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\textbf{\LARGE The Hunting on the Snark}
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The Baker hunts on the Snark on infinite squared paper. In one
move the Snark can jump from square $(x, y)$ to any square $(x',
y')$ such that $|x-x'|\leqslant n$, $|y-y'|\leqslant n$. The
number $n$ is called \emph{velocity} of the Snark. The Baker's
move is placing a cake on any empty square. The Snark hates
cakes, so he never jumps to a square with a cake. The Baker's
goal is to lock the Snark in such a way that the Snark cannot
jump. The Snark's goal is opposite, i.~e., not to let the Baker
lock him.
\begin{itemize}
\item[0.]
Prove that the Baker can lock the Snark of velocity 1.
\item[$1^{**}$.]
\textbf{(Open problem)} For at least one $n>1$
find out who can reach his goal: the Baker or the Snark.
\end{itemize}
We propose several easier questions about somewhat restricted
Snarks.
\begin{itemize}
\item[$\varepsilon$.]
Suppose that with each move the Snark increases his
$y$-coordinate ($y'>y$). Prove that the Baker can lock him.
\item[$2\varepsilon$.]
Suppose that the Snark never decreases his y-coordinate
($y'\geqslant y$). Prove that the Baker can lock him.
\item[$3\varepsilon$.]
Suppose that once the Snark reaches square $(x, y)$ he never
visits squares with $y$-coordinate less than $y-1\,000\,000$.
Prove that the Baker can lock him.
\item[$4\varepsilon$.]
Suppose that with each move the Snark increases his distance
from the origin. (The distance between the origin and square
$(x, y)$ is defined as $\max\{|x|, |y|\}$.) Prove that the Baker
can lock him.
\item[$5\varepsilon$.]
Suppose that once the Snark reaches a square at distance $d$
from the origin he never visits squares located at distance
less than or equal to $d-1\,000\,000$. Prove that the Baker can
lock him.
\item[1/2001.]
Suppose that the Snark is suspicious in the following way: if
once he could have ever jumped to a square but he didn't, then
he will never jump to this square. Prove that if the Baker can
lock the suspicious Snark then he can lock any Snark of the same
velocity.
\end{itemize}
Now the hunting takes place in three-dimensional space divided
into cubes.
\begin{itemize}
\item[$\delta$.]
Suppose that with each move the Snark can jump to any cube
having a common face with his current location (i.~e., he has
six potential moves). Prove that the Baker can lock him.
\item[$1-\delta$.]
Suppose that with each move the Snark can jump from a cube
$(x,y,z)$ to any cube $(x',y',z')$ where $|x-x'|\leqslant1$,
$|y-y'|\leqslant1$, $|z-z'|\leqslant1$ (i.~e., he has 26
potential moves). Can the Baker lock the Snark?
\end{itemize}
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