Varoon and Mahmoud are given two plates of fruit. On one plate, there are \(13\) apples, on the other, there are \(16\) pears. Each of the boys can take any number of fruit from one plate when he moves. The person who takes the last fruit wins. If Mahmoud starts, who will win?
This time, Sally and Fatima have some number of books on a shelf. Every turn, each of them is allowed to take 1, 3 or 4 books from the shelf. The girl that takes the last book wins, Sally goes first. Who will win if there are: a) 14, b) 16, c) 19 books on the shelf?
Danny and Robbie draw diagonals of a regular \(2018\)-gon. They can only draw a diagonal that does not cross any other diagonal that has been already drawn, neither it begins nor ends at a same point as any other drawn diagonal. Robbie starts – can he always win?
Adam and Anthony are playing with a chessboard and a rook. The rook can only be moved either to the bottom or to the left. Each of the boys can move it as far as he wants, but only in a straight line either to the bottom or to the left. The boy who places the rook in the bottom left corner wins. Adam starts, show that he can lose to Anthony only if the rook starts somewhere on the main diagonal.
Nathan and Liam have numbers from \(1\) to \(2018\) written on a board. In each move, one of the players removes a number of their choosing, which is still on the board, together with all its remaining divisors. Liam goes first. The last person to remove a number wins. Who has the winning strategy?
Alex and Priyanka have a chessboard and a queen on it. Each of the players can only move the queen to the top, to the right, or along a diagonal – to the top and right (like the queen moves, but only in three directions out of all eight). The person who places the queen in the top right corner wins. The chessboard is a normal \(8 \times 8\) board. The queen starts four squares to the right from the bottom left corner. If Priyanka starts, who will win the game?
Some inhabitants of the Island of Multi-coloured Frogs speak only the truth, and the rest always lie. Three islanders said:
Bree: There are no blue frogs on our island.
Kevin: Bree is a liar. She herself is a blue frog!
Clara: Of course, Bree is a liar. But she’s a red frog.
Are there any blue frogs on this island?
In the family of happy gnomes there is a father, a mother and a child. The names of the family members: Alex, Charlie and Jo. At the dinner table two gnomes made two statements.
Charlie said: “Alex and Jo are of different genders. Alex and Charlie are my parents”.
Alex said: “I am Jo’s father. I am the daughter of Charlie”.
Who is who? That is, what is the name of the father, the mother and the child, if it is known that each gnome lied once, and each told the truth once.
On the first day of school, in all three of the first year classes (1A, 1B, 1C), there were three lessons: Maths, French and Biology. Two classes cannot have the same lesson at the same time. 1B’s first lesson was Maths. The Biology teacher praised the students in 1B: “You have even better marks than 1A”. 1A’s second lesson was not French. Which class’s last lesson was Biology?
a) A square of area 6 contains three polygons, each of area 3. Prove that among them there are two polygons that have an overlap of area no less than 1.
b) A square of area 5 contains nine polygons of area 1. Prove that among them there are two polygons that have an overlap of area no less than \(\frac{1}{9}\).