Question 7
Let be given an integer $n \gt 1$ and an orthonormal system of coordinates $Oxyz$ in space. Let $T$ be the set of all points $P(x, y, z)$, the coordinates $x, y, z$ of which are integers satisfying the conditions $1 \le x, y, z \le n$. Colour some points in $T$ so that if the point $A(x_0, y_0, z_0)$ had been coloured then every point $B(x_1, y_1, z_1)$ with $x_1 \le x_0, y_1 \le y_0, z_1 \le z_0$ ($B \ne A$) could not be coloured. How many points at most can be coloured?