14. Let $a, b, c$ and $d$ be distinct positive integers such that $a+b, a+c$ and $a+d$ are all odd and are all square. Let $L$ be the least possible value of $a+b+c+d$. What is the value of $10 L$ ?
Solution
670
The numbers $a, b, c$ and $d$ are distinct, so $a+b, a+c$ and $a+d$ must also be distinct. The smallest three odd squares which may be formed in this way are 9,25 and 49 . Therefore, $(a+b)+(a+c)+(a+d)=9+25+49=83$.
We may write $L=a+b+c+d=83-2 a$. So we may minimise $L$ by maximising $a$. Since $a+b=9$, the largest possible value for $a$ is 8 . This means that $L=83-2 \times 8=67$. We must check that a solution exists. The values $a=8, b=1, c=17$ and $d=41$ satisfy all the conditions provided.
If one started with a different set of odd squares, then at least one would be 81 or larger. However, $a+b+c+d$ is greater than each of the odd squares involved. Hence 67 is indeed the least possible value.
Therefore $L$ is indeed 67 and $10 L=670$.