I'm in the space!

Probability Level 3

There are nn distinct lattice points marked in the 3D space.

Find least possible value of nn, such that we can always choose 2 points out of nn points (wherever they may be marked), such that there's at least one more lattice point on the segment joining them.


Details and assumptions:-

\bullet In the 3D space, every point can be represented as coordinates (x,y,z)(x,y,z), where x,y,zRx,y,z \in \mathbb{R}

\bullet Lattice points are points that have integer coordinates.


Easier version 2D

Harder version 5D

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