CTK Insights

The US Constitution mandates fair and equitable distribution of seats among the member states. Starting with 1790, the population census has been conducted every tenth year. Several methods (algorithms, one can say) for the seat distribution (known as apportionment) have been considered; some tried at different times bringing to light curious inconsistencies. Needless to say [...]

Years ago I came across a problem of flipping several items simultaneously: There are 7 glasses on a table–all standing upside down. It is allowed to turn over any 4 of them in one move. Is it possible to reach a situation where all the glasses stand right side up. The solution is rather obviously [...]

Find an integral arithmetic progrssion with an arbitrary large number of terms such that no term is a perfect rth power for r = 2, 3, …, n. Trigg gives two solutions. One is trivial, with the first term a non-power and the difference 0. The second solution is by Azriel Rosenfeld. It is quite [...]

Charles Lutwidge Dodgson – better known to broad public as Lewis Carroll – has developed an algorithm to calculate the exact date of Easter Sunday for every year until 2499. In this he bested the great Carl Friedrich Gauss who also found it a sufficiently important occupation. Gauss devised a formula that gave the correct [...]

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 [...]

I found a remarkable talk given by Robert Lang pointed to at the Math Frolic! blog. Dr. Robert J. Lang is an American physicist who is also one of the foremost origami artists and theorists in the world. Among other achievementsm he is known for having proved the completeness of Huzita–Hatori axioms and developing paper [...]

The formula for the sum of the consecutive cubes of integers is one of the most elegant in elementary mathematics: Taking into account a better known formula for the sum of plain integers the formula for cubes can be rewritten succinctly as The sum of the cubes is a square of a triangular number. There [...]

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 [...]

A human hand carries five fingers; two hands have ten of them. Undoubtedly, this fact is responsible for the universal adoption of the decimal system. Children learn to count by counting fingers, first to 5 on one hand and then to 10 on two hands. However, there is a simple way to count to 10 [...]

George G. Szpiro, author of Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present is a mathematician and journalist living in Switzerland. Numbers Rule focuses on key figures in the development of democracy and on the mathematics of voting, elections, and apportionment that they developed. Szpiro pays particular attention to the paradoxes [...]