256. Find the numbers $a$ and $b$, if the numbers $a, 1, b$ form an arithmetic progression, and the numbers $a^{2}, 1, b^{2}$ form a geometric progression. List all solutions.
$\triangle$ Let's use the characteristic properties of arithmetic and geometric progressions:

$$
\left\{\begin{array} { l } 
{ 1 = \frac { a + b } { 2 } , } \\
{ 1 = \sqrt { a ^ { 2 } b ^ { 2 } } , }
\end{array} \quad \left\{\begin{array}{l}
a+b=2 \\
a^{2} b^{2}=1
\end{array}\right.\right.
$$

It remains to solve the last system of equations.

Answer: $a=b=1 ; \quad a=1+\sqrt{2}, b=1-\sqrt{2} ; \quad a=1-\sqrt{2}, b=1+\sqrt{2}$.