Let a and b be natural numbers such that $2a - b, a - 2b$ and $a + b$ are all distinct squares. What is the smallest possible value of $b$? Let a and b be natural numbers such that $2a - b, a - 2b$ and $a + b$ are all distinct squares.
What is the smallest possible value of $b$?
Let, $2a-b=k^2, a-2b=p^2, a+b=q^2$.
$k^2=p^2+q^2$ after adding any of the two equations. 
How to proceed further?
 A: If you subtract the last two you get $q^2-p^2=3b$.  If you add the first and last you get $k^2+q^2=3a$.  No primitive Pythagorean triangle has legs that differ by a multiple of $3$, so we need a triangle that has a common factor of $3$.  The smallest such is $9-12-15$ and we find
$$3b=144-81=63\\b=21\\3a=225+144=369\\a=123$$
This is the smallest $b$ because the difference of the two legs must be at least $3$.  If the shorter leg is $c$ we have $(c+3)^2-c^2=6c+9$ and $b$ will grow with the shorter leg.  
A: The first few quintuples are
$$(a,b,a+b,2a-b,a-2b)\\\in\big\{
(6,3,9,9,0 ),\quad 
(24,12,36,36,0 ),\quad 
(54,27,81,81,0 ),\quad 
(96,48,144,144,0 ),\\ 
\mathbf{(123,21,144,225,81 )},\quad 
(150,75,225,225,0 ),\quad 
(216,108,324,324,0 ),\\ 
(294,147,441,441,0 ),\quad 
(384,192,576,576,0 ),\quad 
(486,243,729,729,0 ),\\ 
\mathbf{(492,84,576,900,324 )},\quad 
(600,300,900,900,0 ),\quad 
(726,363,1089,1089,0 ),\\ 
(864,432,1296,1296,0 ),\quad 
\mathbf{(939,357,1296,1521,225 )},\quad  \cdots\big\}
$$
The lowest value of $\space b\space $ that works is either $\space 3\space $ or $\space 21\space $ depending on whether or not you allow $\space (a-2b)=0\space $.
