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Lattice Vertices Verification (Posted on 2016-07-02) Difficulty: 3 of 5
The side lengths of a polygon with 1994 sides are given by:
A(n) = √(n2+4) for n= 1,2,…1994

Can all the vertices of this polygon lie on lattice points?

Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
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Solution Solution | Comment 1 of 4
All of the lengths A(n) are irrational.  Each length is the hypotenuse of a right triangle with integral legs.

Assume all the vertices of the 1994-gon lie on the lattice points.  Then the legs of the triangles corresponding to each side must follow the grid lines.  Substitute the legs for each side of the 1994-gon to create a 3988-gon.

The perimeter of the 3988-gon must be even.  Its perimeter is calculated as 2*1994 + 1994*1995/2 = 1993003.  This is odd, a contradiction.  Therefore the vertices of the 1994-gon cannot all lie on lattice points.

  Posted by Brian Smith on 2016-07-02 12:01:29
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