CTK Insights

Six edges of a tetrahedron fall naturally into 3 pairs of the opposites - the non-intersecting edges. From the point of view of the four vertices, the latter can be split into 2 pairs in three ways. Either way, there are exactly three line segments joining the midpoints of the opposite edges. Three such segments [...]

Find the probability that if the digits 0, 1, 2, ..., 9 be placed in random order in the blank spaces of 5_383_8_2_936_5_8_203_9_3_76 the resulting number will be divisible by 396. As a hint, the question is more about divisibility than that of probability. Reference Probability of Divisibility

I've been charmed by the opening allegory in John Rosemond's article I reproduce below (Southern Illinoisan [Carbondale, IL], Sunday, May 24, 2009, p. 6D See http://www.southernillinoisan.com/articles/2009/05/24/lifestyles/family/28992003.txt.) I do believe that parents attitude is one of the factors that affects the process and the results of education system. Growing up, children absorb the surrounding culture, with [...]

There is a class of two-dimensional geometric problems that benefit from being embedded in 3d. Here's one such example: P  is a point on the incircle of an equilateral triangle ABC. Prove that AP² + BP² + CP² is constant, i.e., independent of P. Reference Sum of Squares in Equilateral Triangle at Interactive Mathematics Miscellany and [...]

In one of the very first posts on the blog I wrote about a problem of dividing a circle into parts of equal area. More recently I noticed that the solution to that problem supplies an insight into a puzzle posed first by the British puzzlist H. E. Dudeney at the beginning of the 20th century. [...]