# CTK Insights

• ## Pages

13 Apr

### 7 or 22

Without ever trying to answer such questions, I was always confident that the poster (if not the author) were smugly awaiting a definite reply, although, even with the most benevolent interpretation, the problem has to be considered ill-posed, like that of asserting the next term in a given sequence

15 Feb

### A Problem in Complex Numbers Sets 3 Aside

Let $a,b$ be complex numbers that satisfy $\displaystyle\frac{a}{b}+\frac{b}{a}+1=0.$
Find the value of the expression $\displaystyle\frac{a^{2016}}{b^{2016}}+\frac{b^{2016}}{a^{2016}}+1.$

01 Apr

### Why Learn Mental Math Tricks?

I strongly believe that those tricks are more than anything else convey to the early learners the essence of practicing mathematics. Presh Talwalkar's book may also open the eyes of an older generation on what they missed in the early grades.

18 Jan

### Kordemsky's Palindrome Problem

In his last book Mathematical Allurments (Matematicheskie Zavlekalki), published posthumously in 2000, he tells a story of a 7th grade girl who got tempted to solve that problem and found a solution, too. She informed Kordemsky that her solution was different from the one in the book. Kordemsky encouraged her to look further, for other solutions. Several of her classmates get involved in the search that eventually produced more than 120 solutions. I can imagine Kordemsky's delight in seeing his efforts at attracting young minds to mathematics being born fruit. The kids even came up with something unexpected: many of the numbers they came up lead to other solutions when some pairs of their digits get swapped.

17 Oct

### Elementary Problems that Beg for Generalization

TweetIn a well known puzzle, a father willed to his three sons camels, with the proviso that of the inheritance should go to the oldest among them, with being due to the middle one and to the youngest. Shortly after the father's death, a wise man riding on his camel through the village noticed the […]

22 Oct

### A Property of the Power of 5

My late father was an inveterate human calculator. During the 1930s at the height of the New Economic Policy (NEP) that allowed a degree of post-revolution entrepreneurship in the Soviet Union, he made a living by giving on-stage mental math performances. He became an electrical engineer when the NEP was curtailed.

Browsing through his notes I have recently come across an observation concerning the fifth powers of integers and its relevance to the absence of integer solutions of x^5 + y^5 = z^5 - Fermat's equation for n=5.

10 Jun

### Dominoes and Chessboard Activities

Tiling a chessboard with dominoes is uniquely suitable as an entertaining and edifying activity even for young children. Both implements are widely available, while experimentation with them leads to a good number of problems. Some of the problems admit simple (albeit ingenious) proofs that I would classify as the "very first," in the sense that they require minimal (if any) knowledge of mathematics.

07 Jun

### A Kindergarten Activity As a Problem for Adolescents

TweetTwelve kids stand in a circle, a kid per one of twelve marked spots. Every now and then one moves clockwise, another counterclockwise to an adjacent position (which may be occupied by more than one kid or be empty.) Is it possible that after a while all of them stand at the same spot? The […]

24 May

### Three Checker Game on a One Row Board

TweetThe setup for the 2-players game described below consists of three checkers placed on a K×1 board: A move consists in picking one of the outside checkers and placing it anywhere between the other two. Here's a sequence of two successive moves. As usual, players take turns. The one who can't make a move loses […]

21 May

### Engaging Math Activities for the Summer Break

TweetBelow is a growing collection of resources I designate as "Summer Math Activities". These deal with various (mostly non-computational) aspects of early mathematics and are suitable for young children. There is no good reason not to try these activities at any other time of the year. During summertime, with the kids out of school or […]