Why does this method for finding the square root of a number work? Recently I read this, which teaches a trick for finding the square root of a number very quickly and was quite astounded. How/why does this method work?
For those of you with linkphobia, I copied and pasted part of the page below:

Step-1:  Look at the magnitude of the “hundreds number” (the numbers preceding the last two digits) and find the largest square that is equal to or less than the number. This is the 1st part of the answer.
Step-2:  Now, look at the last (unit’s) digit of the number. If the number ends in a:
$0 \rightarrow$ then the ending digit of the answer is a $0$. 
  $1 \rightarrow$ then the ending digit of the answer is $1$ or $9$.
  $4 \rightarrow$ then the ending digit of the answer is $2$ or $8$.
  $5 \rightarrow$ then the ending digit of the answer is a $5$.
  $6 \rightarrow$ then the ending digit of the answer is $4$ or $6$.
  $9 \rightarrow$ then the ending digit of the answer is $3$ or $7$.
To determine the right answer from $2$ possible answers (other than $0$ and $5$), mentally multiply the findings in step-1 with its next higher number. If the left extremities (the numbers preceding the last two digits) are greater than the product, the right digit would be the greater option $(9,8,7,6)$ and if left extremities are less than the product, the right digit would be the smaller option $(1,2,3,4)$.


Write $n = 10x + y$ where $y \in \{0, 1, \dots, 9\}$. Then
$n^2 = 100x^2 + 20xy + y^2$
Note that the "magnitude of the hundreds numbers" of $n^2$ (as in the link) is $x^2 + 0.2xy$ if $xy \geq 5$ and $x^2$ otherwise. Since $y < 10$, we may characterize $x$ as the largest integer smaller or equal to the square root of the "magnitude of the hundreds number". That's step 1 in the link.
The congruence $y^2 \equiv y \mod 10$ has two solutions. Those are the options for the last digit in step 2.
Finally note that $(x + 1)x$ is $x^2 + x$. The "magnitude of the hundreds" is bigger than $x^2 + x$ if and only $y \geq 5$. This allows you to choose one and only one solution from step 2. That's what the last step in the method is saying.
