## Grouping with Parentheses and Catalan Numbers

Problem Suppose we have $$n$$ numbers $$x_{1},x_{2}, \cdots x_{n}$$, in that order. Compute the number $$C_{n}$$ of ways of positioning the brackets to multiply the product of the $$n$$ numbers, without changing the given order. HintIf $$n=2$$ we have only one case: $$(x_{1} x_{2})$$ If $$n=3$$ Read more…

## Counting n-digits Numbers with Special Properties

Problem Suppose we compose numbers using only the digits in the set $$\{1,2,3\}$$. Compute the total numbers with the following properties: having length $$n$$ beginning and ending with the digit $$\displaystyle 1$$ the adjacent digits are always different from each other We indicate this number with $$Z_{n}$$. For example Read more…

## Matrices whose Rows and Columns form an Arithmetic Progression

Problem Let’s consider the following $$5 \times 5$$ matrix of positive integers: \[ \begin{pmatrix} d & 2d & 3d & ? & 5d \\ 11 & ? & ? & ? & ? \\ ? & 32 & ? & ? & ? \\ ? & ? & ? & Read more…

## The Mystery of Prime Numbers

A prime number is an integer greater than 1 which is divisible only by 1 and by itself. An integer that is not prime is called composite. Each integer greater than 1 is divisible by at least one prime number. The study of integers and their properties began with the Greeks (Pythagoras, Euclid,…), Read more…

## Game Engines

A useful tool to understand the architecture of modern platforms for the development of video games is the layered model. Complex systems are characterized by a set of components (even hundreds or thousands) that interact with each other. The layered model consists of dividing the system into various levels of abstraction: each Read more…