11442 - Linear Diophantine Tidbits

All about problems in Volume 114. If there is a thread about your problem, please use it. If not, create one with its number in the subject.

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11442 - Linear Diophantine Tidbits

Post by pradhanp »

Any hints for this one? It reduces to counting the number of lattice points in a triangle in 3-D space. How do you do that? Is there a generalisation of Pick's theorem which could be applied here?

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Re: 11442 - Linear Diophantine Tidbits

Post by sclo »

Short answer is:
No, there is no known generalization of Pick's theorem for 3d space.

You probably know that the given triangle lies on a certain plane, so it lies in 2d space. All you need to do is to find a parametrization of the plane in such a way that the use of Pick's theorem on the parametrization gives the required answer.
In other words, find linear function F such that (x,y,z)=F(s,t) and (x0,y0,z0) is a solution to the linear diophantine equation iff (s0,t0) is in Z X Z where (x0,y0,z0)=F(s0,t0)

The fact that F is linear is important since it guarantees the transformation preserves straight lines.

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