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Questions and Answers
What is the condition for no two queens to be in the same row in the n-queens problem?
There must be one queen in each row.
What does Q1 assert in the n-queens problem?
Every row contains at least one queen.
What must be true for a column in the n-queens problem?
No column can contain more than one queen.
What does Q4 assert about queens on diagonals?
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How many solutions are there for the n-queens problem when n equals 8?
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What is the structure of a standard Sudoku puzzle?
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What are the assigned numbers in a Sudoku puzzle?
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In Sudoku, each of the ______ cells are called givens.
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Study Notes
The N-Queens Problem
- The goal of the N-Queens problem is to place N chess queens on an N x N chessboard so that no two queens threaten each other.
- Queens can attack horizontally, vertically, and diagonally.
- A predicate p(i, j) is used to represent the presence of a queen in row
iand columnjof the chessboard. - The problem can be solved using logic and propositional equivalences.
Problem Representation
- To express the N-Queens problem logically, a set of propositions is used.
- Proposition
Q1ensures that every row contains at least one queen. - Proposition
Q2ensures that every row contains at most one queen (by using the negation of the predicate). - Proposition
Q3guarantees that every column contains at most one queen. - Propositions
Q4andQ5check for the presence of more than one queen on diagonals. - The solution to the N-Queens problem is provided by the assignments of truth values to the variables
p(i, j)to make the combined propositionQtrue, which is the conjunction of propositions Q1, Q2, Q3, Q4, and Q5.
Problem Complexity
- The number of possible ways to place N queens on a chessboard has been computed for values of N up to 27.
- There are 92 solutions for N = 8.
- For N = 16, the number of solutions grows to a significant 14,772,512.
Sudoku
- Sudoku puzzles are played on a 9 x 9 grid divided into nine 3 x 3 subgrids.
- Some cells are assigned numbers (givens) and others are blank.
- The goal is to fill in the blanks so that every row, column, and 3 x 3 subgrid contains all numbers from 1 to 9.
- The puzzle can be generalized to larger grids of size n^2 x n^2, including n x n subgrids.
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Description
Test your understanding of the N-Queens problem, a classic puzzle in chess where the goal is to place N queens on an N x N board without them attacking each other. This quiz covers logical representations, propositions, and solutions related to this fascinating problem.
