The March Hare and the Dormouse are playing a game. A rook is placed on square a1 on a chessboard. In one go it is allowed to move the rook by any number of squares but only up or to the right. The winner is the one who places the rook on square h8. The Dormouse makes the first move. Who will win the game? (It is assumed that everybody is following the best possible strategy).
The March Hare made three piles of stones of 10, 15, and 20 stones respectively, and invited the Dormouse to play the following game. It is allowed to split any existing pile into two smaller ones in one go. The loser is the one who cannot make a move.
Alice and the Hatter decided to play another game. They found a field with exactly 2016 stones on it. In one go Alice picks 1 or 4 stones, while the Hatter picks 1 or 3 stones. The loser is the one who cannot make a move. Can Alice or the Hatter win irrespective of the other player’s strategy?
Tweedledum and Tweedledee play a game. They have written numbers 1, 2, 3, 4 in a circle. Tweedledum, who makes the first move, can add 1 to any two adjacent numbers; while Tweedledee is allowed to exchange any two adjacent numbers. Tweedledum wins if all the numbers become equal. Can Twedleedee prevent Tweedledum from winning if both must make a move every turn?
While walking through the woods in Wonderland, Alice met three very peculiar hunters. They hunted a hare, which was hiding in one of the vertices of the cube \(ABCDEFGH\).
The three hunters fire simultaneously to hit the vertices of the cube (the hunters are all excellent shooters). If they don’t hit the hare, the hare runs over one of the three adjacent edges to the next vertex and hides there. The hunters ask Alice to help them. They want to shoot the hare firing not more than 4 times, but not sure how to do it. Can you help Alice advise the hunters? (please write four vertex triples to be fired by the hunters).
In the middle of the Dark Forest in Wonderland there is a large square clearing, where a wolf is sitting right is the middle of the square, and four dogs are sitting at the four vertices of the square. The wolf can run inside the square with maximum speed \(v\), while the dogs can run along the edges of the square with the speed \(1.5v\). It is known that the wolf kills a dog if they meet one to one, and two dogs kill the wolf if they overpower it together. Can the wolf escape from that square into the forest?
Assume you have a chance to play the following game. You need to put numbers in all cells of a \(10\times10\) table so that the sum of numbers in each column is positive and the sum of numbers in each row is negative. Once you put your numbers you cannot change them. You need to pay £1 if you want to play the game and the prize for completing the task is £100. Is it possible to win?
Once again consider the game from Example 2.
(a) Will you change your answer if the field is a rectangle?
(b) The rules are changed. Now you win if the sum of numbers in each row is greater than 100 and the sum of the numbers in each column is less than 100. Is it possible to win?
Two clowns A and B are playing the following game. They have 33 tomatoes on a plate. One of the tomatoes is rotten and both clowns know which one. Each move they can choose one, two, or three of the remaining tomatoes from the plate and smash them into their own faces. They take turns and the clown who chooses the rotten tomato looses the game. They cannot skip the moves. Clown A starts the game. Does A or B have a winning strategy? (A winning strategy is a strategy following which you win no matter how your opponent plays.)
A candle that burns out completely in one hour costs \(60\) pence and a candle that lasts for \(11\) minutes costs \(11\) pence. For some reasons which he doesn’t want to explain to the seller, Mr. Fawkes wants to measure exactly \(1\) minute with some of these candles. Can he manage to do that if he has only \(1\) pound and \(50\) pence to spend on the candles?
Mr. Fawkes can only measure the whole interval with each of the candles. He is not guaranteed that any of the candles burns uniformly, so he cannot divide it into two parts. He cannot make it burn faster by igniting them from both ends, but he can extinguish a candle and light it up again later.