Problem of the Month (April 2007)
Pick two types of chess pieces. How many of each can you place on an m × n chessboard so that each piece attacks exactly one other piece, and the attacks form one large loop? We can ask the same question for cycles of 3 or more chess pieces. We allow Pawns to be in the first row.
For a given cycle of chess pieces and given board, what is the maximum number of pieces that can be placed? If we disregard the condition requiring pieces to attack other specific pieces, what is the largest possible attack loop on an m × n chessboard?
ANSWERS
Here are the best results I could find.
Any Pieces
![](0407/6x5.gif) |
|
B → K
![](0407/BK88.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
2 | 1
|
---|
3 | 1 | 1
|
---|
4 | 1 | 2 | 2
|
---|
5 | 1 | 2 | 2 | 4
|
---|
6 | 1 | 2 | 4 | 5 | 6
|
---|
7 | 1 | 2 | 4 | 5 | 6 | 8
|
---|
8 | 1 | 2 | 4 | 6 | 8 | 8 | 10
|
---|
9 | 1 | 2 | 4 | 6 | 8 | 8
|
---|
10 | 1 | 2 | 6 | 6 | 8
|
---|
11 | 1 | 2 | 6 | 8
|
---|
12 | 1 | 2 | 6 | 8
|
---|
|
B → N
![](0407/BN69.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 0 | 2
|
---|
4 | 2 | 2 | 4
|
---|
5 | 2 | 4 | 4 | 4
|
---|
6 | 2 | 4 | 4 | 6 | 6
|
---|
7 | 2 | 4 | 4 | 6 | 6 | 8
|
---|
8 | 2 | 6 | 6 | 6 | 8 | 10 | 10
|
---|
9 | 2 | 6 | 6 | 8 | 10 | 10
|
---|
10 | 2 | 6 | 6 | 8 | 10
|
---|
11 | 2 | 6 | 6 | 8 | 10
|
---|
12 | 2 | 6 | 8 | 10
|
---|
|
B → R
![](0407/BR512.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 2 | 2
|
---|
4 | 2 | 2 | 3
|
---|
5 | 2 | 3 | 4 | 4
|
---|
6 | 2 | 4 | 4 | 5 | 5
|
---|
7 | 2 | 4 | 4 | 6 | 6 | 6
|
---|
8 | 2 | 4 | 5 | 6 | 6 | 7 | 7
|
---|
9 | 2 | 4 | 6 | 6 | 7 | 8
|
---|
10 | 2 | 4 | 6 | 7 | 7
|
---|
11 | 2 | 4 | 6 | 7
|
---|
12 | 2 | 4 | 7 | 8
|
---|
|
K → N
![](0407/KN512.gif) |
|
K → R
![](0407/KR78.gif) |
n\m | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
2 | 1 | 1
|
---|
3 | 1 | 1 | 1
|
---|
4 | 1 | 2 | 2 | 2
|
---|
5 | 1 | 2 | 2 | 3 | 3
|
---|
6 | 1 | 2 | 2 | 3 | 3 | 4
|
---|
7 | 1 | 2 | 2 | 3 | 4 | 4 | 4
|
---|
8 | 1 | 2 | 2 | 3 | 4 | 5 | 6 | 6
|
---|
9 | 1 | 2 | 4 | 4 | 5 | 5 | 6
|
---|
10 | 1 | 2 | 4 | 4 | 5 | 6
|
---|
11 | 1 | 2 | 4 | 4 | 5
|
---|
12 | 1 | 2 | 4 | 4 | 5
|
---|
|
N → Q
![](0407/NQ69.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
4 | 2 | 2 | 2
|
---|
5 | 2 | 2 | 3 | 4
|
---|
6 | 2 | 3 | 3 | 4 | 4
|
---|
7 | 2 | 3 | 4 | 4 | 4 | 5
|
---|
8 | 2 | 3 | 4 | 4 | 5 | 5 | 6
|
---|
9 | 2 | 3 | 4 | 5 | 6 | 6
|
---|
10 | 2 | 3 | 4 | 5 | 6
|
---|
11 | 2 | 3 | 5 | 5
|
---|
12 | 2 | 4 | 5 | 6
|
---|
|
N → R
![](0407/NR512.gif) |
|
P → R
![](0407/PR124.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
|
---|
4 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3
|
---|
5 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4
|
---|
6 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5
|
---|
7 | 2 | 3 | 3 | 4 | 5 | 5 | 5 | 6
|
