# CTK Insights

• ## Pages

24 Feb

### The Joy of Homogeneity

TweetIn a recent blog A Lovely Observation Gary Davis (@RepublicOfMath) elaborated on an observation of Ben Vitale (@BenVitale) to the effect that In the fractions both numerators and denominators are sums of successive odd numbers: the numerators start with 1, the denominators where the numerators leave off. Thus naturally derivation of the formula for the […]

21 Feb

### Mathematical Legends

TweetThe bible takes six words to describe the evolution of the Hebrew tribe in Egypt from 70 souls to a people (Exodus 1.7). The closest translation I found among several is this: "... and the sons of Israel have been fruitful, and they teem, and multiply, and are very very mighty ..." In Hebrew, the […]

20 Feb

### Probability of Two Integers Being Coprime

TweetFor a prime , two integers are both divisible by with the probability , because this only happens when the two integers have the residue 0 (one out of available residues) modulo . Two integers are mutually prime if they have no common nontrivial factors, prime facors in particular. Assuming divisibility by one prime is […]

20 Feb

### Can one learn from people who do not know what they are talking about?

TweetI've been following up on a story according to which George Dantzig - the father of linear programming - once happened to be late to a lecture. When he arrived, he copied down from the blackboard two problems he thought to be a homework assignment. He solved them at home not knowing that the two […]

19 Feb

### Thought Provokers to Start a Class With, VI

TweetHere are a few engaging problems that many a student will be able to solve and, if not, would be able to appreciate the simplicity of a missed solution. Without a measuring tape, is it possible to cut a half-meter piece from the rope of 2/3 m length? Solution What is the digital root of […]

01 Feb

### Existence of the Incenter: a Second Look

TweetThe three angle bisectors of a triangle meet at incenter of the triangle. Reversing the problem we may ask a relevant question: Given three concurrent lines: α, β, and γ. Is there always a triangle with the three lines as the angle bisectors. If so, construct the triangle. Solution Given three concurrent lines: α, β, […]

01 Feb

### Medians in a Triangle Meet at the Center: a Second Look

TweetThe medians of a triangle meet at a point known at the center of the triangle. Reversing the problem we may ask a relevant question: Given three concurrent lines: α, β, and γ. Is there always a triangle with the three lines as the medians. If so, construct the triangle. Solution Given three concurrent lines: […]