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 Read more
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}} = Read more
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 Read more
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 Read more
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 Read more
In this article we propose some exercises on Elementary Number Theory; they don’t require advanced mathematical knowledge. Other articles will follow with exercises related to this beautiful branch of mathematics. We recall that the symbol \(\left\lfloor x \right\rfloor\) denotes the Read more
Problem Let P be a convex polygon with \(n \) sides. Calculate in how many different ways the polygon can be divided into triangles using diagonals that do not intersect each other in the interior of P. This problem was Read more
Problem Suppose we only use the digits of the set \(A = \{1,2,3,4,5 \} \). How many numbers of \(n\) digits can be formed with the set \(A\), if all adjacent digits differ exactly by \(1\)? We denote the number Read more
This article describes the primitive geometric forms used in 2D collisions, with references to the features provided by the Unity’s 2D Physics engine. For a review of vector algebra, necessary to understand the topic, you can also see the article of this Read more
In this article we propose some combinatorial problems related to the chessboard and the game of chess, which have a scope of application in other sectors of Mathematics as well. Exercise 1 We number an \(8 \times 8 \) chessboard Read more