1

First question - suppose we have a knight at the origin, and $k$ rooks positioned far away. How many rooks are needed to guarantee capture the knight on an infinite board? It should be straightforward to show that four rooks suffice, but can it be done with 3?

Note that there is a configuration with two rooks that covers all squares a knight can go to.

What if we have $n$ knights, (that can protect each other), what is the minimum number of rooks that is needed to capture all knights?

Alex Ortiz
  • 24,844
Per Alexandersson
  • 3,626
  • Yeah, so suppose that the person who play knight is allowed to place the rooks, but not so close as to capture one of them in the initial turn.

    Say that knight player can decide rook starting positions, but must be outside some radius R away from all knights.

     – Per Alexandersson Sep 18 '16 at 02:13
  • How do you do it with 4? – Bart Michels Oct 15 '17 at 09:36

1 Answers1

0

You have the knight in the middle, d4, put the rook on c6, and another one on f2, this will cover the knights movement squares. Put another rook on the a-file or 6th rank and this secures capture... So given $n$ knights, the maximum number of rooks required to secure capture $(k)$is $3n$

Xetrov
  • 2,089