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.)
Becky and Rishika play the following game: There are 21 biscuits on the table. Each girl is allowed to take 1, 2 or 3 biscuits at once. The girl who cannot take any more biscuits loses. Rishika starts – show that she can always win.
Alice and Bob play a game, Alice will go first. They have a strip divided into \(2018\) identical squares. In one move, they put a \(2 \times 1\) domino block on the strip, covering two full squares. One that is not able to make their move, loses. Who has the winning strategy?