# How to compute $\sqrt{1+2015\sqrt{1 + 2018 \cdot 2016}}$

I recently found a new math riddle that I couldn't solve, it ask you compute the following without a calculator:

$$\sqrt{1+2015\sqrt{1 + 2018 \cdot 2016}}$$

I tried to rewrite both of 2016 as powers of two but I couldn't get anywhere close .

How wouLd you solve this problem ?

• Haha, gues which year it still is? – Simply Beautiful Art Dec 10 '16 at 19:52

$2016\times 2018 = (2017-1)(2017+1) = 2017^2-1$
In this kind of questions, start by doing what you shouldn't and understand what happens. Using a calculator you see that the final result is $2016$.
Another common thing, is to use standard arithmetic tricks, which seem subtle as the numbers are huge! As Joffan pointed out $2016\times 2018 = (2017-1)\times(2017+1) = 2017^2-1$ you could have seen that by noticing how close the two numbers were, and therefore the standard algebraic formula.
Then you get $2015 \times 2017 = (2016-1)(2016+1) = 2016^2 - 1$ and therefore you get 2016 as the final answer.
You could have seen that from the beginning as you see that $1$s are involved in both squares. You knew the result was $2016$ so you know that $2015\sqrt{1+2018\times 2016}$ must have been $2016^2 - 1$ and from this you could have worked backwards.