Discussion: This student has found the essential key to the generalized problem, the case of nine lattice points in three dimensional space. The exposition is not flawless by any means but this student uses notation in a clever way to represent the thinking that is going on. As in the other solution, the parity pattern of the three tuple divides the lattice points into different classes, in this case eight and consequently by the pigeonhole principle two have the same pattern and the mid point of the segment joining them is a lattice point.

Written response: Basically an elegant solution to this problem but your exposition needs work.

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