CTK Insights

Recently, I've been reading and solving problems from a Russian publication by A. V. Shapovalov. As is now customary with popular books, in this book too every chapter is preceded by a suggestive epigraph cuing the reader to the content of the chapter. Every chapter in the book contains solved examples and exercises that come [...]

This is a beautifully illustrated collection of interviews and biographical etudes of 16 mathematicians of different backgrounds, varied professional interests, diverse level of achievement - all incredibly interesting as human beings. The sixteen interviewees lived and were active in the 1900s, though some are yet alive; the stories throw light - if only in the [...]

In a recent post, I have implied that Socrates new how to dissect a 2×1 rectangle into a square. There is actually no evidence that he did. However, he certainly knew how to produce a square half the area of a given one. How would he relate the two problems? A sangaku tablet has preserved [...]

By Wit of Woman is a novel by Arthur Marchmont (1852-1923), a popular author who wrote around the turn of the 20th century. He penned several best-sellers, By Wit of Woman among them. The book is written in the name of a young woman who set out to clear her father's name. I just came [...]

A frequently cited math curiosity relates the relative increase in the rope length between the rope laid on the Earth's equator and that around an average size watermelon. In both case the sought increase in length is due to the uniform expansion of the rope to, say, 1 ft away from the surface. It often [...]

A hyperbola has two axes of symmetry: one that crosses the hyperbola while the other does not. Two different 3D shapes are obtained when a hyperbola is made to rotate around its axes. Two sheet hyperboloid One sheet hyperboloid The equation of one is and that of the other is The one sheet hyperboloid has [...]

According to [P. Beckmann, p. 12], by 2000 BC, the ancient Babilonians already new that π is close to 25/8 (≈ 3.125), while the Egyptians estimated it as 4(8/9)² (≈ 3.1605). Beckmann, as many others, do not conceal his astonishment that in the Bible the apparent estimate is a simplistic 3. This is based on [...]

What's the task? The task is to combine several 3- and 4-pyramids into larger 3- and 4-pyramids. What's the setup? You'll need 4 tetrahedra and 6 square pyramids. Having 8 tetrahedra and 8 square pyramids will allow to complete 3- and 4-pyramids simultaneously. For your convenience, here are the maps of the pyramids. Just cut, [...]

Much of problem solving in mathematics is about finding a representation in a way that simplifies if not trivializes a given problem. Come to think of it, putting a word problem into algebraic terms - as an equation or a system of equations - is ultimately finding another representation of the problem, a representation more [...]

To an average student (and, perhaps, an average teacher used to teaching from a textbook) it may come as a surprise that there are numerous problems with multiple known solutions. "How come?" - may wonder the average student, "Who would want the drudge of solving a problem whose solution is already known?" Why, it's a [...]