Problem of the Month (November 2011)

This month we are interested in another chessboard problem. Given a positive integer n, a type of chess piece P, and an interval of non-negative integers I, is there a position on an n × n chessboard using only copies of P so that the unoccupied squares are attacked some number of times in I, each number in I being used equally. If so, what is the minimum number of pieces necessary?

ANSWERS

Solutions were sent by George Sicherman, Mark Mammel, and Ambrus Zsbán.

The 2×2, 3×3, 4×4 and 5×5 solutions are optimal, modulo typos. The 6×6 have not yet been completely computer searched yet. The 7×7 and larger boards are probably too large to be completely searched.

2 × 2 Chessboards
[0]
[0]
[1]
[0]
[1]
[2]
[0]
[1]
[2]
[3]
[0]
[1]
[2]
[3]

3 × 3 Chessboards
[0]
[1]
[2]
[0,1]
[1,2]
[0,2]
[0]
[1]
[2]
[4]
[0,1]
[1,2]
[0,2]
[1,3]
[0]
[1]
[2]
[3]
[4]
[0,1]
[1,2]
[2,3]
[0]
[1]
[3]
[4]
[5]
[8]
[2,3]
[3,4]
(GS) 		[1,3]
[3,5]
{0}
{1}
[3]
[4]
[5]
[8]
[2,3]
(GS) 		[2,4]
(GS) 		[1,4]
[0,4]

4 × 4 Chessboards

Knights
N01234
0
none
1
2
(GS)
3
4

Bishops
B01234
0
none
1
2
3 none none
4

Rooks
R01234
0
none
none none
1
none
2
3
4

Queens
Q012345678
0
none none none none none none none none
1 none
none
none none none
2
none none none
3
4
none none
5
none
6 none none none
7 none none
8

Kings
K012345678
0
none
none
none none
1
none none
2 none
none none
3
none
4
none
none
5
none none none
6
none none
7
(GS) 		none
8

5 × 5 Chessboards

Knights
N012345678
0
(GS)
(GS)
none
(AZ) 		none none
1 none
(AZ)
none none
2
(GS)
(GS)
(GS)
(GS) 		none
3
(GS)
(GS)
(GS)
(GS) 		none none
4
none none none none
5 none none none none
6
none none
7 none none
8

Bishops
B01234
0
(GS)
(GS)
1
(GS)
2
(GS)
3 none none
4

Rooks
R01234
0
none none
(AZ) 		none
(AZ) 		none
(AZ)
1
(GS)
2
(AZ)
(AZ)
3
(AZ)
4

Queens
Q012345678
0
none
(AZ) 		none
(AZ) 		none
(AZ) 		none
(AZ) 		none
(AZ) 		none
(AZ) 		none
(AZ)
1 none
none
(AZ) 		none
(AZ) 		none
(AZ)
2
(AZ)
none
(AZ)
3
4
(GS)
(GS)
5
(GS)
(GS)
(GS)
6 none none none
7 none none
8

Kings
K012345678
0
1
(GS)
2
3
(GS)
(GS)
4
(GS)
(GS)
5
none
6
none
7
8

6 × 6 Chessboards

Knights
N012345678
0
(GS)
(MM)
(MM)
(MM)
(GS)
(MM)
(MM)
1 none
(MM)
(MM)
(GS)
(MM)
(GS)
(MM)
(GS)
(MM)
2
(MM)
(MM)
(GS)
(MM)
(GS)
(MM)
3
(MM)
(GS)
(GS)
(GS)
(MM)
4
(MM)
(MM)
(MM)
(MM)
5
(MM)
(GS)
(MM)
6
none none
7 none none
8

Bishops
N01234
0
(GS)
(GS)
(MM)
1
(GS)
(GS)
(GS)
(GS)
2
(MM)
(MM)
(MM)
3 none
(MM) 		none
(MM)
4

Rooks
N01234
0
none
(MM) 		none
(MM)
1
(GS)
(MM)
(GS)
2
3
4

Queens
N012345678
0
none none
(MM) 		none
(MM) 		none
(MM) 		none
(MM) 		none
(MM) 		none
(MM) 		none
(MM)
1 none
(GS)
(GS)
(GS)
(GS) 		none
(MM) 		none
(MM) 		none
(MM)
2 ?
(GS)
(GS)
(GS)
(GS)
(GS) <
(GS)
3
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
4
(MM)
(MM)
(MM)
(MM)
(MM)
5
(MM)
(MM)
(MM)
(MM)
6 none none none
7 none none
8

Kings
N012345678
0
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(MM)
1
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(MM)
2 none
(MM)
(GS)
(GS)
(GS)
(GS)
(GS)
(MM)
3
(MM)
(MM)
(MM)
(MM)
(MM)
(MM)
4
(MM)
(MM)
(MM)
(MM)
(MM)
5
6
7
8

7 × 7 Chessboards

Knights
N012345678
0
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
1 ?
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
2
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
3
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
4
(GS)
(GS)
(GS)
(GS)
(GS)
5
(GS)
(GS)
(GS)
(GS)
6
(GS) 		?
7
8

Bishops
N01234
0
(GS)
(GS)
(GS) 		?
1
(GS)
(GS)
(GS)
2
(GS)
(GS)
(GS)
3 none none
4

Rooks
N01234
0
? ? ? ?
1
(GS)
(GS)
(GS)
(GS)
2
(GS)
(GS)
3
(GS)
4

Queens
N012345678
0
(GS) 		? ? ? ? ? ? ?
1 ? ?
(GS)
(GS) 		? ? ? ?
2
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
3
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
4
(GS)
(GS)
(GS)
(GS)
(GS)
5
(GS)
(GS)
(GS)
(GS)
6 none none none
7 none none
8

Kings
N012345678
0
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
1
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
2 ?
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
3
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
4
(GS)
(GS)
(GS)
(GS)
(GS)
5
(GS)
(GS)
(GS)
6
7
8

8 × 8 Chessboards
Knights
N012345678
0
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
1 ?
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
2
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
3
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
4
(GS)
(GS)
(GS)
(GS)
(GS)
5
(GS)
(GS)
(GS)
(GS)
6
(GS)
(GS)
(GS)
7
(GS)
(GS)
8
(GS)

Bishops
B01234
0
(GS)
(GS)
(GS)
(GS)
(GS)
1
(GS)
(GS)
(GS)
(GS)
2
(GS)
(GS)
(GS)
3 none none
4
(GS)

Rooks
N01234
0
(GS) 		?
(GS) 		? ?
1
(GS)
(GS)
(GS)
(GS)
2
(GS)
(GS)
(GS)
3
(GS)
(GS)
4
(GS)

Queens
Q012345678
0
(GS) 		none? none? none? none? none? none? none? none?
1 none? none?
(GS)
(GS)
(GS) 		none? none? none?
2
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
3
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
4
(GS)
(GS)
(GS)
(GS)
(GS)
5
(GS)
(GS)
(GS)
(GS)
6 none none none
7 none none
8
(GS)

Kings
K012345678
0
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
1
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
2
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
3
(GS)
(GS)
(GS)
(GS)
(GS)
(GS)
4
(GS)
(GS)
(GS)
(GS)
(GS)
5
(GS)
(GS)
(GS)
(GS)
6
(GS)
(GS)
(GS)
7
(GS)
(GS)
8
(GS)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 11/12/11.