<s>[INST] 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.
 [/INST] 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.
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