---|
8 | 2 | 4 | 4 | 4 | 5 | 5 | 6
|
---|
9 | 2 | 4 | 5 | 5 | 5 | 6
|
---|
10 | 2 | 4 | 5 | 5 | 6
|
---|
11 | 2 | 4 | 5 | 5
|
---|
12 | 2 | 4 | 6 | 6
|
---|
|
B → K → N
![](0407/BKN512.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 1
|
---|
4 | 1 | 1
|
---|
5 | 2 | 2 | 2
|
---|
6 | 2 | 2 | 2 | 3
|
---|
7 | 2 | 2 | 3 | 4 | 4
|
---|
8 | 2 | 2 | 4 | 4 | 5 | 5
|
---|
9 | 2 | 2 | 4 | 4 | 6
|
---|
10 | 2 | 4 | 4 | 5
|
---|
11 | 2 | 4 | 5
|
---|
12 | 2 | 5 | 6
|
---|
|
B → K → R
![](0407/BKR512.gif) |
n\m | 4 | 5 | 6 | 7 | 8
|
---|
5 | 2 | 2
|
---|
6 | 2 | 2 | 3
|
---|
7 | 2 | 3 | 3 | 3
|
---|
8 | 2 | 3 | 3 | 4 | 4
|
---|
9 | 2 | 4 | 4 | 4
|
---|
10 | 2 | 4 | 4
|
---|
11 | 2 | 4
|
---|
12 | 2 | 5
|
---|
|
B → N → K
![](0407/BNK610.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 1 | 1
|
---|
4 | 1 | 1 | 1
|
---|
5 | 1 | 2 | 2 | 2
|
---|
6 | 1 | 2 | 2 | 3 | 3
|
---|
7 | 1 | 2 | 2 | 3 | 4 | 4
|
---|
8 | 1 | 2 | 2 | 4 | 4 | 4 | 5
|
---|
9 | 1 | 2 | 2 | 4 | 4 | 5
|
---|
10 | 1 | 2 | 2 | 4 | 6
|
---|
11 | 1 | 2 | 2 | 4
|
---|
12 | 1 | 2 | 2 | 5
|
---|
|
B → N → R
![](0407/BNR78.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 1 | 1
|
---|
4 | 1 | 2 | 2
|
---|
5 | 1 | 2 | 2 | 2
|
---|
6 | 2 | 2 | 2 | 3 | 3
|
---|
7 | 2 | 2 | 3 | 3 | 4 | 4
|
---|
8 | 2 | 2 | 3 | 4 | 4 | 5 | 5
|
---|
9 | 2 | 3 | 4 | 4 | 4 | 5
|
---|
10 | 2 | 4 | 4 | 4 | 5
|
---|
11 | 2 | 4 | 4 | 4
|
---|
12 | 2 | 4 | 4 | 5
|
---|
|
B → N → Q
![](0407/BNQ68.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 1 | 1
|
---|
4 | 1 | 1 | 1
|
---|
5 | 1 | 2 | 2 | 2
|
---|
6 | 2 | 2 | 2 | 2 | 2
|
---|
7 | 2 | 2 | 2 | 2 | 3 | 4
|
---|
8 | 2 | 2 | 2 | 3 | 4 | 4 | 4
|
---|
9 | 2 | 2 | 3 | 3 | 4 | 4
|
---|
10 | 2 | 3 | 3 | 3 | 4
|
---|
11 | 2 | 3 | 3 | 4
|
---|
12 | 2 | 3 | 4 | 4
|
---|
|
B → P → N
![](0407/PNB511.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
|
---|
3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4
|
---|
5 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8
|
---|
6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 6
|
---|
7 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6
|
---|
8 | 2 | 2 | 2 | 4 | 6 | 6 | 6
|
---|
9 | 2 | 2 | 2 | 4 | 6 | 6
|
---|
10 | 2 | 2 | 2 | 4 | 6
|
---|
11 | 2 | 2 | 2 | 4
|
---|
12 | 2 | 2 | 2 | 4
|
---|
|
B → P → R
![](0407/PRB106.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
|
---|
3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
5 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4
|
---|
6 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4
|
---|
7 | 1 | 2 | 3 | 3 | 3 | 4 | 4 | 5
|
---|
8 | 1 | 2 | 3 | 3 | 4 | 4 | 4
|
---|
9 | 1 | 2 | 3 | 3 | 4 | 5
|
---|
10 | 1 | 2 | 3 | 4 | 5
|
---|
11 | 1 | 2 | 4 | 4
|
---|
12 | 1 | 2 | 4 | 4
|
---|
|
B → R → K
![](0407/BRK69.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
4 | 0 | 2 | 2
|
---|
5 | 0 | 2 | 2 | 2
