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?

$$\color{Green}{n^2=1+3+5+\cdots+(2n-1)}$$
Therefore
$$\begin{align}
10^2+11^2+12^2+13^2+14^2
&=(1\cdot+19)+(1\cdot+21)+(1\cdot+23)+(1\cdot+25)+(1\cdot+27)\\
&=5\times(1\cdot+19)+(4\times21)+(3\times23)+(2\times25)+27
\end{align}$$
Also,  $1+3+\cdot+19=10^2=100.$  
Hence
$$10^2+11^2+12^2+13^2+14^2=500+84+69+50+27=730$$
$$\color{Green}{10^2+11^2+12^2+13^2+14^2=2\times365.}$$
