12 Knights to Attack or Occupy All Squares on Chessboard

Theorem

On a standard chessboard, a minimum of $12$ knights are needed to ensure all squares are either occupied or under attack.

Proof

First, we show that fewer than $12$ knights are not enough to occupy or attack each square.

Consider the $12$ squares $\text a 1$, $\text a 2$, $\text b 2$, $\text a 8$, $\text b 8$, $\text b 7$, $\text h 8$, $\text h 7$, $\text g 7$, $\text h 1$, $\text g 1$ and $\text g 2$.

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

No knight can occupy or attack more than one of these squares.

Hence, fewer than $12$ knights are not sufficient.

$\Box$

Second, we show that $12$ knights can be placed to ensure all squares are either occupied or under attack.

For example, with the knights being on $\text b 3$, $\text c 3$, $\text c 4$, $\text c 7$, $\text c 6$, $\text d 6$, $\text g 6$, $\text f 6$, $\text f 5$, $\text f 2$, $\text f 3$ and $\text e 3$, all squares are either occupied or under attack:

	a	b	c	d	e	f	g	h
8									8
7									7
6									6
5									5
4									4
3									3
2									2
1									1
	a	b	c	d	e	f	g	h

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12$