# Find the Sums of Perfect Squares in Java

## The challenge

The task is simply stated. Given an integer `n`

(`3 < n < 10<sup>9</sup>`

), find the length of the smallest list of *perfect squares* which add up to `n`

. Come up with the best algorithm you can; you’ll need it!

**Examples:**

`sum_of_squares(17) = 2`

17 = 16 + 1 (4 and 1 are perfect squares).`sum_of_squares(15) = 4`

15 = 9 + 4 + 1 + 1. There is no way to represent 15 as the sum of three perfect squares.`sum_of_squares(16) = 1`

16 itself is a perfect square.

**Time constraints:**

5 easy (sample) test cases: `n < 20`

5 harder test cases: `1000 < n < 15000`

5 maximally hard test cases: `5 * 1e8 < n < 1e9`

300 random maximally hard test cases: `1e8 < n < 1e9`

## The solution in Java code

Option 1:

```
import java.lang.Math;
import java.util.*;
public class SumOfSquares {
public static int nSquaresFor(int n) {
if(Math.sqrt(n)%1==0) return 1;
while(n % 4 == 0) n /= 4;
if(n % 8 == 7) return 4;
for(int i = 0; i*i < n; ++i) {
int a = n - i*i;
if(Math.sqrt(a)%1==0) return 2;
}
return 3;
}
}
```

Option 2:

```
public class SumOfSquares {
public static int nSquaresFor(int n) {
if (Math.sqrt(n) % 1 == 0) {
return 1;
}
for (int t = 1; t * t <= n; t++) {
if (Math.sqrt(n - t * t) % 1 == 0) {
return 2;
}
}
while (n % 4 == 0) {
n /= 4;
}
if(n%8 == 7) {return 4;}
return 3;
}
}
```

Option 3:

```
import java.util.ArrayList;
import java.util.List;
import java.util.Set;
import java.util.TreeSet;
import java.util.stream.Collectors;
import java.util.stream.IntStream;
public class SumOfSquares {
public static int nSquaresFor(int n) {
List<Integer> counts = new ArrayList<>();
int end = (int) Math.floor(Math.sqrt(n));
if (end*end == n){
return 1;
}
for (int start = 1; start < end; start++) {
int sum = start*start;
int s = n - start*start;
int count = 1;
while (sum != n) {
int p = (int) Math.floor(Math.sqrt(s));
sum += p * p;
s = n - sum;
count++;
}
counts.add(count);
}
return counts.stream().min(Integer::compareTo).get();
}
}
```

## Test cases to validate our solution

```
import org.junit.Test;
import static org.junit.Assert.assertEquals;
import org.junit.runners.JUnit4;
public class SolutionTest {
@Test
public void easyTests() {
assertEquals(4, SumOfSquares.nSquaresFor(15));
assertEquals(1, SumOfSquares.nSquaresFor(16));
assertEquals(2, SumOfSquares.nSquaresFor(17));
assertEquals(2, SumOfSquares.nSquaresFor(18));
assertEquals(3, SumOfSquares.nSquaresFor(19));
}
}
```