There are numbers \(1,2,3,4,5,6,7,8,9\) and \(10\) written on a board. Each time you make a “move” you can erase three of the remaining numbers, \(a,b\) and \(c\), and replace them with the numbers \(2a+b\), \(2b+c\) and \(2c+b\). The goal is to make all the \(10\) numbers left on the board equal. Is it possible?
On a certain island there are 17 grey, 15 brown and 13 crimson chameleons. If two chameleons of different colours meet, both of them change to the third colour. No other colour changes are allowed. Is it possible that after a few such colour transitions all the chameleons have the same colour?
Sixteen lightbulbs are arranged in a \(4 \times 4\) grid. Some are on, some are off. You are allowed to change the state of all the bulbs in a column, in a row, or along any diagonal (note: there are 14 diagonals in total!). Is it possible to go from the arrangement in the left to the one on the right by repeating this operation?
There are numbers from \(1\) to \(2018\) written on a board. In one go, we can erase two numbers and replace them with an absolute value of their difference. Can we achieve a sequence consisting only of several numbers \(0\) after some number of moves?
A broken calculator can only do several operations: multiply by 2, divide by 2, multiply by 3, divide by 3, multiply by 5, and divide by 5. Using this calculator any number of times, could you start with the number 12 and end up with 49?
The numbers 1 through 12 are written on a board. You can erase any two of these numbers (call them \(a\) and \(b\)) and replace them with the number \(a+b-1\). After 11 such operations, there will be just one number left. What could this number be?
If a magician puts 1 dove into his hat, he pulls out 2 rabbits and 2 flowers from it. If the magician puts 1 rabbit in, he pulls out 2 flowers and 2 doves. If he puts 1 flower in, he pulls out 1 rabbit and 3 doves. The magician starts with 1 rabbit. Could he end up with the same number of rabbits, doves, and flowers after performing his hat trick several times?
There are real numbers written on each field of a \(m \times n\) chessboard. Some of them are negative, some are positive. In one move we can multiply all the numbers in one column or row by \(-1\). Is that always possible to obtain a chessboard where sums of numbers in each row and column are nonnegative?
Tom found a large, old clock face and put 12 sweets on the number 12. Then he started to play a game with himself. In each move he moves one sweet to the next number clockwise, and some other to the next number anticlockwise. Is it possible that after finite number of steps there is exactly 1 of the sweets on each number?
a) There is an unlimited set of cards with the words “abc”, “bca”, “cab” written. Of these, the word written is determined according to this rule. For the initial word, any card can be selected, and then on each turn to the existing word, you can either add on a card to the left or to the right, or cut the word anywhere (between the letters) and put a card there. Is it possible to make a palindrome with this method?
b) There is an unlimited set of red cards with the words “abc”, “bca”, “cab” and blue cards with the words “cba”, “acb”, “bac”. Using them, according to the same rules, a palindrome was made. Is it true that the same number of red and blue cards were used?