This document discusses the N-Queens problem, which involves placing N chess queens on an N×N chessboard so that no two queens attack each other. It provides an overview of the problem statement, history, backtracking algorithm used to solve it, data flow diagram, pseudocode, sample outputs, and advantages of the backtracking approach. The backtracking algorithm places queens column-by-column, checking for valid placements and backtracking when it reaches an invalid configuration. The time complexity increases exponentially with board size N as the number of possible solutions grows.

1. Presented by : G.SRILEKHA
B16CS045
Counsellor : B.SRINIVAS sir
N-QUEENS PROBLEM

2. CONTENTS
• Problem statement
• History
• Backtracking
• Data flow diagram
• Algorithm
• Output
• Advantages over other methods
• Conclusion

3. PROBLEM STATEMENT
• The N Queen is the problem of placing N chess
queens on an N×N chessboard so that no two
queens attack each other.
• That means , no two queens are placed in same
row or in same column or diagonal to each other.

4. HISTORY
• Chess composer Max Bezzel published the eight queens
puzzle in 1848.Franz Nauck published the first solutions
in 1850.Nauck also extended the puzzle to the n queens
problem, with n queens on a chessboard of n × n squares.
• Since then, many mathematicians, including Carl
Friedrich Gauss, have worked on both the eight queens
puzzle and its generalized n-queens version. In
1874, S.Gunther proposed a method using determinants to
find solutions.J.W.L. Glaisher refined Gunther's approach.

5. BACKTRACKING
• Backtracking is finding the solution of a problem whereby
the solution depends on the previous steps taken.
• Thus, the general steps of backtracking are:
• start with a sub-solution
• check if this sub-solution will lead to the solution or not
• If not, then come back and change the sub-solution and
continue again

6. startt
Place a Queen and
check for next queen
Is
possi
ble or
not
Backtrack and
again check other
column
stop
No
Yes
FLOW CHART

7. ALGORITHM
• Place the queens column wise, start from the left most column
• If all queens are placed.
• return true and print the solution.
• Else
• Try all the rows in the current column.
• Check if queen can be placed here safely if yes mark the current cell in
solution matrix as 1 and try to solve the rest of the problem recursively.
• If placing the queen in above step leads to the solution return true.
• If placing the queen in above step does not lead to the solution ,
BACKTRACK, mark the current cell in solution matrix as 0 and return false.
• If all the rows are tried and nothing worked, return false and print NO
SOLUTION.

8. OUTPUT ( IF N=4)

9. If we continue the process , we get another solution as :
First Solution

10. TIME TAKEN FOR EACH INTEGER N OF A QUEEN

11. ADVANTAGES OVER OTHER METHODS
• The major advantage of the backtracking algorithm is the
abillity to find and count all the possible solutions rather
than just one while offering decent speed. In fact this is
the reason it is so widely used. Also one can easily
produce a parallel version of the backtracking
algorithm increasing speed several times just by starting
multiple threads with different starting positions of the
first queens.

12. CONCLUSION
• In N-Queen, as N value increases the time of processing
also takes more time .This happens when we try to display
all the possible solutions.
• Of course, we could make it much faster if we wanted to
only find one solution instead of all of them: not more
than a few milliseconds for board sizes up to 50.

13. THANK YOU !!