How do people perform mental arithmetic for complicated expressions? 
This is the famous picture "Mental Arithmetic. In the Public School of S. Rachinsky." by the Russian artist Nikolay Bogdanov-Belsky.
The problem on the blackboard is:
$$
\dfrac{10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2}}{365}
$$
The answer is easy using paper and pencil: $2$.
However, as the name of the picture implies, the expression ought be simplified only mentally.
My questions: 


*

*Are there general mental calculation techniques useful for performing basic arithmetic and exponents?

*Or is there some trick which works in this case? 

*If so, what is the class of problems this trick can be applied to?

I wasn't smart enough to group them the way Barry Cipra did, so I just did the arithmetic: $169 + 121 = 290$; adding $144$ makes $434$; adding $196=200-4$ makes $630$, adding $100$ makes $730$; wow, that's twice $365$, so the answer is $2$.
