## Mean Value of Permutation Sums

Problem Let $$(a_{1},a_{2}, \cdots ,a_{n})$$ be a permutation of the set $$\{1,2, \cdots,n \}$$. Compute the average value, indicated with $$M_{n}$$ , of the following sum: $(a_{1}- a_{2})^2 + (a_{2}-a_{3})^2 + \cdots + (a_{n-1} – a_{n})^2$ taken on all permutations. If $$n=2$$ the set of permutations is $$\{12,21\}$$ Read more…

## The Problem of the Four Liars

Problem There are four people A, B, C, D. A box with a red ball inside it is given to A, who can leave things unchanged with probability $$p$$ or replace the red ball with a white one with probability $$q=1-p$$. The box is passed to B who can leave Read more…

## 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…