Does there exist a polygon intersecting each of its own sides only once (each side is intersected only once by a different side) and has all together (a) 6 sides; (b) 7 sides.
Is it possible to make a hole in a wooden cube in such a way that one can drag another cube of the same size through that hole?
Is it true that for any point inside any convex quadrilateral the sum of the distances from the point to the vertices of the quadrilateral is less than the perimeter?
The March Hare bought seven drums of different sizes and seven drum sticks of different sizes for his seven little leverets. If a leveret sees that their drum and their drum sticks are bigger than a sibling’s, they start drumming as loud as they can. What is the largest number of leverets that may be drumming together?
Mary Ann and Alice went to buy some cupcakes. There are at least five different types of cupcakes for sale (all different types are priced differently). Mary Ann says, that whatever two cupcakes Alice buys, Mary Ann can always buy another two cupcakes spending the same amount of money as Alice. What should be the smallest number of cupcakes available for sale at the shop if Mary Ann is not lying?
The March Hare and the Dormouse also decided to play a game. They made two piles of matches on the table. The first pile contains 7 matches, and the second one 8. The March Hare set the rules: the players divide a pile into two piles in turns, i.e. the first player divides one of the piles into two, then the second player divides one of the piles on the table into two, then the first player divides one of the piles into two and so on. The loser is the one who cannot not find a pile to divide. The March Hare starts the game. Can the March Hare play in such a way that he always wins?
A board \(7\times7\) is coloured in chessboard fashion in such a way that all the corners are black. The Queen orders the Hatter to colour the board white but sets the rule: in one go it is allowed to repaint only two adjacent cells into opposite colours. The Hatter tries to explain that this is impossible. Can you help the Hatter to present his arguments?
A snail climbs a 10-meter high tree. In day time the snail manages to climb 4 meters, but slips down 3 meters during the night time. How long would it take the snail to reach the top of the tree if it started the journey on a Monday morning?
A girl and a boy are sitting on a long playground bench. Twenty other children approach them, and one by one sit down in-between two already sitting children until everybody is sitting comfortably on the bench. We call a boy “brave” if he sat between two girls, and we call a girl “brave” if she sat between two boys. How many boys and girls are “brave” if the boys and the girls who sit on the bench alternate?
A natural number \(n\) can be exchanged to number \(ab\), if \(a+b=n\) and \(a\) and \(b\) are natural numbers. Is it possible to receive 2017 from 22 after such manipulations?