Question 15
A police officer tries to catch a thief on a $2001 \times 2001$ board. They play alternately. Each player, on their turn, must move one square in one of the following three directions:
$\downarrow$ (below); $\rightarrow$ (right); $\vee$ (upper left diagonal).
If the policeman is in the lower right corner square, he can use his move to move directly to the upper left corner square (the thief cannot make this move). Initially the policeman is in the central square and the thief is in the upper right diagonal neighboring square to the policeman. The policeman starts the game. Show that:
a. The thief manages to move at least $10000$ times without being caught.
b. The police officer has a strategy to capture the thief.
$\downarrow$ (below); $\rightarrow$ (right); $\vee$ (upper left diagonal).
If the policeman is in the lower right corner square, he can use his move to move directly to the upper left corner square (the thief cannot make this move). Initially the policeman is in the central square and the thief is in the upper right diagonal neighboring square to the policeman. The policeman starts the game. Show that:
a. The thief manages to move at least $10000$ times without being caught.
b. The police officer has a strategy to capture the thief.