the queen’s movement

THE QUEEN’S MOVEMENTBy: Milka Kusturica

INTRODUCTION• How many ways can you put two queens on an nxn chessboard so that

no queen attacks any other?• I used n = 2 and n =3 to find out the formula for using any n where the

queen doesn’t attack any other

CHESS• Chess is a two player boardgame• Has 6 different pieces (king, queen, rook, bishop, knight, and pawn)• All of them have a different movement and average number of squares that they control on an nxn chess board

• Average of rook is always

n + n - 1

• Average for queen depends on where the queen is placed on the

board

QUEEN IN CORNER• I used a 5x5 board• Queen controls forn + (n – 2) + n squares

13 13 13 13 13

13 13

13 13

13 13

13 13 13 13

• The further we move toward the center, the average number of squares controlled by the queen increases by 2

• Queen controls forn + (n – 2) + n + 2

• Queen controls forn + (n – 2) + n + 2 + 2

13 13 13 13 13

13 15 15 15 13

13 15 17 15 13

13 15 15 13

13 13 13 13 13

13 13 13 13 13

13 15 15 15 13

13 15 17

15 13

13 15 15 15 13

13 13 13 13 13

• For a queen, it depends on where the queen is placed on the board.• Average number of squares controlled with queen on chess board where

we have n = 5 is• = 13.8

• =13.8

13 13 13 13 13

13 15 15 15 13

13 15 17

15 13

13 15 15 15 13

13 13 13 13 13

• The number of ways that a queen placed on an nxn board so that no queen is attacking another is

• Examplen = 2• Not possible

COMBINATION FORMULA• Number of ways to choose k objects from n objects

• = 𝑛 !

𝑘! (𝑛−𝑘 ) !

• Number of ways of placing a rook on an nxn board so that no rook is attacking another is

• Where rows and columns is

• The number of ways that the queen placed on an nxn board so that no queen is attackinganother is• nxn• Unknown

MOVEMENT NUMBER• Number of ways of placing queen on an nxn board so that no queen

touches each other• First I worked with the stone problem• Second I worked with the rooks problem• These two problems have given me an idea of what to do on queen

problem

STONE PROBLEM• Method 1• There are n2 tiles on an nxn chessboard.• We can choose two stones in ( ) ways.n

2

STONE PROBLEM• Method 2• The first stone can be chosen from n2 tiles.• Second stone from n2 – 1 tiles.• Then both stones can be chosen in ( )

2n2

ROOKS PROBLEM• Method 1• Choose the two rows for the rooks in ( ) ways.• Choose the two columns for the rooks in ( ) ways.

n2 n

2

ROOKS PROBLEM• There are two ways to place the rooks in the intersection of the rows and

columns for 2( )( ) total ways.n2

n2

ROOKS PROBLEM• Method 2• The first rook can be placed on any of n2 tiles.• This rook controls n + n – 1 total tiles.• The second rook can be placed on any of n2 – (n + n – 1) tiles.

ROOKS PROBLEM• The total number of ways of place the two rooks is 2( )( ).

2n

2n

QUEEN PROBLEM• Method 1• We don’t know (nobody proved it).• Method 2• I used method 2 to find how many ways we can put two queens on an nxn

chessboard so that no queen attacks any other.

• n even goes from 1 ≤ j ≤ • The answer will be .• Sq represents the average number of squares on an nxn chessboard.• Sq =

• n is an odd number• We used the same formula .• The number of squares that a queen controls on average is .

THEOREM• Number of ways to place two queens on an nxn chessboard so that no

queen attacks another.

FORMULA

• Sq =

• =

• n = 2

• = 0

• = 8

Q

Q

QQ

Q

Q

Q

Q

Q

Q

QQQ

QQ

Q

n = 3

• N = 4• Sq =

2 3 4 5 6 7 8 9 10 11 120100020003000400050006000700080009000

0 8 44 144 340 7001288

2184

3480

5280

7700

Number of Possibilities

n

Num

ber o

f Pos

sibilit

ies

The graph rises exponentially. The graph shows the number of possibilities that two queens could be placed on an nxn chessboard without attacking each other.