## Rounded Roots

#### Problem

A special number machine will accept any whole number, find its square root and then output the result rounded off to the nearest whole number.

For example,

Input | Output | |||

32 | 32 5.6568542... | 6 |

If the output is 6, one possible input is 32, but which other numbers could have gone into the machine to give an output of 6?

#### Solution

5.5^{2} = 30.25, so 30 5.5 and 6.5^{2} = 42.25, so 43 > 6.5.

Hence the solution set is 31, 32, 33, ..., 42

Find the solution sets for each of the integers from 1 to 10.

Is there a rule connecting the output with the solution set?

Problem ID: 47 (May 2001) Difficulty: 2 Star