Strugle with the problem for many days, but can not find the correct solution. But I think it may use the pigeonhole principle to get the proof. Because there are nine points, and the number of combination of even and odd exactly is 2^3=8, that means 8 to 9 may fits the pigeonhole principle.
There are still something I can think about. The lattice point lies on the segment joining a pair of the other nine points. Set a point as the origin, then the other will be
(a_i, b_i, c_i), i=1,2,3,...,8. Then the statement equivalent to there is at least one of the greate common divisor of the co-oridinates not equals to 1. But I do not know how to prove it.
I also illustrate a figure above, considering the pattern like the upper corner. I believe that if there any four points lie on a same plane, they have the same pattern, or otherwise there will be a transformation. And it is know that this kind of transformation can be separated into a scale multiple and a rotation, which is just matter the vector
i, j, and
k. The coordinates of the point remain unchange. Since there only 8 points, adding one more point will make the GCD of coordinates of its difference(vector) with the original 8 points greater than 1 (at least one vector).
[hard to express my idea in english] This method may solve the problem, but I was wondering am I think it too complex. One hand, because my boss gave me a lot of data to analysis, and the other hand I feel the books of number theory were a little hard to read, so I stop thinking this problem.
Maybe there is easier approach, but take long time to waite the announcement of the answer.
By the way Problem 7 is much easier than Problem 6.
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