Sum of odd integers
Starting with 1, add up sequential odd integers and the sum will always be a perfect square and the number of integers added will be the square root. Ex. 1 + 3 + 5 + 7 = 16. Number of integers = 4.
With the speed of todays cpu's it should not be difficult to write a small repetative subtration routine. There are ways of extending the process to as many decimal places as you wish, but it does get a little more involved.
This was the method used by the first electronic calculators back in the 60's.