Roots near 9
For how many integers 𑛠is the difference between √𑛠and 9 is less than 1?
Age
14 to 16
Challenge level
Problem
Given that $n$ is an integer, and the difference between $\sqrt n$ and $9$ is less than $1$, how many different possibilities are there for $n$?
This problem is taken from the World Mathematics Championships
Student Solutions
Answer: $35$ numbers
$\sqrt n $ is less than $1$ from $9$
$\sqrt n$ is between $8$ and $10$
$n$ is between $64$ and $100$ (but not $64$ or $100$)
$65,66,67,68,...,99$
$\underbrace{\underbrace{1, 2, 3, ..., 64}_{\text{64 numbers}}, 65, 66, 67, 68, ..., 99}_{\text{99 numbers}}$
$99-64=35$