1)
A frog started from the origin of the coordinate plane and made three jumps.
Each time the frog jumped a distance of 5 units and landed at a point with integer coordinates.
How many different possibilities of the final position of the frog are there?
All 264 different possibilities of the final position of the frog:
\scalebox{0.5}{% scale
\begin{tikzpicture}
\draw[thick] (-15,-15) grid (15,15);
% Axes
\foreach \x in {-15,...,-1,0,1,2,...,15} {%
\draw (\x, -.1) -- (\x, 0) node[below left=2pt] {$\scriptstyle\x$};
}
\foreach \y in {-15,...,-1,1,2,...,15} {%
\draw (-.1,\y) -- (0,\y) node[below left=2pt] {$\scriptstyle\y$};
};
\foreach \x/\y in {5/0,4/3,3/4,0/5,-5/0,-4/3,-3/4,4/-3,3/-4,0/-5/,-4/-3/,-3/-4} %{\x \y,}
{
\foreach \v/\w in {5/0,4/3,3/4,0/5,-5/0,-4/3,-3/4,4/-3,3/-4,0/-5/,-4/-3/,-3/-4} %{\v \w,}
{
\foreach \r/\s in {5/0,4/3,3/4,0/5,-5/0,-4/3,-3/4,4/-3,3/-4,0/-5/,-4/-3/,-3/-4} %{\r \s,}
{
\node[draw,circle,inner sep=2pt,fill] at (\x+\v+\r,\y+\w+\s) {};
}
}
}
\end{tikzpicture}
}