TITLE: How do people perform mental arithmetic for complicated expressions?
QUESTION [181 upvotes]: 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?

REPLY [2 votes]: I am surprised that nobody mentioned mnemonic techniques for doing calculations in your head. Instead of trying to explain basic techniques myself I am linking this video where famous mathemagician Arthur Benjamin explaining techniques he uses for multiplying huge numbers in his head.
For those who prefer to read rather than watch and listen I also link one of the articles where A.Benjamin covers basics of mnemonic techniques, for squaring large numbers, memorizing digits of $\pi$, and more.
Several relevant textbooks are also referenced below.


Ringel', D.E., Mnemonic numbers and mnemonic vectors, Vestn. Beloruss. Gos. Univ., Ser. 1, Fiz. Mat. Inform. 2003, No. 1, 71--76 (2003). ZBL1272.46035. 
Benjamin, Arthur. T., The Secrets of Mental Math (2008), (Amazon link). 
Toma, Marina, Mathematics and mental calculus, J. Sci. Arts 10, No. 1, 39-42 (2010). ZBL1220.00025.