Short Square Sum
Every positive integer can be expressed as a sum of perfect squares. For example, 13 can be expressed as , but can also be ineffeciently expressed as .
What is the least number of perfect squares (including repeated ones) that must be added together to equal 197?
Bonus : Try to create a proof that determines the solution for any positive integer.
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1 solution
We can immediately elimate 1, because 197 is not a perfect square itself, so it cannot br expressed with just one perfect square.
The solution is greater than 1.
Let's try to eliminate 2.
Using the basic odd-even rules, we know that:
odd + even = odd
odd + odd = even
even + even = even
Since 197 is odd, if 197 can be expressed with just 2 squares, one must be even, and one must be odd, since that is the only way to achieve an odd sum.
Let's go through every perfect square and determine what value would need to be added to it in order to achive 197.
4 + 1 9 3 : Second value is not a perfect square.
9 + 1 8 8 : Second value is not a perfect square.
1 6 + 1 8 1 : Second value is not a perfect square.
...
1 9 6 + 1 : Correct!
Because an accurate solution exists that only uses 2 perfect squares,
The least number of perfect squares that must be added together to achieve 197 is 2.