Inspired by normal numbers I created the simple following problem: First take a rational number, for example $\frac{3}{4}$ which is equal to $0.75$; now add the first digit after the decimal separator to the digit $4$ we get $\frac{3}{4.7}$ which is equal $0.63829\cdots$. Now take the second digi...

so what is the average of the sequence made from cos(2)? $\cos(2)=0.4161468365471423869975682295007621897660...$ the first number in the sequence is the first number in the decimal expansion until it hits a $0$ so $f(1)=4161468365471423869975682295$ The next number is the next number in the expan...

I have only started linear algebra, I was watching the lectures of Gilbert Strang, he gave us two linear equation $2x-y=0$ $x-2y=3$ . He wrote this in the way of coefficient matrices and it became $\begin{bmatrix} 2 & -1 \\ 1 & -2 \end{bmatrix} $ $\begin{bmatrix} x \\ y \end{bmatrix} $=$\b...

Let s be the set of five positive integer the maximum of which is at most 9 prove that the sums of the elements in all the non empty subset of as cannot be distinct? Note: I know this is similar to another question with 6 elements with max 14. I saw this in Quora, and it annoys me that my proof i...

I am currently working on a chess engine with a friend and to break down my question: For the chess engine, there is something called the evaluation function $s(p)$ which takes a position $p$ and outputs a numeric value. The idea is to tune this evaluation function to accurately predict the outco...

There are quite a lot of questions about chess, some of them are classified as combinatorial-game-theory, which is (sometimes) incorrect. Chess it not a mathematical field; however, I can easily imagine people (like myself) that are interested in mathematical questions having chess as a "backgro...

Let $\rho$ be an Hermitian matrix with unit trace (this is the context I've found this result stated in, but I don't know if these restrictions are necessary for the result). Suppose $p_i\ge0$ with $\sum_i p_i=1$, and let $U_i$ be unitaries. Consider $$\rho'= \sum_i p_i \, U_i \rho U_i^\dagger.$$...