HMMT February

For this problem, it is useful to know the following square root algorithm that allows for digit-by-digit extraction of $\sqrt{x}$$and gives one decimal place of \sqrt{x}$ for each two decimal places of $x$. We will illustrate how to extract the second digit after the decimal point of $\sqrt{\pi}$$, knowing that \pi=3.1415 \cdots$ and $\sqrt{\pi}=1.7 \cdots$$. Let$d be the next decimal digit. Then d should be the largest digit such that (1.7+0.01 $d)^{2}<\pi$, which in this case we will treat as (1.7+0.01 d)^{2}<3.1415$. Expanding this, we get$2.89+0.034 d+0.0001 d^{2}< 3.1415, from which we get the value of d$to be approximately \left\lfloor\frac{3.1415-2.89}{0.034}\right\rfloor=\left\lfloor\frac{0.2515}{0.034}\right\rfloor=7$, since the $0.0001 d^{2}$ term is negligible. Indeed, 7 is the largest such digit, and so $d=7$ is the second digit of $\sqrt{\pi}\$.$ Because we are constantly subtracting the square of our extracted answer so far, we can record the difference in a manner similar to long division, which yields a quick method of extracting square roots by hand.