CTK Insights

Here is a simple problem I found in a recent book from MAA (C. Alsina, R. B. Nelsen, Charming Proofs, p. 89.) Geometric constructions of a square inscribed in a triangle are well known. There are two ways to inscribe a square into a right isosceles triangle: Which square has the larger area? The problem [...]

Theorem For any greater than is irrational. The following proof has been submitted by Richard Ehrenborg, University of North Carolina, Charlotte and published in the American Mathematical Monthly (May 2003, 423) with a remark that the proof was found by William Henry Schultz, at the time an undergraduate at UNC-Charlotte. The proof has been reproduced [...]

Years ago I came across a problem of flipping several items simultaneously: There are 7 glasses on a table–all standing upside down. It is allowed to turn over any 4 of them in one move. Is it possible to reach a situation where all the glasses stand right side up. The solution is rather obviously [...]

Find an integral arithmetic progrssion with an arbitrary large number of terms such that no term is a perfect rth power for r = 2, 3, …, n. Trigg gives two solutions. One is trivial, with the first term a non-power and the difference 0. The second solution is by Azriel Rosenfeld. It is quite [...]

Three ropes have been fastened to a horizontal plunk, tangled a little as if one tried to make a braid and, lastly, loosely attached to an auxiliary plank to keep them braided. Down below there is a third plank. The task is to attach three additional ropes to the first ones at one end and [...]

The formula for the sum of the consecutive cubes of integers is one of the most elegant in elementary mathematics: Taking into account a better known formula for the sum of plain integers the formula for cubes can be rewritten succinctly as The sum of the cubes is a square of a triangular number. There [...]

In a related post I have shown that Without resorting to this proof, I am going to show that Let and Then which gives us a third degree equation in : By direct verification is one of the roots of that equation. Factoring gives The second factor which is a quadratic polynomial is always positive [...]

A human hand carries five fingers; two hands have ten of them. Undoubtedly, this fact is responsible for the universal adoption of the decimal system. Children learn to count by counting fingers, first to 5 on one hand and then to 10 on two hands. However, there is a simple way to count to 10 [...]

The NCTM holds to his promise to post interesting Problems to Ponder in the NCTM’s newsletter. Here’s one from the July 15 issue. In the figure below, quadrilateral ABCD is a square, and E is the midpoint of the side AD. How do the areas of regions I, II, III, and IV compare? Another way [...]

The fact that π exists is due to the similarity of all circles. The ratio of the circumference to the diameter would not be constant otherwise. The simple idea of similarity is commonly being reported as overly hard on children. I am confident that the problem must lie with the manner of presentation and not [...]