the queen’s movement
TRANSCRIPT
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
Q
Q
Q
Q
Q
Q
QQQ
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.