|
---|
6 | 0 | 2 | 2 | 3 | 3
|
---|
7 | 2 | 2 | 3 | 3 | 4 | 4
|
---|
8 | 2 | 2 | 3 | 4 | 4 | 5 | 5
|
---|
9 | 2 | 3 | 3 | 4 | 5 | 5
|
---|
10 | 2 | 3 | 4 | 4 | 5
|
---|
11 | 2 | 3 | 4 | 4
|
---|
12 | 2 | 3 | 4 | 5
|
---|
|
B → R → N
![](0407/BRN610.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 1 | 1
|
---|
4 | 1 | 1 | 1
|
---|
5 | 2 | 2 | 2 | 2
|
---|
6 | 2 | 2 | 2 | 2 | 2
|
---|
7 | 2 | 2 | 2 | 3 | 4 | 4
|
---|
8 | 2 | 2 | 2 | 4 | 4 | 4 | 5
|
---|
9 | 2 | 2 | 3 | 4 | 4 | 5
|
---|
10 | 2 | 2 | 3 | 4 | 5
|
---|
11 | 2 | 2 | 4 | 4
|
---|
12 | 2 | 2 | 4 | 4
|
---|
|
K → N → Q
![](0407/NQK77.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 1 | 1
|
---|
4 | 1 | 1 | 1
|
---|
5 | 1 | 1 | 2 | 2
|
---|
6 | 1 | 2 | 2 | 2 | 2
|
---|
7 | 1 | 2 | 2 | 2 | 3 | 4
|
---|
8 | 1 | 2 | 2 | 2 | 3 | 4 | 4
|
---|
9 | 2 | 2 | 2 | 3 | 4 | 4
|
---|
10 | 2 | 2 | 3 | 3 | 4
|
---|
11 | 2 | 2 | 3 | 3
|
---|
12 | 2 | 3 | 3 | 4
|
---|
|
K → N → R
![](0407/NRK79.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 1 | 1
|
---|
4 | 1 | 1 | 1
|
---|
5 | 1 | 1 | 2 | 2
|
---|
6 | 1 | 2 | 2 | 2 | 3
|
---|
7 | 1 | 2 | 2 | 2 | 3 | 4
|
---|
8 | 1 | 2 | 2 | 3 | 4 | 4 | 5
|
---|
9 | 2 | 2 | 3 | 3 | 4 | 5
|
---|
10 | 2 | 2 | 3 | 4 | 4
|
---|
11 | 2 | 3 | 3 | 4
|
---|
12 | 2 | 3 | 4 | 4
|
---|
|
K → P → R
![](0407/PRK95.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
|
---|
3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3
|
---|
5 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4
|
---|
6 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4
|
---|
7 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4
|
---|
8 | 1 | 2 | 2 | 3 | 3 | 4 | 4
|
---|
9 | 1 | 2 | 2 | 4 | 4 | 4
|
---|
10 | 1 | 2 | 2 | 4 | 4
|
---|
11 | 1 | 2 | 2 | 4
|
---|
12 | 1 | 2 | 4 | 4
|
---|
|
K → R → N
![](0407/NKR512.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 1 | 1
|
---|
4 | 1 | 1 | 1
|
---|
5 | 1 | 1 | 1 | 1
|
---|
6 | 1 | 1 | 2 | 2 | 2
|
---|
7 | 1 | 1 | 2 | 2 | 2 | 3
|
---|
8 | 1 | 2 | 2 | 2 | 2 | 3 | 4
|
---|
9 | 1 | 2 | 2 | 2 | 2 | 3
|
---|
10 | 1 | 2 | 2 | 2 | 3
|
---|
11 | 1 | 2 | 2 | 2
|
---|
12 | 1 | 2 | 2 | 4
|
---|
|
N → P → R
![](0407/PRN106.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
|
---|
2 | 0 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
3 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
4 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
5 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
6 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3
|
---|
7 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3
|
---|
8 | 1 | 2 | 2 | 3 | 3 | 3 | 3
|
---|
9 | 1 | 2 | 2 | 3 | 3 | 4
|
---|
10 | 1 | 2 | 2 | 3 | 4
|
---|
11 | 1 | 3 | 3 | 3
|
---|
12 | 1 | 3 | 3 | 3
|
---|
|
N → R → P
![](0407/PNR78.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
|
---|
2 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
