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

n\m 1 2 3 4 5 6 7 8 9
1 0 2 2 2 2 2 2 2 2
2 2 2 4 5 6 8 9 10 12
3 2 6 6 8 10 10 11 14 14
4 2 7 8 8 10 12 13
5 2 8 8 10 13 14
6 2 8 9 11 15
7 2 9 11 13
8 2 10 12
9 2 12 13

B → K

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

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

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

n\m 2 3 4 5 6 7 8
3 2 2
4 2 2 2
5 2 2 3 4
6 2 2 4 4 5
7 2 2 4 6 6 8
8 2 2 4 6 8 9 10
9 2 2 6 7 8 10
10 2 2 6 8 9
11 2 2 6 9
12 2 2 6 10

K → R

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

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

n\m 2 3 4 5 6 7 8
3 0 2
4 2 3 3
5 2 3 4 4
6 4 4 4 4 5
7 4 4 4 5 6 6
8 4 5 5 6 6 7 8
9 4 5 6 6 7 8
10 4 5 6 7 7
11 4 5 6 7
12 4 5 7 8

P → R

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

n\m 8 9
7 2 2
8 2

P → N → B → R

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

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

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

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

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

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

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

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

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

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

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:

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.