CTK Insights

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 [...]

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 [...]

Due to a freak accident, the 28th issue of the Math teachers at play carnival went out two days before a submission deadline. With sincerest apologies I’d like to mention the submissions that arrived after the accident. John Golden, a.k.a. Math Hombre, celebrates a full circle day (6/28) with an article on Similarity and π. [...]

One of my favorite examples of simple mathematics that seems to benefit from the tools that only appeared at the end of the last century is the problem of breaking chocolate bars. Given a rectangular chocolate bar that consists of a number of small squares. The task is to break it up along the “fault” [...]

The identity although simple, is rather surprising. It could be verified by raising the right-hand side to the power of 3: One way of accidently running into this identity is by means of Cardano’s cubic formula applied to the third degree equation The formula will produce terms that include cubic roots. On the other hand, [...]

Steve Fisk, 63, passed away on January 31, 2010 after a long battle with leukemia. Fisk earned his PhD from Harvard and accepted a post-doctoral teaching postion at MIT. He moved to Bowdoin College in 1977 where he remained until his death. Many in the mathematics community will remember him through his proof of the [...]

A Mathematician’s Lament by Paul Lockhart is one book I wish I had written. The points concerning mathematics and math education raised in the book are very close to those in my manifesto. Mathematics plays a tremendously important role in science and engineering as a tool and often the vehicle of progress (especially in modern [...]

The trigonometric form of the Pythagorean theorem is well known: sin²(α) + cos²(α) = 1. If α and β are two acute angles in a right triangle, then, since α = 90° – β and sin(90° – β) = cosβ, that identity can be rewritten as cos²(α) + cos²(β) = 1. Taking into account that [...]

How to divide evenly 7 apples between 12 boys? An obious answer is to divided each apple into twelve parts and give each boy seven of them. But assume there is a restriction: one does not know how to divide an apple into more than 5 parts. What do you do then? See Why 1/3 [...]

Wonderment at the identity -1 = 1 + 2 + 4 + 8 + … dwarfes the perennial question as to whether .9999… equals 1 or not. Still, it holds in exactly same sense as the limit of a convergent series. The sums of the successive powers of 2 converge in the 2-adic norm! Moreover, [...]