Re: lattice points question

On Wed, 17 Aug 2005 09:57:13 +0300, Jyrki Lahtonen <lahtonen@xxxxxx>
wrote:

>quasi wrote:
>> On 16 Aug 2005 20:58:51 -0700, "David M Einstein" <Deinst@xxxxxxxxx>
>> wrote:
>> Let's phrase that as a question:
>>
>> Does there exist a convex n-gon such that all sides are of equal
>> length, all vertices are on integer lattice points, and no 2 sides are
>> parallel?
>
>Wouldn't the followinig example fit the bill:
>
>The complex numbers 8+i, -1+8i, -7+4i, -7-4i, -1-8i, and 8-i
>sum up to zero, and all have modulus sqrt{65}. Here they are
>sorted in the order of increasing argument (constrained
>into the interval [0,2Pi) ). The negative of any number from
>this list is not included in the list, so these form a
>pairwise non-parallel set of vectors.
>
>So if these numbers are used as edges, we get a closed
>hexagon. Because the arguments of the edges increase
>monotonously, the boundary of the hexagon is always "turning
>to the left", and hence the hexagon is convex.
>
>Starting from the origin we get the following vertices
>(0,0),(8,1),(7,9),(0,13),(-7,9),(-8,1) (and back to
>the origin).
>
>As an odd number of sides was proven to be impossible,
>six is the smallest number of vertices tha can be used here.
>
>Cheers,
>
>Jyrki Lahtonen, Turku, Finland

Ok, the goes that conjecture -- nice example.

So now let's make the requirements harder.

Here are a few more challenges:

(1) For which positive integers k does there exist a set S of k
vectors in R^2 with integer coordinates and equal lengths, such that
the sum of the elements of S is 0 but the sum of any nonempty proper
subset of S is nonzero?

(1') Same question as (1) except we require the vectors to have
integer length.

(2) Same question as (1) except we require the stronger condition that
the sum of any nonempty subset S is not parallel to any of the 4
vectors <1,0>, <0,1>, <1,1>, <1,-1>.

(2') Same question as (2) except we require the vectors to have
integer length.

Next, consider the same questions in R^n, n>1, except that for
question (2) we require the proper sums to avoid being parallel to any
vector all of whose nonzero coordinates have absolute value 1. Does
the set of possible values of k get larger as n increases?

quasi
.

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