1. The problem of placing the largest number of chess pieces of one type on an n×n chessboard so that none of them attacks any other has been well studied. You can fit n queens (for n≥4), n rooks, 2n–2 bishops, n/2² kings, or n²/2 knights (for n≥3).

The problem for placing two types of pieces has only been studied for n=8. What about other square chessboards? What are the collections of two types of pieces that are optimal, in the sense that an additional type of either piece will not fit? For example, the 2×2 optimal solutions are show below.

2. The April 2007 problem investigated the smallest rectangular boards that contained loops of pieces that could attack the next piece in the loop but no other piece. What if each piece requires a minimum of n moves to capture the next piece in the loop? We do not require the pieces to be different, or to not threaten other pieces in the loop. For example, the position below is an example of N → K → R → N, where each capture requires 3 moves.

3. How can the 5 different chess pieces be put on a square chessboard so that each piece attacks the same number of unoccupied squares? Below is one solution. How many others are there?

1.

	N	K	B	R
Q	none	none	none	none
R		none	none
B		none
K	none

	N	K	B	R
Q	none			(MM)
R			(MM)
B
K

	N	K	B	R
Q
R
B
K

	N	K	B
Q
R	(MM) (MM)
B		(MM) (MM)
K

The 5×5 chessboards with Rook and Queen are trivial, with 5 pieces being possible regardless of the combination.

B → B K → K N → N Q → Q R → R

B → K B → N B → Q B → R K → N

K → Q K → R N → Q N → R (MM) Q → R

B → B → B K → K → K N → N → N Q → Q → Q R → R → R all these by (BZ)

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

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

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

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

B → B B → K K → K K → N N → N Impossible: B → N Q → x R → x

B → B → K B → B → Q B → B → R Impossible: B → B → B (JD) B → B → N N → N → N Q → Q → x Q → R → x R → Q → x R → R → x

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

N → N → B N → N → K N → N → Q N → N → R (BH)

B → K → N B → K → Q B → K → R

B → N → K B → N → Q B → N → R

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

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