CTK Insights

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 + 1/4 [...]

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, they converge [...]

Here’s a lively problem that was offered at the 1975 USA Math Olympiad (Porblem 4): Circle C(E), with center E, and C(F) with center F, meet in points P and Q. A on C(E) and B on C(F) are such that AB passes through P. Find the position of A and B for which AP*PB [...]

3² + 4² = 5² is the first and the simplest member in an infinite sequence if the Pythagorean triples. It is also the first and the simplest member in an infinite sequence of identities of the sums of successive squares, the next ones being
10² + 11² + 12² = 13² + 14²,
21² + [...]

Here is an engaging problem for all ages: Six points are given in the space such that the pairwise distances between them are all distinct. Consider the triangles with vertices at these points. Prove that the longest side of one of these triangles is at the same time the shortest side of another
(See, http://www.cut-the-knot.org/proofs/ShortestIsLongest.shtml)

Two material points are in equilibrium if their distances from the fulcrum are inversely proportional to their weights. This is known as the Law of the Lever. The law serves as an engaging exercise for finding the greatest common divisor of two integers. Its proof by Archimedes is a captivating example – accessible in [...]

How can be one so wrong? The common objection to the existence of trigonometric proofs of the Pythagorean theorem stems from the assertion that the most important trigonometric identity
sin²α + cos²α = 1,
being equivalent to the Pythagorean theorem, can;t be employed to proof the latter. However, as Jason Zimba has recently observed, that identity admits [...]

M. C. Escher, the Dutch graphic artist, now famous for his lithographs replete with paradoxes of shifting view point, tessellations by uncommon shapes and unseen animals, metamorphoses of seemingly unrelated figures, put down in his Notebooks a simple theorem he was unable to prove but that guided some of his works. The theorem relates to [...]

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 [...]

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 Puzzles