-the-knot (CTK) was a free, advertisement-funded educational website which Bogomolny maintained from 1996 to 2018. It was devoted to popular exposition of various topics in mathematics. The site was designed for teachers, children and parents, and anyone else curious about mathematics, with an eye to educating, encouraging interest, and provoking curiosity. Its name is a reference to the legend of Alexander the Great's solution to the Gordian knot.

CTK won more than 20 awards from scientific and educational publications, including a Scientific American Web Award in 2003, the Encyclopædia Britannica's Internet Guide Award, and Science's NetWatch award.

The site was remarkably prolific and contained extensive analysis of many of the classic problems in recreational mathematics including the Apollonian gasket, Napoleon's theorem, logarithmic spirals, The Prisoner of Benda, the Pitot theorem, and the monkey and the coconuts problem. Once, in a remarkable tour de force, CTK published 122 proofs of the Pythagorean theorem.

Bogomolny did indeed entertain but his deeper goal was to educate. He wrote a manifesto for CTK in which he said that "Judging Mathematics by its pragmatic value is like judging symphony by the weight of its score."[11] He describes the site as "a resource that would help learn, if not math itself, then, at least, ways to appreciate its beauty." And he wonders why it is acceptable among otherwise well-educated people "to confess a dislike and misunderstanding of Mathematics as a whole."

Many mathematical ideas are illustrated by applets. CTK wiki (powered by PmWiki) extends the main site with additional mathematical content, especially that with more complicated formulae than available on the main site.

Bogomolny had to leave academia because he had an uncorrectable hearing problem and was practically deaf in latter years. He is survived by his wife Svetlana Bogomolny, two sons David (Israel) and Eli (USA) Bogomolny, and granddaughter Liorah Shaindel Bogomolny.