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 6 18 753 294 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 6 18 793 254 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.
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 6 18 753 294 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 6 18 793 254 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.
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Transcribed Image Text: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
6 18
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
713
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.
Transcribed Image Text: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.
3. Let h(n) be a heuristic function for a state n 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
11
11
h(n) = Σ (lk - Σtij¹) + Σ(k-Σtÿj¹) + (k-Σtiil)
i=1
j=1
j=1
i=1
i=1
n
+ (k - Σ tip | )
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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