How to prove that player I can always win a Nim game in which hte number...

How to prove that player I can always win a Nim game in which hte number of heaps with an odd number of coins is odd.

I understand the objective, having trouble formuating a more formula argument for the problem.

Expert Solution

Colby Messinger answered 2 years ago

Using induction: Show that player 1 can always win a Nim game in which the number...

Using induction: Show that player 1 can always win a Nim game in which the number of heaps with an odd number of coins is odd.

The game of Nim. This is a well-known game with a number of variants. The following...

The game of Nim. This is a well-known game with a number of variants. The following variant has an interesting winning strategy. Two players alternately take marbles from a pile. In each move, a player chooses how many marbles to take. The player must take at least one but at most half of the marbles. Then the other player takes a turn. The player who takes the last marble loses. Write a program in which the computer plays against a...

Game of dice - fair dice: If I throw a number >= 5 I win. If...

Game of dice - fair dice: If I throw a number >= 5 I win. If he throws a number =< 4 he wins. I throw the first dice. Given that he loses(throws a number >4) what is the probability of me winning ? What is the expected number of throws before either of us wins?

In the game of roulette, a player can place a $ 4 bet on the number...

In the game of roulette, a player can place a $ 4 bet on the number 27 and have a StartFraction 1 Over 38 EndFraction 1 38 probability of winning. If the metal ball lands on 27, the player gets to keep the $ 4 paid to play the game and the player is awarded an additional $ 140 . Otherwise, the player is awarded nothing and the casino takes the player's $ 4 . What is the expected...

Consider a game in which a player shoots 3 free throws; if the player makes i...

Consider a game in which a player shoots 3 free throws; if the player makes i free throws, she draws one bill at random from a bag containing i + 1 ten-dollar bills and 5 − (i + 1) one-dollar bills. Let X be the number of free throws she makes and Y be the amount of money she wins and assume that she makes free-throws with probability 1/2. (a) Tabulate the marginal probabilities P(X = x) for x ∈...

In the game of roulette, a player can place a $4 bet on the number 14...

In the game of roulette, a player can place a $4 bet on the number 14 and have a StartFraction 1 Over 38 EndFraction probability of winning. If the metal ball lands on 14, the player gets to keep the $4 paid to play the game and the player is awarded an additional $140. Otherwise, the player is awarded nothing and the casino takes the player's $4. What is the expected value of the game to the player? If...

In the game of roulette, a player can place a $9 bet on the number 6...

In the game of roulette, a player can place a $9 bet on the number 6 and have a 1 /38 probability of winning. If the metal ball lands on 6, the player gets to keep the $9 paid to play the game and the player is awarded an additional $315 Otherwise, the player is awarded nothing and the casino takes the player's $9. Find the expected value E(x) to the player for one play of the game. If x...

Prove the following statement: Suppose it's your turn and the Nim sum of the number of...

Prove the following statement: Suppose it's your turn and the Nim sum of the number of coins in the heaps is equal to 0. Then whatever you do, the Nim sum of the number of coins after your move will not be equal to 0.

In order to win a game, a player must throw two fair dice and the sum...

In order to win a game, a player must throw two fair dice and the sum of the dice needs to be either 4 or less or 10 or more for the player to win. What is the probability that the sum of the dice is 4 or less? What is the probability that the sum of the dice is 10 or more? What is the probability that the player will win the game?

Consider a game in which, simultaneously, player 1 selects any real number x and player 2...

Consider a game in which, simultaneously, player 1 selects any real number x and player 2 selects any real number y. The payoffs are given by: u1 (x, y) = 2x − x2 + 2xy u2 (x, y) = 10y − 2xy − y2. (a) Calculate and graph each player’s best-response function as a function of the opposing player’s pure strategy. (b) Find and report the Nash equilibria of the game. (c) Determine the rationalizable strategy profiles for this game.

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