# A Microblog by Gaurav Tiwari

Thoughts, discoveries and administrative updates from Gaurav Tiwari. All opinions are personal and are not endorsed by any brand or person.

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1. In how many ways can two queens, two rooks, one white bishop, one black bishop, and a knight be placed on a standard $8 \times 8$ chessboard so that every position on the board is under attack by at least one piece?
Note: The color of a bishop refers to the color of the square on which it sits, not to the color of the piece.
2. Can you attack every position on the board with fewer than seven pieces?

# Solution

1. Two ways as follow:
Let $\mathbf {Q}$ be the set of rational numbers. It is well known that $\mathbf {Q}$ is an ordered field, i.e., $\mathbf {Q}$ is an Algebraic Structure on which the operations of addition, subtraction, multiplication and division by a non-zero number can be carried out. Also the set $\mathbf {Q}$ is equipped with a relation called “less than” which is an order relation. Between two rational numbers there exist infinite number of elements of $\mathbf {Q}$. Thus the system of rational numbers seems to be dense and so apparently complete. But it is quite easy to show that there exist some numbers which are not rational. Such numbers are called irrational numbers. The set/collection of rational and irrational numbers combined is called the set of real numbers. Study of these numbers is Real Analysis.