IIT JEE , AIEEE Forum - Mathematics , Algebra

prior1_2

find the number of ways in which 8queen can be arranged on chess board so that none of them cut each other??

shrey

40320

shrey

shrey

edison

The eight queens puzzle is the problem of putting eight chess queens on an 8×8 chessboard such that none of them is able to capture any other using the standard chess queen's moves.

Constructing a solution

There is a simple algorithm yielding a solution to the n queens puzzle for n = 1 or any n ? 4:

Divide n by 12. Remember the remainder (it's 8 for the eight queens puzzle).
Write a list of the even numbers from 2 to n in order.
If the remainder is 3 or 9, move 2 to the end of the list.
Write the odd numbers from 1 to n in order, but, if the remainder is 8, switch pairs (i.e. 3, 1, 7, 5, 11, 9, ?).
If the remainder is 2, switch the places of 1 and 3, then move 5 to the end of the list.
If the remainder is 3 or 9, move 1 and 3 to the end of the list.
Place the first-column queen in the row with the first number in the list, place the second-column queen in the row with the second number in the list, etc.

For n = 8 this results in the solution shown above. A few more examples follow.

14 queens (remainder 2): 2, 4, 6, 8, 10, 12, 14, 3, 1, 7, 9, 11, 13, 5.
15 queens (remainder 3): 4, 6, 8, 10, 12, 14, 2, 5, 7, 9, 11, 13, 15, 1, 3.
20 queens (remainder 8): 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 3, 1, 7, 5, 11, 9, 15, 13, 19, 17.

Solutions to the eight queens puzzle

The eight queens puzzle has 92 distinct solutions. If solutions that differ only by symmetry operations (rotations and reflections) of the board are counted as one, the puzzle has 12 unique solutions.

Ask & Discuss Questions with Community & Experts

Solutions to the eight queens puzzle