3 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
5 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3
|
---|
6 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3 | 4
|
---|
7 | 1 | 2 | 2 | 2 | 3 | 4 | 5 | 5
|
---|
8 | 1 | 2 | 3 | 3 | 3 | 4 | 5
|
---|
9 | 1 | 2 | 3 | 3 | 3 | 4
|
---|
10 | 1 | 2 | 3 | 3 | 4
|
---|
11 | 1 | 2 | 3 | 3
|
---|
12 | 1 | 2 | 3 | 4
|
---|
|
N → Q → P
![](0407/PNQ88.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
|
---|
2 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
3 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
4 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
5 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3
|
---|
6 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3
|
---|
7 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3
|
---|
8 | 1 | 2 | 2 | 2 | 3 | 3 | 4
|
---|
9 | 1 | 2 | 2 | 2 | 3 | 3
|
---|
10 | 1 | 2 | 3 | 3 | 3
|
---|
11 | 1 | 2 | 3 | 3
|
---|
12 | 1 | 2 | 3 | 3
|
---|
|
N → Q → R
![](0407/NQR68.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 1 | 1
|
---|
4 | 1 | 1 | 1
|
---|
5 | 1 | 1 | 1 | 2
|
---|
6 | 1 | 1 | 1 | 2 | 2
|
---|
7 | 1 | 1 | 2 | 2 | 2 | 3
|
---|
8 | 1 | 1 | 2 | 2 | 3 | 3 | 3
|
---|
9 | 1 | 1 | 2 | 2 | 3 | 3
|
---|
10 | 1 | 1 | 2 | 2 | 3
|
---|
11 | 1 | 1 | 2 | 2
|
---|
12 | 1 | 1 | 2 | 2
|
---|
|
P → P → R
![](0407/PPR76.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
5 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
6 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
7 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3
|
---|
8 | 1 | 2 | 2 | 2 | 3 | 3 | 3
|
---|
9 | 1 | 2 | 2 | 2 | 3 | 3
|
---|
10 | 1 | 2 | 2 | 2 | 3
|
---|
11 | 1 | 2 | 2 | 2
|
---|
|
B → N → R → K
![](0407/BNRK68.gif) |
n\m | 4 | 5 | 6 | 7 | 8
|
---|
4 | 1
|
---|
5 | 1 | 1
|
---|
6 | 1 | 1 | 2
|
---|
7 | 1 | 2 | 2 | 2
|
---|
8 | 1 | 2 | 3 | 3 | 3
|
---|
9 | 1 | 2 | 3 | 3
|
---|
10 | 1 | 2 | 3
|
---|
11 | 2 | 2
|
---|
|
B → N → K → R
![](0407/BNKR88.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8
|
---|
4 | 0 | 1
|
---|
5 | 0 | 1 | 1
|
---|
6 | 1 | 1 | 2 | 2
|
---|
7 | 1 | 1 | 2 | 2 | 2
|
---|
8 | 1 | 2 | 2 | 2 | 2 | 4
|
---|
9 | 2 | 2 | 2 | 3 | 3
|
---|
10 | 2 | 2 | 3 | 3
|
---|
11 | 2 | 2 | 3
|
---|
|
B → R → N → K
![](0407/BRNK78.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 0 | 1
|
---|
4 | 0 | 1 | 1
|
---|
5 | 1 | 1 | 1 | 2
|
---|
6 | 1 | 1 | 2 | 2 | 2
|
---|
7 | 1 | 2 | 2 | 2 | 3 | 3
|
---|
8 | 1 | 2 | 2 | 2 | 3 | 4 | 4
|
---|
9 | 1 | 2 | 2 | 3 | 3 | 4
|
---|
10 | 1 | 2 | 2 | 3 | 4
|
---|
11 | 1 | 2 | 3 | 3
|
---|
|
B → R → K → N
![](0407/BRKN68.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8
|
---|
4 | 1 | 1
|
---|
5 | 1 | 1 | 1
|
---|
6 | 1 | 1 | 2 | 2
|
---|
7 | 1 | 2 | 2 | 2 | 2
|
---|
8 | 1 | 2 | 2 | 3 | 3 | 3
|
---|
9 | 1 | 2 | 2 | 3 | 3
|
