I'm in the space!

Discrete Mathematics Level 3

There are \(n\) distinct lattice points marked in the 3D space.

Find least possible value of \(n\), such that we can always choose 2 points out of \(n\) 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 \((x,y,z)\), where \(x,y,z \in \mathbb{R}\)

\(\bullet\) Lattice points are points that have integer coordinates.

Easier version 2D

Harder version 5D


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