How many unique distances are there in a 5 x 5 grid? I cannot figure this out:
I have a square in the plane with side length $5$. $A$ and $B$ are points in the square. The coordinates of $A$ and $B$ are always integers. 
I want to know how many unique Euclidean distances are possible between $A$ and $B$.
I thought $15$? 
Editor note The original ambiguous phrasing of this question and the consequent edits to clarify resulted in conflicting solutions. Please take this into consideration as you vote on the answers.

If you count vertical or horizontal distances the values are 0,1,2,3,4,5; 
If the horizontal (or vertical) has length 1 and the vertical has length 1, 2, 3, 4, 5 then the distances are $\sqrt{1^2+1^2}=\sqrt{2}, \sqrt{1^2+2^2}=\sqrt{5}, \sqrt{1^2+3^2}=\sqrt{10}, \sqrt{1^2+4^2}=\sqrt{17}, \sqrt{1^2+5^2}=\sqrt{26}$, 
If the horizontal (or vertical) has length 2 and the vertical has length  2, 3, 4, 5 then the distances are $\sqrt{2^2+2^2}=\sqrt{8}, \sqrt{2^2+3^2}=\sqrt{13}, \sqrt{2^2+4^2}=\sqrt{20}, \sqrt{2^2+5^2}=\sqrt{29}$,
If the horizontal (or vertical) has length 3 and the vertical has length  3, 4, 5 then the distances are  $\sqrt{3^2+3^2}=\sqrt{18}, \sqrt{3^2+4^2}=\sqrt{25}=5, \sqrt{3^2+5^2}=\sqrt{34}$.
If the horizontal (or vertical) has length 4 and the vertical has length   4, 5 then the distances are  $\sqrt{4^2+4^2}=\sqrt{32}, \sqrt{4^2+5^2}=\sqrt{41}$.
If the horizontal (or vertical) has length 5 and the vertical has length  5 then the distances are  $\sqrt{5^2+5^2}=\sqrt{50}$.
So there are 20 distinct values for the distances between two points.