---|
10 | 2 | 2 | 3 | 3
|
---|
11 | 2 | 2 | 3
|
---|
|
B → K → N → R
![](0407/BKNR88.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8
|
---|
4 | 1 | 1
|
---|
5 | 1 | 1 | 2
|
---|
6 | 1 | 2 | 2 | 2
|
---|
7 | 1 | 2 | 2 | 3 | 3
|
---|
8 | 1 | 2 | 3 | 3 | 3 | 4
|
---|
9 | 1 | 2 | 3 | 3 | 3
|
---|
10 | 1 | 2 | 3 | 3
|
---|
11 | 1 | 2 | 3
|
---|
|
B → K → R → N
![](0407/BKRN78.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8
|
---|
6 | 1 | 1 | 2 | 2
|
---|
7 | 1 | 1 | 2 | 2 | 2
|
---|
8 | 1 | 1 | 2 | 2 | 3 | 3
|
---|
9 | 1 | 1 | 2 | 2 | 3
|
---|
10 | 1 | 2 | 2 | 3
|
---|
11 | 1 | 2 | 2
|
---|
|
N → Q → B → K
![](0407/NQBK77.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8
|
---|
4 | 1 | 1
|
---|
5 | 1 | 1 | 2
|
---|
6 | 1 | 2 | 2 | 2
|
---|
7 | 1 | 2 | 2 | 2 | 3
|
---|
8 | 1 | 2 | 2 | 2 | 3 | 3
|
---|
9 | 1 | 2 | 2 | 3 | 3
|
---|
10 | 1 | 2 | 2 | 3
|
---|
11 | 1 | 2 | 3
|
---|
|
N → Q → B → R
![](0407/NQBR68.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
3 | 0 | 1
|
---|
4 | 1 | 1 | 1
|
---|
5 | 1 | 1 | 1 | 1
|
---|
6 | 1 | 1 | 2 | 2 | 2
|
---|
7 | 1 | 1 | 2 | 2 | 2 | 3
|
---|
8 | 1 | 2 | 2 | 2 | 3 | 3 | 3
|
---|
9 | 1 | 2 | 2 | 2 | 3 | 3
|
---|
10 | 1 | 2 | 2 | 3 | 3
|
---|
11 | 1 | 2 | 3 | 3
|
---|
|
N → Q → K → B
![](0407/NQKB78.gif) |
n\m | 4 | 5 | 6 | 7 | 8
|
---|
4 | 1
|
---|
5 | 1 | 1
|
---|
6 | 1 | 1 | 2
|
---|
7 | 1 | 2 | 2 | 2
|
---|
8 | 2 | 2 | 2 | 3 | 3
|
---|
9 | 2 | 2 | 2 | 3
|
---|
10 | 2 | 2 | 3
|
---|
11 | 2 | 2
|
---|
|
N → Q → K → R
![](0407/NQKR47.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8
|
---|
6 | 1 | 1 | 1 | 1 | 2
|
---|
7 | 1 | 1 | 2 | 2 | 2 | 2
|
---|
8 | 1 | 1 | 2 | 2 | 2 | 2 | 2
|
---|
9 | 1 | 1 | 2 | 2 | 2 | 2
|
---|
10 | 1 | 2 | 2 | 2 | 2
|
---|
11 | 1 | 2 | 2 | 2
|
---|
|
N → Q → R → B
![](0407/NQRB58.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8
|
---|
4 | 0 | 1
|
---|
5 | 1 | 1 | 1
|
---|
6 | 1 | 1 | 1 | 1
|
---|
7 | 1 | 1 | 1 | 2 | 2
|
---|
8 | 1 | 1 | 2 | 2 | 2 | 2
|
---|
9 | 1 | 1 | 2 | 2 | 2
|
---|
10 | 1 | 1 | 2 | 2
|
---|
11 | 1 | 2 | 2
|
---|
|
N → Q → R → K
![](0407/NQRK67.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8
|
---|
5 | 1 | 1 | 1
|
---|
6 | 1 | 1 | 1 | 1
|
---|
7 | 1 | 1 | 1 | 2 | 2
|
---|
8 | 1 | 1 | 1 | 2 | 2 | 2
|
---|
9 | 1 | 1 | 2 | 2 | 2
|
---|
10 | 1 | 1 | 2 | 2
|
---|
11 | 1 | 2 | 2
|
---|
|
N → Q → B → P
![](0407/NQBP69.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2
|
---|
5 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2
|
---|
6 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3
|
---|
7 | 1 | 1 | 1 | 1 | 2 | 2 | 3
|
---|
8 | 1 | 2 | 2 | 2 | 2 | 3
|
---|
9 | 1 | 2 | 2 | 2 | 2
|
---|
10 | 1 | 2 | 2 | 2
|
---|
11 | 1 | 2 | 2
|
---|
|
N → Q → K → P
![](0407/NQKP58.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
3 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
