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?

If you know your squares out to $14$ (which students used to memorize) and do some simple three-digit arithmetic in your head, you can see that
$$100+121+144=365$$
and
$$169+196=365$$
