Question 75
Real numbers $x_1$, $x_2$, $x_3$, $x_4$ in $[0; 1]$ are such that the product
$$
K = |x_1 - x_2| \cdot |x_1 - x_3| \cdot |x_1 - x_4| \cdot |x_2 - x_3| \cdot |x_2 - x_4| \cdot |x_3 - x_4|
$$
is as large as possible. Prove that $\frac{1}{27} \gt K \gt \frac{4}{243}$.
$$
K = |x_1 - x_2| \cdot |x_1 - x_3| \cdot |x_1 - x_4| \cdot |x_2 - x_3| \cdot |x_2 - x_4| \cdot |x_3 - x_4|
$$
is as large as possible. Prove that $\frac{1}{27} \gt K \gt \frac{4}{243}$.