There is a chequered board of dimension (a) \(9\times 10\), (b) \(9\times 11\). In one go you are allowed to cross out any row or column if it contains at least one square which was not crossed out yet. The loser is the player who cannot make a move. Is there a winning strategy for any player?
Alice and the Hatter play a game. Alice takes a coin in each hand: 2p coin and 5p coin, one coin per hand. Then she multiplies the value of the coin in the left hand by 4, 10, 12, or 26, and the value of the coin in the right hand by 7, 13, 21, or 35. Finally, she adds the two products together, and tells the result to the Hatter. To her surprise, the Hatter immediately knows in which hand she has the 2p coin. How does he do it?
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?
You have two sticks and matchbox. Each stick takes exactly an hour to burn from one end to the other. The sticks are not identical and do not burn at a constant rate. As a result, two equal lengths of the stick would not necessarily burn in the same amount of time. How would you measure exactly 45 minutes by burning these sticks?
One of your employees insists on being paid daily in gold. You have a gold bar whose value is that of seven days’ salary for this employee. The bar is already segmented into seven equal pieces. If you are allowed to make just two cuts in the bar, and must settle with the employee at the end of each day, how do you do it?