Problems

Numbers $1,2,\dots ,20$ are written on a whiteboard. In one go Louise is allowed to wipe out any two numbers $a$ and $b$, and write their sum $a+b$ instead. Louise enjoys erasing the numbers, and continues the procedure until only one number is left on the whiteboard. What number is it?

Three tablespoons of milk from a glass of milk are poured into a glass of tea, and the liquid is thoroughly mixed. Then three tablespoons of this mixture are poured back into the glass of milk. Which is greater now: the percentage of milk in the tea or the percentage of tea in the milk?

Louise has a chessboard $8\times 8$ without two opposite corners (see the picture), and 31 dominoes $2\times 1$. Can she tile the crippled chessboard with dominoes she got?

Numbers $1,2,\dots ,20$ are written on a whiteboard. In one go Louise is allowed to wipe out any two numbers $a$ and $b$, and write instead

(a) $a+b-1$; (b) $a\times b$.

As you already know, Louise enjoys erasing the numbers, and has fun until only one number is left on the whiteboard. What number is it?

There is a $3\times 3$ grid filled with zeros. Louise is allowed to add 1 to each small square inside any $2\times 2$ grid. Can she ever get the following table as a result of her actions?

Prove that a $10\times 10$ board cannot be covered by T-shaped tiles (shown below)

Zara has an $8\times 8$ chessboard, in the usual coloring. She can repaint all the squares of a row or column, i.e., all white squares become black, and all black squares become white. Can she get exactly one black square?

A rectangular floor is to be covered by $2\times 2$ and $1\times 4$ tiles (everything is arranged). Unfortunately one tile got smashed, but we have one more tile of the other kind available. Can we retile the floor perfectly?

There are 13 green, 15 blue, and 17 red chameleons on an island. Whenever two chameleons of different colours meet, they both swap to the third colour (i.e., a green and blue would both become red). Is it possible for all chameleons to become one colour?

Numbers 1 and 2 are written on a whiteboard. Every day Louise’s friend Zara changes these numbers to their arithmetic mean $a_m$ and harmonic mean $h_m$.

(The arithmetic mean of two numbers $a$ and $b$ is $a_m=\frac{a+b}{2}$, and harmonic mean of two numbers $a$ and $b$ is $h_m=\frac{2}{\frac{1}{a}+\frac{1}{b}}$ ).

(a) At some point Zara wrote $\frac{941664}{665857}$ as one of the two numbers (it is not known which). What was the other number written on the whiteboard at that time?

(b) Can $\frac{35}{24}$ be ever written by Zara on the whiteboard?