1. Formulate the 3 x 3 game as a search problem, i.e. define the states, the moves, etc2. Draw the state space that would be generated by the Breadth-First Search algorithm for theinitial state given in Table 2. You can stop when the first goal state is reached. Perform themoves strictly in the following sequence: Up; Down; Left; Right. Do not create more thanone copy of any particular state, and identify the goal state. 3rd question in second picture Moving Magic Square is the name of a game that is based on the concept of a magic square. Amagic square is any square array of numbers, usually positive integers, in which the sums ofthe numbers in each row, each column, and both main diagonals are the same. For example,the 3 x 3 square in Table 1 is a magic square because the sum of every row, every column andthe two diagonals is 15.Table 1618753294The game, Moving Magic Square, is played on any n x n grid containing positive integernumbers from 1, ..., n². The number n² is the movable number. You can move the number n²in one of four directions (up/down/left/right), and swap n² with the number that is currentlyoccupying that cell. The player wants to move the number n² to reach a goal state such that thesum of the n numbers in every row, column, and both diagonals is equal to k. There are multiplestates that satisfy this condition, and you can stop the game when you find the first goal state.In a 3 x 3 game, 9 is the movable number.The grid in Table 2 shows an initial state for a 3 x 3 game.Table 26987 13254The grid in Table 3 shows the result of a "down" move on the initial state shown in Table 2.Table 3618793254After another "down" move on the grid in Table 3, the goal state (i.e. the magic square)shown in Table 1 results. The game has been completed. 3. Let h(n) be a heuristic function for a staten in the game where h(n) is the sum of alldifferences between k and the total of the entries in any row, column or diagonal:nnnh(n)= [(lk-[tijl) + Σ(k-tj) + (k-Σtal) + (k-Σtl)ΣHij-l Fli-1i-1where p=n-i+1For the state depicted by the grid in Table 2, (with k = 15 and n = 3)h(n) = (8 + 4 + 4) + (0+0+0)+4+4=24Show the heuristic function values for each of the states in the drawing of your answer toquestion 2.

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Answered: 1. Formulate the 3 x 3 game as a search…

Programming Logic & Design Comprehensive

9th Edition

ISBN:9781337669405

Author:FARRELL

Publisher:FARRELL

Chapter9: Advanced Modularization Techniques

Section: Chapter Questions

Problem 2GZ

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1. Formulate the 3 x 3 game as a search problem, i.e. define the states, the moves, etc

2. Draw the state space that would be generated by the Breadth-First Search algorithm for the
initial state given in Table 2. You can stop when the first goal state is reached. Perform the
moves strictly in the following sequence: Up; Down; Left; Right. Do not create more than
one copy of any particular state, and identify the goal state.

3rd question in second picture

Moving Magic Square is the name of a game that is based on the concept of a magic square. A
magic square is any square array of numbers, usually positive integers, in which the sums of
the numbers in each row, each column, and both main diagonals are the same. For example,
the 3 x 3 square in Table 1 is a magic square because the sum of every row, every column and
the two diagonals is 15.
Table 1
618
753
294
The game, Moving Magic Square, is played on any n x n grid containing positive integer
numbers from 1, ..., n². The number n² is the movable number. You can move the number n²
in one of four directions (up/down/left/right), and swap n² with the number that is currently
occupying that cell. The player wants to move the number n² to reach a goal state such that the
sum of the n numbers in every row, column, and both diagonals is equal to k. There are multiple
states that satisfy this condition, and you can stop the game when you find the first goal state.
In a 3 x 3 game, 9 is the movable number.
The grid in Table 2 shows an initial state for a 3 x 3 game.
Table 2
698
7 13
254
The grid in Table 3 shows the result of a "down" move on the initial state shown in Table 2.
Table 3
618
793
254
After another "down" move on the grid in Table 3, the goal state (i.e. the magic square)
shown in Table 1 results. The game has been completed.

3. Let h(n) be a heuristic function for a staten in the game where h(n) is the sum of all
differences between k and the total of the entries in any row, column or diagonal:
n
n
n
h(n)= [(lk-[tijl) + Σ(k-tj) + (k-Σtal) + (k-Σtl)
Σ
Hi
j-l Fl
i-1
i-1
where p=n-i+1
For the state depicted by the grid in Table 2, (with k = 15 and n = 3)
h(n) = (8 + 4 + 4) + (0+0+0)+4+4=24
Show the heuristic function values for each of the states in the drawing of your answer to
question 2.

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