5 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2
|
---|
6 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2
|
---|
7 | 1 | 1 | 1 | 1 | 2 | 2 | 2
|
---|
8 | 1 | 1 | 1 | 1 | 2 | 2
|
---|
9 | 1 | 1 | 2 | 2 | 2
|
---|
10 | 1 | 1 | 2 | 2
|
---|
11 | 1 | 2 | 2
|
---|
|
N → Q → R → P
![](0407/NQRP68.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
3 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
6 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2
|
---|
7 | 1 | 1 | 1 | 1 | 2 | 2 | 2
|
---|
8 | 1 | 1 | 1 | 1 | 2 | 2
|
---|
9 | 1 | 1 | 1 | 2 | 2
|
---|
10 | 1 | 1 | 1 | 2
|
---|
11 | 1 | 1 | 1
|
---|
|
N → Q → P → R
![](0407/NQPR105.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
2 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
3 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2
|
---|
5 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3
|
---|
6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3
|
---|
7 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3
|
---|
8 | 1 | 1 | 2 | 2 | 2 | 2 | 3
|
---|
9 | 1 | 1 | 2 | 2 | 3 | 3
|
---|
10 | 1 | 1 | 2 | 3 | 3
|
---|
11 | 1 | 2 | 2 | 3
|
---|
|
P → N → B → K
![](0407/PNBK78.gif) |
|
P → N → B → R
![](0407/PNBR79.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
3 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 0 | 0 | 1 | 1 | 1 | 1 | 2 | 2 | 2
|
---|
5 | 0 | 0 | 1 | 2 | 2 | 2 | 2 | 3 | 3
|
---|
6 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3
|
---|
7 | 1 | 1 | 2 | 2 | 3 | 3 | 4
|
---|
8 | 1 | 2 | 2 | 3 | 3 | 3
|
---|
9 | 1 | 2 | 2 | 3 | 3
|
---|
10 | 1 | 2 | 2 | 3
|
---|
11 | 1 | 3 | 3
|
---|
|
P → N → K → B
![](0407/PNKB69.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
5 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3
|
---|
6 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4
|
---|
7 | 1 | 1 | 2 | 2 | 3 | 3 | 4
|
---|
8 | 1 | 1 | 2 | 2 | 3 | 3
|
---|
9 | 1 | 1 | 2 | 2 | 3
|
---|
10 | 1 | 1 | 2 | 2
|
---|
11 | 1 | 1 | 2
|
---|
|
P → N → K → R
![](0407/PNKR69.gif) |
n\m | 4 | 5 | 6 | 7 | 8 | 9 | 10
|
---|
6 | 0 | 0 | 0 | 2 | 2 | 3 | 3
|
---|
7 | 0 | 2 | 2 | 2 | 2 | 3
|
---|
8 | 0 | 2 | 2 | 2 | 2
|
---|
9 | 2 | 2 | 2 | 2
|
---|
10 | 2 | 2 | 2
|
---|
11 | 2 | 2
|
---|
|
P → N → R → B
![](0407/PNRB106.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2
|
---|
5 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3
|
---|
6 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 3
|
---|
7 | 1 | 1 | 2 | 2 | 2 | 3 | 3
|
---|
8 | 1 | 1 | 2 | 2 | 3 | 3
|
---|
9 | 1 | 1 | 2 | 2 | 4
|
---|
10 | 1 | 1 | 3 | 4
|
---|
11 | 1 | 1 | 3
|
---|
|
P → N → R → K
![](0407/PNRK34.gif) |
n\m | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
6 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
7 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
8 | 1 | 1 | 1 | 1 | 1
|
---|
9 | 1 | 1 | 1 | 1
