How many knights does it take to attack or control the board?
competition/games/chess/knight.control Added material
Problem
How many knights does it take to attack or control the board?
Show solution
Fourteen knights are required to attack every square:
1 2 3 4 5 6 7 8
___ ___ ___ ___ ___ ___ ___ ___
h | | | | | | | | |
--- --- --- --- --- --- --- ---
g | | | N | N | N | N | | |
--- --- --- --- --- --- --- ---
f | | | | | | | | |
--- --- --- --- --- --- --- ---
e | | N | N | | | N | N | |
--- --- --- --- --- --- --- ---
d | | | | | | | | |
--- --- --- --- --- --- --- ---
c | | N | N | N | N | N | N | |
--- --- --- --- --- --- --- ---
b | | | | | | | | |
--- --- --- --- --- --- --- ---
a | | | | | | | | |
--- --- --- --- --- --- --- ---
Three knights are needed to attack h1, g2, and a8; two more for b1, a2,
and b3, and another two for h7, g8, and f7.
The only alternative pattern is:
1 2 3 4 5 6 7 8
___ ___ ___ ___ ___ ___ ___ ___
h | | | | | | | | |
--- --- --- --- --- --- --- ---
g | | | N | | | N | | |
--- --- --- --- --- --- --- ---
f | | | N | N | N | N | | |
--- --- --- --- --- --- --- ---
e | | | | | | | | |
--- --- --- --- --- --- --- ---
d | | | N | N | N | N | | |
--- --- --- --- --- --- --- ---
c | | N | N | | | N | N | |
--- --- --- --- --- --- --- ---
b | | | | | | | | |
--- --- --- --- --- --- --- ---
a | | | | | | | | |
--- --- --- --- --- --- --- ---
Twelve knights are needed to control (attack or occupy) the board:
1 2 3 4 5 6 7 8
___ ___ ___ ___ ___ ___ ___ ___
a | | | | | | | | |
--- --- --- --- --- --- --- ---
b | | | N | | | | | |
--- --- --- --- --- --- --- ---
c | | | N | N | | N | N | |
--- --- --- --- --- --- --- ---
d | | | | | | N | | |
--- --- --- --- --- --- --- ---
e | | | N | | | | | |
--- --- --- --- --- --- --- ---
f | | N | N | | N | N | | |
--- --- --- --- --- --- --- ---
g | | | | | | N | | |
--- --- --- --- --- --- --- ---
h | | | | | | | | |
--- --- --- --- --- --- --- ---
Each knight can control at most one of the twelve squares a1, b1, b2,
h1, g1, g2, a8, b8, b7, h8, g8, g7. This position is unique up to
reflection.
References
Martin Gardner, _Mathematical Magic Show_.
Additional material
[ Editor: a variant in which every square must be occupied or attacked and no knight may attack another, posted in 1994 under the subject "Minimum # of Knights on 8x8 chessboard". ] Due to the lack of response, here's the quick answer... For a 8x8 chessboard the minimum number of knights required so that all squares are dominated or occupied by a knight, but no knight is attacking any other is 14. The following are two solutions: (notice the symmetry of the 2nd) --------------------------------- | K | K | K | | | | | | --------------------------------- | | | | | | K | | | --------------------------------- | | | | | K | K | | | --------------------------------- | K | | | | | | | | --------------------------------- | | | | | | | | K | --------------------------------- | | | K | K | | | | K | --------------------------------- | | | K | | | | | | --------------------------------- | | | | | | K | | K | --------------------------------- --------------------------------- | K | K | K | | | | | | --------------------------------- | | | | | | K | | | --------------------------------- | | | | | K | K | | | --------------------------------- | K | | | | | | | | --------------------------------- | | | | | | | | K | --------------------------------- | | | K | K | | | | | --------------------------------- | | | K | | | | | | --------------------------------- | | | | | | K | K | K | --------------------------------- Maybe this puzzle is too hard? -- Randall Rathbun <randall@coyote.csusm.edu>