# Clever calculation

Find the shortcut to do this calculation quickly!

Without using a calculator, or performing any long multiplication calculations, find the value of

$$2017^2-2015\times2019$$

*This problem is taken from the World Mathematics Championships*

**Answer**: $4$

**Using a symbol**

Suppose we use a symbol to represent $2017$, such as $n$.

Then if $n=2017$, $2015=n-2$ and $2019=n+2$.

So $$\begin{split}2017^2-2015\times2019&=n^2-(n-2)(n+2)\\

&=n^2-(n^2+2n-2n-4)\\

&=n^2-(n^2-4)\\

&=n^2-n^2+4\\

&=4.\end{split}$$

**Using just numbers**

Write all the numbers in terms of $2017$, so $2015=2017-2$ and $2019=2017+2$.

ThenĀ $$\begin{split}2017^2-2015\times2019&=2017^2-(2017-2)(2017+2)\\

&=2017^2-(2017^2+2017\times2-2\times2017-2^2)\\

&=2017^2-(2017^2-4)\\

&=2017^2-2017^2+4\\

&=4.\end{split}$$