Problem 1 There are two plates with 7 candies on each plate. Pooh and Tiger take turns removing any number of candies from one plate. The player who takes the last candy wins. Pooh makes the first move. Who can always win and how must he play?
Answer Tiger can always win. Whatever Pooh does, Tiger should always take the same number of candies from the other plate.
Problem 2 There are two plates with 7 candies on the first and 10 candies on the second. Pooh and Tiger take turns removing any number of candies from one plate. The player who takes the last candy wins, and Pooh makes the first move. Who can always win and how must he play?
Answer Pooh can always win. Step 1. Pooh takes 3 candies from the second plate. Now they have two plates with seven candies, as in Problem 1. Step 2. Whatever Tiger does, Pooh must take the same number of candies from the other plate.
Problem 3 John and Peter play a game. John chooses a number: 1, 2, 3, or 4. Peter adds 1, 2, 3, or 4 to that number and says the sum. Then John adds 1, 2, 3, or 4 and says the sum, etc. Whoever says 40 wins the game. Who can always win and how should he choose numbers?
Answer Peter can always win the game. Regardless of Johns choice, Peter should choose his numbers so that the sum he says is divisible by 5. For example, if John says 3, Peter should add 2 and say 5. After the second round he says 10, and so on until he reaches 40.
Problem 4 There are 10 candies on a table. Allison and Mary decided to play a game. Allison goes first and can take 1, 2, or 3 candies. After that Mary can also take 1, 2, or 3 candies, and so on. The winner is the girl who gets the last candy. Who can win the game and how should she play to win?
Answer Allison can always win the game. She must take 2 candies and leave 8 on the table. The second time, regardless of Marys choice, she must leave 4 candies on the table. Whatever Mary does, Alison takes the last candy and wins.
Problem 5 There are 35 ping-pong balls in a box. Two gamers can take turns removing 1, 2, 3, 4, or 5 balls from the box. The gamer who picks the last ball loses. How must the first gamer play to win the game?
Answer The first gamer must take 4 balls and leave 31 in the box. At the next steps he must leave 25, 19, 13, 7 and 1 balls in the box, regardless of the second gamers choices. The second gamer takes the last ball and loses.
Problem 6 Tiger and Piglet play a game on a round table. They have a lot of quarter coins. Tiger places the first coin on the table, Piglet places the second, Tiger places the third and so on. Whoever cannot find a place for a coin loses. How can Tiger play to win?
Answer Tiger can place the first coin at the center of the table. After Piglet places the second coin in some place, Tiger can place the third coin in exactly the opposite place with respect to the center of the table. If Piglet can find a place for his coin, Tiger can always find a symmetrical place for his, so Tiger will eventually win.
Problem 7 There are three plates with 1 apple on the first, 2 apples on the second and 3 apples on the third plate. Bob and Jim play a game. They take turns picking any number of apples from one plate. The winner is the player who picks the last apple. Bob makes the first move. How can Jim play to win the game?
Answer Bob has two choices: either empty one plate, or make the number of apples on two plates equal. In his turn Jim can make one plate empty and leave the other two with the same number of apples. In the second round, if Bob takes apples from one plate, Jim takes the same number of apples from the other. Jim will always take the last apple and win.
Problem 8 A 6 x 8 chocolate bar is shown on the picture. Ann and Nicole play a game. Ann makes one cut, then Nicole cuts any of the pieces once, then Ann cuts once, and so on. They can only cut along the white lines. Whoever makes the last cut wins the game. How can Ann play to win the game?
Answer After Ann makes a move, there will always be an even number of chocolate pieces left. After Nicole makes a move, there will be an odd number left. The game ends when the original bar is cut into 48 pieces of size 1x1. Regardless of how Ann and Nicole play, Ann will always be the person to make the last cut.
Problem 9 Shrek and Donkey play a game. They put 40 coins on a table. In turn, each player can take up to 10 coins from the table. The winner is the player who takes the last coin. Shrek makes the first move. How should he play to win?
Answer Step 1. Shrek must take 7 coins and leave 33 on the table. Step 2. If Donkey takes X coins, Shrek must take (11-X) coins and leave 22 on the table. Step 3. Shrek leaves 11 coins on the table. Step 4. Shrek takes the last coin, regardless of Donkeys choice. He wins the game.
Problem 10 Shrek and Donkey play a game. They put 100 coins on the table. Each player can take up to 10 coins from the table. The winner is the player who takes the last coin. Shrek makes the first move. How should he play to win?
Answer Let us consider the end of the game. Shrek wins if he leaves 11 coins to Donkey. That means that at the previous round he has to leave 22 coins on the table. And so on – 33, 44, 55, 66, 77, 88, 99. Now let us look from the very beginning. Shrek takes 1 coin and leaves 99 on the table. At his next move, he leaves 88 coins on the table, regardless of Donkeys choice. At the next rounds he leaves 77,66, 55, 44, 33, 22, 11 coins on the table, and ultimately wins a game.