|
---|
10 | 1 | 1 | 1
|
---|
11 | 1 | 1
|
---|
|
P → R → B → K
![](0407/PRBK106.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
5 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3
|
---|
6 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3
|
---|
7 | 1 | 1 | 2 | 3 | 3 | 3 | 4
|
---|
8 | 1 | 1 | 2 | 3 | 3 | 3
|
---|
9 | 1 | 1 | 3 | 3 | 4
|
---|
10 | 1 | 2 | 3 | 4
|
---|
11 | 1 | 2 | 3
|
---|
|
P → R → B → N
![](0407/PRBN69.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
5 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2
|
---|
6 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3
|
---|
7 | 1 | 2 | 2 | 2 | 2 | 3 | 3
|
---|
8 | 1 | 2 | 2 | 2 | 3 | 3
|
---|
9 | 1 | 2 | 2 | 2 | 3
|
---|
10 | 1 | 2 | 2 | 3
|
---|
11 | 1 | 2 | 3
|
---|
|
P → R → K → B
![](0407/PRKB610.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
3 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2
|
---|
5 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3
|
---|
6 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4
|
---|
7 | 1 | 2 | 2 | 2 | 3 | 3 | 3
|
---|
8 | 1 | 2 | 2 | 3 | 3 | 3
|
---|
9 | 1 | 2 | 3 | 3 | 3
|
---|
10 | 1 | 2 | 3 | 3
|
---|
11 | 1 | 2 | 3
|
---|
|
P → R → K → N
![](0407/PRKN115.gif) |
n\m | 4 | 5 | 6 | 7 | 8 | 9 | 10
|
---|
6 | 0 | 2 | 2 | 2 | 2 | 2 | 3
|
---|
7 | 0 | 2 | 2 | 2 | 2 | 2
|
---|
8 | 2 | 2 | 2 | 2 | 3
|
---|
9 | 2 | 2 | 2 | 3
|
---|
10 | 2 | 2 | 3
|
---|
11 | 2 | 3
|
---|
|
P → R → N → B
![](0407/PRNB68.gif) |
n\m | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
3 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
4 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2
|
---|
5 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3
|
---|
6 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3
|
---|
7 | 1 | 1 | 2 | 2 | 2 | 3 | 3
|
---|
8 | 1 | 1 | 2 | 2 | 3 | 3
|
---|
9 | 1 | 1 | 2 | 3 | 3
|
---|
10 | 1 | 1 | 2 | 3
|
---|
11 | 1 | 2 | 2
|
---|
|
P → R → N → K
![](0407/PRNK106.gif) |
n\m | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11
|
---|
2 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1
|
---|
3 | 0 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2
|
---|
4 | 0 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2
|
---|
5 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3
|
---|
6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3
|
---|
7 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3
|
---|
8 | 1 | 2 | 2 | 3 | 3 | 3 | 3
|
---|
9 | 1 | 2 | 2 | 3 | 3 | 4
|
---|
10 | 1 | 2 | 3 | 3 | 4
|
---|
11 | 1 | 2 | 3 | 3
|
---|
|
Gavin Theobald realized that I had omitted Pawn and Knight cycles, since no more than 4 of each can be used, but he thought I should show the cycles anyway:
![](0407/NP52.gif)
If you can extend any of these results, please
e-mail me.
Click here to go back to Math Magic. Last updated 4/22/07.