MathDB

What is the largest number of horses that you can put on a chessboard without there being two horses that can beat each other? a. Describe an arrangement with that maximum number. b. Prove that a larger number is not possible.
(A chessboard consists of

8 \times 8

spaces and a horse jumps from one field to another field according to the line "two squares vertically and one squared horizontally" or "one square vertically and two squares horizontally")
[asy] unitsize (0.5 cm);
int i, j;
for (i = 0; i <= 7; ++i) { for (j = 0; j <= 7; ++j) { if ((i + j) % 2 == 0) { if ((i - 2)^2 + (j - 3)^2 == 5) { fill(shift((i,j))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), red); } else { fill(shift((i,j))*((0,0)--(1,0)--(1,1)--(0,1)--cycle), gray(0.8)); } } }}
for (i = 0; i <= 8; ++i) { draw((i,0)--(i,8)); draw((0,i)--(8,i)); }
label("

a

", (0.5,-0.5), fontsize(10)); label("

b

", (1.5,-0.5), fontsize(10)); label("

c

", (2.5,-0.5), fontsize(10)); label("

d

", (3.5,-0.5), fontsize(10)); label("

e

", (4.5,-0.5), fontsize(10)); label("

f

", (5.5,-0.5), fontsize(10)); label("

g

", (6.5,-0.5), fontsize(10)); label("

h

", (7.5,-0.5), fontsize(10)); label("

1

", (-0.5,0.5), fontsize(10)); label("

2

", (-0.5,1.5), fontsize(10)); label("

3

", (-0.5,2.5), fontsize(10)); label("

4

", (-0.5,3.5), fontsize(10)); label("

5

", (-0.5,4.5), fontsize(10)); label("

6

", (-0.5,5.5), fontsize(10)); label("

7

", (-0.5,6.5), fontsize(10)); label("

8

", (-0.5,7.5), fontsize(10)); label("

P

", (2.5,3.5), fontsize(10)); [/asy]

3

Problems(1)