## Cardano, Gambling and the dawn of Probability Theory

The birth of probability theory is usually set in the mid-seventeenth century. At that time the two great mathematicians Blaise Pascal (1623–1662) and Pierre de Fermat (1601– 1665) discussed together some gambling problems and defined the theoretical basis of the mathematical theory of classical probability.However, the first studies on the Read more…

## Exercises in Elementary Number Theory (III)

Exercise 1 The Fermat numbers are defined as follows: $F_{n} = 2^{2^{n}} + 1 \quad n =0,1,2,\cdots$ Prove that all Fermat numbers with $$n \gt 1$$ have the last digit equal to $$7$$. HintThe number $$2^{2^{2}} = 16$$ ends with the digit $$6$$. The same Read more…

## Exercises in Elementary Number Theory (II)

Exercise 1 Let $$x_{n}$$ be a sequence of positive integers so defined: $\begin{array}{l} x_{1}= 2 \\ x_{n + 1} = x_{n}^{2} – x_{n} +1 \quad n \gt 1 \\ \end{array}$ Prove that the numbers $$x_{n}$$ are pairwise relatively prime. SolutionThe values of the sequence are obtained Read more…

## Euler Angles, Hamilton’s Quaternions and video games

Programming video games, as many areas of science and technology, requires computing the coordinates of an object in different reference systems, constructed by combining together translations, rotations or scale changes.The problem is particularly complex with the rotations. In a previous article we have studied the representation of rotations by means of matrices, Read more…

## The Principle of Inclusion-Exclusion

The principle of inclusion-exclusion is an important result of combinatorial calculus which finds applications in various fields, from Number Theory to Probability, Measurement Theory and others. In this article we consider different formulations of the principle, followed by some applications and exercises. For further information, see for example [1] and Read more…