How many squares?
How many 4-digit numbers are perfect squares?
Age
14 to 16
Challenge level
Problem
How many four digit whole numbers are perfect squares?
This problem is taken from the World Mathematics Championships
Student Solutions
Answer: 68
The smallest integer with 4 digits is 1000, 1 more than 999 which only has 3 digits.
The largest integer with 4 digits is 9999, 1 less than 10 000 which has 5 digits.
10 000 = 100$^2$, so all integers whose squares have 4 digits must be less than 100.
30$^2$ = 900, 31$^2$ = 961 and 32$^2$ = 1024.
So all integers $n$ with 31$\lt n\le$99 have 4-digit squares.
So there are 99 $-$ 31 = 68 such numbers.