# Problem of the Month (December 2010)

This month we explore 4 "chess on polyominoes" problems:

1.

Given a collection of 2 or more chess pieces, what is the smallest polyomino with a starting square and an ending square so that it takes the same minimum number of moves by each piece to get from start to finish? What about 3 pieces? What about including other fairy chess pieces, such as these leapers: Alfil (2,2), Fers (1,1), Threeleaper (0,3), Dabbaba (0,2), Wazir (0,1), Giraffe (1,4), Zebra (2,3), Camel (1,3)?

2.

Given a polyomino containing one chess piece P, what is the smallest polyomino that can be tiled with copies of P so that 1) no chess piece threatens any other and 2) every vacant square is threatened by at least one piece? What about larger polyominoes? What if the polyominoes contain more than one chess piece?

3.

Given a polyomino containing one chess piece P, what is the smallest polyomino that can be tiled with copies of P so that every square is attacked the same number of times? What about larger polyominoes? What if we only require the empty squares to be attacked the same number of times?

4.

What is the smallest polynomial that contains a "mate in n" chess problem with a unique solution? What if a unique solution is not required?

Solutions were sent by George Sicherman, Maurizio Morandi, Richard Sabey, Corey Plover, Giovanni Resta, Joe DeVincentis, and Bryce Herdt.

1.

The best-known results are shown below.

Smallest Chessboards Requiring Equal Travel Time Between Two Pieces
Alfil
(2,2)
Fers
(1,1)
Threeleaper
(0,3)
Dabbaba
(0,2)
Wazir
(0,1)
Giraffe
(1,4)
Zebra
(2,3)
Camel
(1,3)
RookkNightBishopQueen
King
(GS)
Queen
(GR)
Bishop
(GS)
kNight
Rook
Camel
(1,3)
Zebra
(2,3)

(GR)
Giraffe
(1,4)

(GS)

(GR)
Wazir
(0,1)
Dabbaba
(0,2)
Threeleaper
(0,3)

(BH)
Fers
(1,1)

(GS)

Smallest Chessboards Requiring Equal Travel Time Between Three Pieces
King+Alfil
(2,2)
Fers
(1,1)
Threeleaper
(0,3)
Dabbaba
(0,2)
Wazir
(0,1)
Giraffe
(1,4)
Zebra
(2,3)
Camel
(1,3)
RookkNightBishop
Queen
(BH)

(GR)

(GS)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
Bishop
(GS)

(GR)

(GS)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
kNight
(JD)

(GR)

(GS)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
Rook
(GS)

(GR)

(GS)

(GR)

(GR)

(GR)

(GR)

(GR)
Camel
(1,3)

(GS)

(GR)

(GS)

(GR)

(GR)

(GR)

(GR)
Zebra
(2,3)

(GS)

(GR)

(GS)

(GR)

(GR)

(GR)
Giraffe
(1,4)

(GS)

(GR)

(GS)

(GS)

(GR)
Wazir
(0,1)

(GS)

(GR)
none
(JD)
none
(JD)
Dabbaba
(0,2)

(GS)

(GR)

(JD)
Threeleaper
(0,3)

(GS)
Fers
(1,1)

(JD)

Queen+Alfil
(2,2)
Fers
(1,1)
Threeleaper
(0,3)
Dabbaba
(0,2)
Wazir
(0,1)
Giraffe
(1,4)
Zebra
(2,3)
Camel
(1,3)
RookkNight
Bishop
(GR)

(GR)

(GR)

(GR)
none
(BH)

(GR)

(GR)

(GR)

(GR)

(GR)
kNight
(GR)

(GR)

(GR)

(GR)
none
(BH)

(GR)

(GR)

(GR)

(GR)
Rook
(GR)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
Camel
(1,3)

(GR)

(GR)

(GR)

(GR)
none
(BH)

(GR)

(GR)
Zebra
(2,3)

(GR)

(BH)

(GR)

(GR)
none
(BH)

(GR)
Giraffe
(1,4)

(GR)

(GR)

(GR)

(GR)
none
(BH)
Wazir
(0,1)
none
(BH)
none
(BH)
none
(BH)
none
(BH)
Dabbaba
(0,2)

(GR)
none
(BH)

(BH)
Threeleaper
(0,3)

(BH)

(GS)
Fers
(1,1)
none
(BH)

Bishop+Alfil
(2,2)
Fers
(1,1)
Threeleaper
(0,3)
Dabbaba
(0,2)
Wazir
(0,1)
Giraffe
(1,4)
Zebra
(2,3)
Camel
(1,3)
Rook
kNight
(GR)

(GR)

(GS)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
Rook
(GR)

(GR)

(GR)

(GR)
none
(GS)

(GR)

(GR)

(GR)
Camel
(1,3)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
Zebra
(2,3)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
Giraffe
(1,4)

(GS)

(BH)

(GS)

(GR)

(GR)
Wazir
(0,1)

(GS)

(GR)

(GS)

(GS)
Dabbaba
(0,2)

(GR)
none
(GS)

(GS)
Threeleaper
(0,3)

(GS)

(GR)
Fers
(1,1)

