# 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?

Here are the best results I could find.

Any Pieces

n\m123456789
1022222222
222456891012
326681010111414
42788101213
5288 101314
62891115
7291113
821012
921213

B → K

n\m2345678
21
311
4122
51224
612456
7124568
812468810
9124688
1012668
111268
121268

B → N

n\m2345678
302
4224
52444
624466
7244668
8266681010
926681010
10266810
11266810
1226810

B → R

n\m2345678
322
4223
52344
624455
7244666
82456677
9246678
1024677
112467
122478

K → N

n\m2345678
322
4222
52234
622445
7224668
822468910
92267810
1022689
112269
1222610

K → R

n\m12345678
211
3111
41222
512233
6122334
71223444
812234566
91244556
10124456
1112445
1212445

N → Q

n\m2345678
4222
52234
623344
7234445
82344556
9234566
1023456
112355
122456

N → R

n\m2345678
302
4233
52344
644445
7444566
84556678
9456678
1045677
114567
124578

P → R

n\m23456789101112
422233333333
523334444444
6233445555
723345556
82444556
9245556
1024556
112455
122466

B → K → N

n\m345678
31
411
5222
62223
722344
8224455
922446
102445
11245
12256

B → K → R

n\m45678
522
6223
72333
823344
92444
10244
1124
1225

B → N → K

n\m2345678
311
4111
51222
612233
7122344
81224445
9122445
1012246
111224
121225

B → N → R

n\m2345678
311
4122
51222
622233
7223344
82234455
9234445
1024445
112444
122445

B → N → Q

n\m2345678
311
4111
51222
622222
7222234
82223444
9223344
1023334
112334
122344

B → P → N

n\m23456789101112
322222222222
422222222444
522244444488
6222444446
722244466
82224666
9222466
1022246
112224
122224

B → P → R

n\m23456789101112
311111111111
411222222222
511222334444
6122333344
712333445
81233444
9123345
1012345
111244
121244

B → R → K

n\m2345678
4022
50222
602233
7223344
82234455
9233455
1023445
112344
122345

B → R → N

n\m2345678
311
4111
52222
622222
7222344
82224445
9223445
1022345
112244
122244

K → N → Q

n\m2345678
311
4111
51122
612222
7122234
81222344
9222344
1022334
112233
122334

K → N → R

n\m2345678
311
4111
51122
612223
7122234
81223445
9223345
1022344
112334
122344

K → P → R

n\m23456789101112
311111111111
411122223333
512222233344
6122333344
712233444
81223344
9122444
1012244
111224
121244

K → R → N

n\m2345678
311
4111
51111
611222
7112223
81222234
9122223
1012223
111222
121224

N → P → R

n\m23456789101112
201122222222
311122222222
411122222222
511122222222
6122222233
712222333
81223333
9122334
1012234
111333
121333

N → R → P

n\m23456789101112
201112222222
311112222222
411112222222
511122333333
6111223334
712223455
81233345
9123334
1012334
111233
121234

N → Q → P

n\m23456789101112
201112222222
311122222222
411122222222
511122222223
6111222233
712222333
81222334
9122233
1012333
111233
121233

N → Q → R

n\m2345678
311
4111
51112
611122
7112223
81122333
9112233
1011223
111122
121122

P → P → R

n\m234567891011
31111111111
41111111111
51112222222
6122222222
712223333
81222333
9122233
1012223
111222

B → N → R → K

n\m45678
41
511
6112
71222
812333
91233
10123
1122

B → N → K → R

n\m345678
401
5011
61122
711222
8122224
922233
102233
11223

B → R → N → K

n\m2345678
301
4011
51112
611222
7122233
81222344
9122334
1012234
111233

B → R → K → N

n\m345678
411
5111
61122
712222
8122333
912233
102233
11223

B → K → N → R

n\m345678
411
5112
61222
712233
8123334
912333
101233
11123

B → K → R → N

n\m345678
61122
711222
8112233
911223
101223
11122

N → Q → B → K

n\m345678
411
5112
61222
712223
8122233
912233
101223
11123

N → Q → B → R

n\m2345678
301
4111
51111
611222
7112223
81222333
9122233
1012233
111233

N → Q → K → B

n\m45678
41
511
6112
71222
822233
92223
10223
1122

N → Q → K → R

n\m2345678
611112
7112222
81122222
9112222
1012222
111222

N → Q → R → B

n\m345678
401
5111
61111
711122
8112222
911222
101122
11122

N → Q → R → K

n\m345678
5111
61111
711122
8111222
911222
101122
11122

N → Q → B → P

n\m34567891011
3111111111
4111111222
5111122222
611112233
71111223
8122223
912222
101222
11122

N → Q → K → P

n\m34567891011
3011111111
4111111111
5111112222
611111222
71111222
8111122
911222
101122
11122

N → Q → R → P

n\m34567891011
3001111111
4001111111
5111111111
611111222
71111222
8111122
911122
101112
11111

N → Q → P → R

n\m234567891011
20011111111
30111111111
41111111222
51111122223
6111122223
711222223
81122223
9112233
1011233
111223

P → N → B → K

n\m89
722
82

P → N → B → R

n\m34567891011
3001111111
4001111222
5001222233
611222 333
71122334
8122333
912233
101223
11133

P → N → K → B

n\m34567891011
3111111111
4112222222
5112222233
611223344
71122334
8112233
911223
101122
11112

P → N → K → R

n\m45678910
60002233
7022223
802222
92222
10222
1122

P → N → R → B

n\m34567891011
3111111111
4111122222
5111122233
611122333
71122233
8112233
911224
101134
11113

P → N → R → K

n\m4567891011
311111111
411111111
511111111
61111111
7111111
811111
91111
10111
1111

P → R → B → K

n\m34567891011
3111111111
4112222222
5112222333
611222333
71123334
8112333
911334
101234
11123

P → R → B → N

n\m34567891011
5000002222
611222233
71222233
8122233
912223
101223
11123

P → R → K → B

n\m34567891011
3011111111
4111122222
5112222223
612222334
71222333
8122333
912333
101233
11123

P → R → K → N

n\m45678910
60222223
7022222
822223
92223
10223
1123

P → R → N → B

n\m34567891011
3011111111
4011111122
5011122223
611222333
71122233
8112233
911233
101123
11122

P → R → N → K

n\m234567891011
20001111111
30111112222
40111222222
51111222333
6112222233
711222333
81223333
9122334
1012334
111233

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.