SG. 1 If $a * b=a b+1$, and $s=(2 * 4) * 2$, find $s$.
SG. 2 If the $n^{\text {th }}$ prime number is $s$, find $n$.
SG. 3 If $K=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right) \cdots\left(1-\frac{1}{50}\right)$, find $K$ in the simplest fractional form.
SG. 4 If $A$ is the area of a square inscribed in a circle of radius 10 , find $A$.
Reference: 1984 FG6.4
$$
\begin{array}{c}
2 * 4=2 \times 4+1=9 \\
s=(2 * 4) * 2=9 * 2 \\
\quad=9 \times 2+1=19
\end{array}
$$

Reference: 1989 FSG.3, 1990 FI5.4
$$
\begin{array}{l}
2,3,5,7,11,13,17,19 \\
n=8
\end{array}
$$

Reference: 1984 FG9.1, 1986 FG10.4
$$
K=\frac{1}{2} \times \frac{2}{3} \times \frac{3}{4} \times \cdots \times \frac{49}{50}=\frac{1}{50}
$$

Reference: 1984 FG10.1, 1989 FI3.3
Let the square be $B C D E$.
$$
\begin{array}{l}
B C=20 \cos 45^{\circ}=10 \sqrt{2} \\
A=(10 \sqrt{2})^{2}=200
\end{array}
$$