(GS)

kNight+Alfil
(2,2)
Fers
(1,1)
Threeleaper
(0,3)
Dabbaba
(0,2)
Wazir
(0,1)
Giraffe
(1,4)
Zebra
(2,3)
Camel
(1,3)
Rook
(GR)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
Camel
(1,3)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
Zebra
(2,3)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
Giraffe
(1,4)

(GS)

(GR)

(GR)

(GR)

(GR)
Wazir
(0,1)

(GS)

(GR)

(GS)

(GS)
Dabbaba
(0,2)

(GR)

(GR)

(BH)
Threeleaper
(0,3)

(BH)

(GS)
Fers
(1,1)

(GS)

Rook+Alfil
(2,2)
Fers
(1,1)
Threeleaper
(0,3)
Dabbaba
(0,2)
Wazir
(0,1)
Giraffe
(1,4)
Zebra
(2,3)
Camel
(1,3)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
Zebra
(2,3)

(GR)

(GR)

(GR)

(GR)

(GR)

(GR)
Giraffe
(1,4)

(GR)

(GR)

(GR)

(GR)

(GR)
Wazir
(0,1)
none
(GS)
none
(GS)

(GS)

(GS)
Dabbaba
(0,2)

(GR)

(GR)

(BH)
Threeleaper
(0,3)

(BH)

(GS)
Fers
(1,1)

(GS)

Camel+Alfil
(2,2)
Fers
(1,1)
Threeleaper
(0,3)
Dabbaba
(0,2)
Wazir
(0,1)
Giraffe
(1,4)
Zebra
(2,3)

(GS)

(GR)

(GR)

(GR)

(GR)

(GR)
Giraffe
(1,4)

(GS)

(GR)

(GS)

(GR)

(GR)
Wazir
(0,1)

(GS)

(GR)

(GS)

(GS)
Dabbaba
(0,2)

(GR)

(GR)

(BH)
Threeleaper
(0,3)

(GS)

(GS)
Fers
(1,1)

(GS)

Zebra+Alfil
(2,2)
Fers
(1,1)
Threeleaper
(0,3)
Dabbaba
(0,2)
Wazir
(0,1)
Giraffe
(1,4)

(GS)

(GR)

(GS)

(GR)

(GR)
Wazir
(0,1)

(GS)

(GR)

(GS)

(GS)
Dabbaba
(0,2)

(GR)

(GR)

(GS)
Threeleaper
(0,3)

(GS)

(GS)
Fers
(1,1)

(GS)

Giraffe+Alfil
(2,2)
Fers
(1,1)
Threeleaper
(0,3)
Dabbaba
(0,2)
Wazir
(0,1)

(GS)

(GR)

(GS)

(GS)
Dabbaba
(0,2)

(GS)

(GS)

(GS)
Threeleaper
(0,3)

(GS)

(GS)
Fers
(1,1)

(GS)

Wazir+Alfil
(2,2)
Fers
(1,1)
Threeleaper
(0,3)
Dabbaba
(0,2)

(GS)

(GS)

(GS)
Threeleaper
(0,3)

(GS)

(GS)
Fers
(1,1)

(GS)

Dabbaba+Alfil
(2,2)
Fers
(1,1)
Threeleaper
(0,3)

(GS)

(GS)
Fers
(1,1)

(GS)

Threeleaper+Alfil
(2,2)
Fers
(1,1)

(GS)

Smallest Chessboards Requiring Equal Travel Time
Between Four or More Traditional Pieces
 BKNQ(GR) BKNR(GR) BKQR(GR) BNQR(GR) KNQR(GR) BKNQR(GS)

2.

The best-known results are shown below.

Smallest Chessboards With Every Unoccupied Square Attacked and No Pieces Attacked
 (GS)
 (GS) (GS)
 (GS)
 (CP) (GS)
 (JD) (JD)
 (GS) (GS)

3.

The best-known results are shown below.

Smallest Chessboards With Every Square Attacked an Equal Number of Times
 ? ? ? ? none(JD) ?
 (GS)
 ? ? ? ? ? none(JD)
 ? ? ? ? ?
 ? ? ? ? ? ?
 (GS) ? ? ? ? ?
 ? none(JD) ?
 (GS) ? ? ? ? ? (GS) ? ?
 ? ? ? ? ? ? ? ?

4.

Richard Sabey pointed out that a board with a "mate in n" also has all smaller mates, so we only show the longest mate on any size board. The best-known results are shown below.

Longest Unique Mate Problem on a Chessboard of a Given Size
 4 squaresmate in 1 5 squaresmate in 2 6 squaresmate in 9(MM) 7 squaresmate in 14(MM) 9 squaresmate in 24(MM) 10 squaresmate in 27(MM) 11 squaresmate in 29(MM) 12 squaresmate in 31(MM)

Longest (Not Necessarily Unique) Mate
Problem on a Chessboard of a Given Size
 4 squaresmate in 1 5 squaresmate in 2 6 squaresmate in 9(MM) 7 squaresmate in 14(MM) 9 squaresmate in 25(MM) 10 squaresmate in 28(MM) 12 squaresmate in 32(MM)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 12/27/10.