CTK Insights

Archive for the 'Algebra' Category

26 Apr

Not too easy - not too difficult

TweetThe other day, while driving my HS senior son to school (he could have taken a bus, but, for one, his time is at a premium; also, the drive gives us an opportunity for a small chat), we talked about how words with different basic meanings may mean the same thing in certain contexts. As […]

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

04 Apr

Weekly report, the week of March 28, 2016

The present collection supplies a perfect illustration to the fact that even the most simplest of the problems may be looked at from various angles. I wish that the authors of school textbooks that often include only answers or "solutions to the odd-numbered problems" paid more attention to the possibility of alternative view points.

01 Apr

Doodling on April 1st

TweetI came across the following problem several days ago but hesitated to write about it until April 1st. It is simple, practically trivial, and still, after doodling with it for some time, I was left with an open question. If it appears too trivial, even unworthy of mention, do please make an allowance for the […]

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 May

A wrapping surprise

As you may surmise, the path will behave - if I may say so - in a more rational way. Given the incommensurate dimensions of the box it was rational to expect an endless path. This is what you get on the second attempt. But there remains a question to ponder: Why was the first path so short? Jim Henle leaves to his readers to find the answer.

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 […]

19 Sep

Radical Simplification - Not That Simple!

While the cube root of 2+sqrt(5) is in the extension field Q[sqrt(5)], the square root of 2+sqrt(3) is not in Q[sqrt(3)] but rather in Q[sqrt(2), sqrt(3)]

15 Sep

An identity in radicals

This amounts to a cubic equation for x=a+b: x^3+3x-36=0. The sum of the roots of this equation is -3, their product is 36, and the sum of their squares is 0. The latter implies that two of these are complex conjugates, say, u ± iv and one is a real number w (that is supposed to be 3.)

20 May

Probabilities in a Painted Cube

Now generalize: cut the cube into nxnxn smaller cubes and ask the same question. The problem is not awfully difficult but needs some figuring out. Following the foregoing pattern, we eventually arrive at 1/n. But here is a delightful shortcut

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