How to prove that it is possible to make rhombuses with any number of interior points? I was given some square dot paper which can be found on this link: http://lrt.ednet.ns.ca/PD/BLM/pdf_files/dot_paper/sq_dot_1cm.pdf  and was told to draw a few rhombuses with the vertices on the dots but no other dots on the edges.
Here is an example:
This is not a rhombus it just shows how the shape must be lay out. 

                   A = 6

The number of interior dots would be represented by A. I was then told that it is possible to draw rhombuses with any number of interior points. All I had to do was prove this.
I started by trying to draw these shapes and came up with some of the below:

A = 2

A = 4

A = 9  
However, I am still unsure of how you can prove something like this since there may be numerous ways of drawing these rhombuses.
I would appreciate support.
Thank you :) 

The square dot paper you describe can be thought of as something called the integer lattice, that is, the set of points $(m,n)$ in the plane where $m,n$ are integers. In order to make this identification, pick an arbitrary point on the square dot paper and call it $(0,0)$. If $m,n$ are positive integers, then $(m,n)$ is the point which is $m$ points to the right and $n$ points up from $(0,0)$. If $m$ is negative, move to the left instead of the right. If $n$ is negative move down instead of up.
To rephrase what we wish to prove with this terminology, we could say that for any number of points $s$, there is a rhombus $R$ with vertices on the integer lattice whose boundary (except for the vertices) contains no points of the integer lattice, and $R$ contains $s$ points of the integer lattice in its interior.
Your first example generalizes naturally in the following way.
Let $n$ be a positive integer and consider the rhombus $R$ whose vertices consist of the points $(1,0)$, $(-1,0)$, $(0,n)$ and $(0,-n)$. The rhombus $R$ contains no points of the integer lattice on its boundary except its vertices since its sides have $x$-values between $0$ and $1$. Moreover, $R$ contains the points $(0,k)$ for $-n < k < n$ within its interior, of which there are $2n-1$. This proves that you can find a rhombus containing a desired odd number of points of the integer lattice in its interior. If you are having trouble seeing this, try drawing a rhombus like this on your dot paper.
To find a rhombus which contains a desired even number of points of the integer lattice in its interior, try using a diagonally placed rhombus like yours in the first example (note: that one has 2 points in the interior). Have a shot at working out the rest of this argument for yourself; it is similar to the one I have above.
