Q=
Math Magic
Problem of the Month (February 2007)
This month we investigate chess positions containing two types of pieces A and B subject to the condition: each A attacks exactly n B's (and no A's), and each B attacks exactly m A's (and no B's). Can you find smaller solutions? Can you solve one of the unsolved cases? What are the smallest solutions for other pairs of pieces?
Given a collection of chess pieces, here are more questions we can ask: What is the smallest arrangement so that each piece attacks exactly one piece of each type, including its own? What is the smallest arrangement so that each piece attacks one piece of each other type, not including its own? How about two of each piece? Given a cycle of pieces, what is the smallest arrangement so that each piece attacks only one of the piece next in the cycle? How about two of the next piece?
ANSWERS
Here are the smallest known solutions for 2 pieces A and B (possibly equal) where A attacks exactly n B's (and no A's), and each B attacks exactly m A's (and no B's).
Bishops and Knights
 B=1  B=2  B=3  B=4


N=1  
 


N=2  
 see below 


N=3   ?
 


N=4  
 


B=3 N=2

Bishops and Kings
 B=1  B=2  B=3  B=4


K=1  
 


K=2  
 


K=3  
 (proved by Joe DeVincentis) 


K=4   ?
 


K=5  
 



Bishops and Rooks
 B=1  B=2  B=3  B=4


R=1  
  see below


R=2  
 


R=3  
 


R=4  see below 
 


B=4 R=1
B=1 R=4

Bishops and Queens
 B=1  B=2  B=3  B=4


Q=1  
 


Q=2  
  see below


Q=3  
 see below  ?


Q=4  
 


Q=5  (James Wilson)  ?
 


Q=6  
 


Q=7  
 


Q=8  see below 
 


B=4 Q=2
B=3 Q=3
B=1 Q=8

Kings and Knights
 K=1  K=2  K=3


N=1  
 (Johan de Ruiter)


N=2  
 (James Wilson)



Kings and Queens
 K=1  K=2


Q=1  


Q=2  


Q=3  


Q=4  


Q=5   ?


Q=6  



Kings and Rooks
 K=1  K=2  K=3  K=4


R=1  
 


R=2  
 (proved by Joe DeVincentis) 


R=3   (proved by Joe DeVincentis)
 



Knights and Rooks
 N=1  N=2  N=3  N=4


R=1  
 


R=2  
 see below  ?


R=3  (George Sicherman) 
 ? 


R=4  
 


N=3 R=2
(George Sicherman)

Knights and Queens
 N=1  N=2  N=3


Q=1  
 see below


Q=2   see below
 ?


Q=3  (George Sicherman) 
 ?


Q=4   see below



N=2 Q=2
N=2 Q=4
N=3 Q=1
(James Wilson)

Queens and Rooks
 R=1  R=2  R=3  R=4


Q=1  
 


Q=2  
  see below


Q=3  
  ?


Q=4  
 


Q=5   see below
 


Q=6  
 


Q=7  
 


Q=8  see below 
 


Q=2 R=4
Q=5 R=2
(James Wilson)
Q=8 R=1

Knights and Knights
 N=1  N=2  N=3  N=4


N=1 


N=2  


N=3   


N=4   (Johan de Ruiter)  


N=5    ?  ?


N=6    ? 


N=7   ?  


N=8    



Queens and Queens
 Q=1  Q=2  Q=3  Q=4


Q=1 


Q=2  


Q=3   


Q=4    


Q=5    ?  ?


Q=6    ?  ?


Q=7  (James Wilson)  see below  ?  ?


Q=8  (James Wilson)  ?  ?  ?


Q=2 Q=7
(George Sicherman)

Here are the smallest known solutions for collections of pieces (possibly repeated) where each piece attacks exactly 1 of every other type of piece.
3 Pieces Without Repetition
 no B  no K  no N  no Q


no K 


no N  


no Q  



no R  
 



3 Pieces With Repetition
 B  K  N  Q  R


BB  
  


KK  
 (James Wilson)  


NN  
  


QQ  
  


RR  
  



Here are the smallest known solutions for collections of pieces (possibly repeated) where each piece attacks exactly 1 of each type of piece, including its own.
3 Pieces Without Repetition
 no B  no K  no N  no Q


no K 


no N  


no Q  ?  ?



no R  
 see below  ?


no N, no R

3 Pieces With Repetition
 B  K  N  Q  R


BB  
 ?  


KK  
 see below  (James Wilson)  see below


NN  ? 
   ?


QQ  
  


RR  
 ?  


KKN
 KKR


Here are the smallest known solutions for collections of pieces (possibly equal) where each piece attacks exactly 2 of each type of piece, including its own.
2 Pieces
 B  K  N  Q  R


B 


K  (proved by James Wilson)  (proved by James Wilson)


N  ?  ?



Q  
 ? 


R  
 ?  



Here are the smallest known solutions for cycles of 3 or more pieces where each piece attacks exactly 1 piece of the next type in the cycle (and no others of any type).
3 Pieces Without Repetition
B → K → N →
 B → K → R →
 B → N → K →
 B → N → R →

B → R → K →
 B → R → N →
 K → N → R →
 K → R → N →

N → Q → B →
 N → Q → K →
 N → Q → R →


4 Pieces Without Repetition
B → K → N → R →
 B → K → R → N →
 B → N → K → R →
(Johan de Ruiter)
 B → N → R → K →
(Johan de Ruiter)

B → R → K → N →
 B → R → N → K →
 N → Q → B → K →
 N → Q → B → R →

N → Q → K → B →
 N → Q → K → R →
 N → Q → R → B →
 N → Q → R → K →


4 Pieces With Repetition
B → K → B → K →
 B → K → B → N →
 B → K → B → R →
 B → N → B → N →

B → N → B → R →
 B → R → B → R →
 K → B → K → N →
 K → B → K → R →

K → N → K → N →
 K → N → K → R →
 K → R → K → R →

N → B → N → K →
 N → B → N → R →
 N → K → N → R →
 N → R → N → R →

N → Q → N → B →
 N → Q → N → K →
 N → Q → N → R →
 N → Q → N → Q →

R → B → R → K →
 R → B → R → N →
 R → K → R → N →


5 Pieces Without Repetition
N → Q → B → K → R →
 N → Q → B → R → K →
(Johan de Ruiter)
 N → Q → K → B → R →
(Johan de Ruiter)

N → Q → K → R → B →
 N → Q → R → B → K →
 N → Q → R → K → B →
(Johan de Ruiter)
 
Here are the smallest known solutions for cycles of 3 or more pieces where each piece attacks exactly 2 pieces of the next type in the cycle (and no others of any type).
3 Pieces Without Repetition
B → K → N →
 B → K → R → ?
 B → N → K →
 B → N → R →

B → R → K → ?
 B → R → N →
 K → N → R → ?
 K → R → N →
 N → Q → B →

N → Q → K →
 N → Q → R →


4 Pieces Without Repetition
B → K → N → R → ?
 B → K → R → N → ?
 B → N → K → R → ?

B → N → R → K → ?
 B → R → K → N → ?
 B → R → N → K → ?

N → Q → B → K → ?
 N → Q → B → R →

N → Q → K → B → ?
 N → Q → K → R → ?
 N → Q → R → K → ?

N → Q → R → B →


If you can extend any of these results, please
email me.
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