Problem 6: Sum square difference

This problem is fairly simple, however, it will allow us to explore our good friend recursion. We will use a combination of that and functional programming to solve this fella with relative ease...hopefully. Alright, enough yammering, let's get to it!

Discussion

The sum of the squares of the first ten natural numbers is: 1^2 + 2^2 + ... + 10^2 = 385
The square of the sum of the first ten natural numbers is: (1 + 2 + ... + 10)^2 = 55^2 = 3025
Hence the difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 − 385 = 2640.

Statement

Find the difference between the sum of the squares of the first n natural numbers and the square of the sum.

Video Version

If you like to watch rather than read, check out the video that accompanies this article. If not, keep reading!

Solution

As I mentioned earlier, I chose to use these two things:

1. