A variation of the two player game Nim starts with a single stack of 6 tokens. At eachmove a player divides it into two non-empty stacks. A player who cannot move loses thegame. Min plays first.i. Draw the complete search tree for this game.[6 Marks]ii. The utility function assigns a value 0 to a terminal state if Min wins the game and 1 ifMax wins the game.Apply the MinMax algorithm to the search tree to assign utility values to all states in thesearch tree.[4 Marks]iii. If both Min and Max play a perfect game (always makes correct moves), who will win?Explain your answer and show the winning path(s) for Max or Min.[2 Marks]
Question
A variation of the two player game Nim starts with a single stack of 6 tokens. At eachmove a player divides it into two non-empty stacks. A player who cannot move loses thegame. Min plays first.i. Draw the complete search tree for this game.[6 Marks]ii. The utility function assigns a value 0 to a terminal state if Min wins the game and 1 ifMax wins the game.Apply the MinMax algorithm to the search tree to assign utility values to all states in thesearch tree.[4 Marks]iii. If both Min and Max play a perfect game (always makes correct moves), who will win?Explain your answer and show the winning path(s) for Max or Min.[2 Marks]
Solution
i. The search tree for this game would look something like this:
6
โโโ 5-1
โ โโโ 4-1-1
โ โ โโโ 3-1-1-1
โ โ โ โโโ 2-1-1-1-1
โ โ โ โ โโโ 1-1-1-1-1-1 (End)
โ โ โ โโโ 2-1-1-1-1
โ โ โ โ โโโ 1-1-1-1-1-1 (End)
โ โ โโโ 3-1-1-1
โ โ โ โโโ 2-1-1-1-1
โ โ โ โ โโโ 1-1-1-1-1-1 (End)
โ โโโ 4-1-1
โ โ โโโ 3-1-1-1
โ โ โ โโโ 2-1-1-1-1
โ โ โ โ โโโ 1-1-1-1-1-1 (End)
โโโ 4-2
โ โโโ 3-1-2
โ โ โโโ 2-1-1-2
โ โ โ โโโ 1-1-1-1-2 (End)
โ โโโ 3-2-1
โ โ โโโ 2-1-2-1
โ โ โ โโโ 1-1-1-2-1 (End)
โโโ 3-3
โ โโโ 2-1-3
โ โ โโโ 1-1-1-3 (End)
โ โโโ 2-2-2
โ โ โโโ 1-1-2-2 (End)
ii. Using the MinMax algorithm, we assign utility values to all states in the search tree. The terminal states are assigned a value of 0 if Min wins and 1 if Max wins. Since Min always plays first, the terminal states at the end of Min's turn will have a value of 0 and the terminal states at the end of Max's turn will have a value of 1.
iii. If both Min and Max play a perfect game, Min will always win. This is because Min always plays first and can always make a move that leads to a terminal state with a utility value of 0. The winning path for Min would be: 6 -> 5-1 -> 4-1-1 -> 3-1-1-1 -> 2-1-1-1-1 -> 1-1-1-1-1-1.
Similar Questions
Problem Statement:Nim is a popular two-player game with simple rules:Start with several piles of stones, each pile may have a different number of stones.Players take turns picking 1 or more stones from any single pile.The player who takes the last stone wins.Your task is to determine who wins the game given the number of piles and the number of stones in each pile, assuming both players play optimally.Input Format:The first line contains an integer n representing the total number of piles.The second line contains n space-separated integers representing the number of stones in each pile.Constraints:1 <= n <= 1001 <= pile[i] <= 100Output Format:Print the winner.Example:Imagine a game with 3 piles of stones: [3, 2, 4]. Here's how the game might go:Player 1 takes 2 stones from pile 1, leaving [3, 4].Player 2 takes 2 stones from pile 1, leaving [3, 2].Player 1 takes 1 stone from pile 0, leaving [2, 2].Player 2 takes 1 stone from pile 0, leaving [1, 2].Player 1 takes 1 stone from pile 1, leaving [1, 1].Player 2 takes 1 stone from pile 0, leaving [0, 1].Player 1 takes 1 stone from pile 1, winning the game.Sample Test CasesTest Case 1:Expected Output:33 2 4Playerยท1Test Case 2:Expected Output:21 1Playerยท2
Two kittens, Max and Min, play with a pair of non-negative integers x and y. As you can guess from their names, kitten Max loves to maximize and kitten Min loves to minimize. As part of this game Min wants to make sure that both numbers, x and y became negative at the same time, and kitten Max tries to prevent him from doing so.Each kitten has a set of pairs of integers available to it. Kitten Max has n pairs of non-negative integers (ai,โbi) (1โโคโiโโคโn), and kitten Min has m pairs of non-negative integers (cj,โdj) (1โโคโjโโคโm). As kitten Max makes a move, it can take any available pair (ai,โbi) and add ai to x and bi to y, and kitten Min can take any available pair (cj,โdj) and subtract cj from x and dj from y. Each kitten can use each pair multiple times during distinct moves.Max moves first. Kitten Min is winning if at some moment both numbers a, b are negative simultaneously. Otherwise, the winner of the game is kitten Max. Determine which kitten wins if both of them play optimally.InputThe first line contains two integers, n and m (1โโคโn,โmโโคโ100โ000) โ the number of pairs of numbers available to Max and Min, correspondingly.The second line contains two integers x, y (1โโคโx,โyโโคโ109) โ the initial values of numbers with which the kittens are playing.Next n lines contain the pairs of numbers ai,โbi (1โโคโai,โbiโโคโ109) โ the pairs available to Max.The last m lines contain pairs of numbers cj,โdj (1โโคโcj,โdjโโคโ109) โ the pairs available to Min.OutputPrint ยซMaxยป (without the quotes), if kitten Max wins, or "Min" (without the quotes), if kitten Min wins.
General algorithm applied on game tree for making decision of win/lose is
Consider the following centipede game. At Player 1's first node, he chooses between down (D), which immediately ends the game with payoffs (1,1), or across (A). If A, then Player 2 chooses between down (d), which ends the game with payoffs (0,4), and across (a). If a, then Player 1 moves, choosing between going across (ฮฑ) or down (ฮด). If he goes across, payoffs are (2,5); if he goes down, payoffs are (3,3). The game that we have constructed is as follows:ย Which of the following statements is FALSE? If Player 1 acts rationally at the last node, he will choose down Player 2, expecting that Player 1 will act rationally at the last node, will play down The equilibrium outcome is (3,3) Assuming both players play rationally, the game ends at the first node
General algorithm applied on game tree for making decision of win/lose is ____________a.DFS/BFS Search Algorithmsb.Heuristic Search Algorithmsc.Greedy Search Algorithmsd.MIN/MAX Algorithms
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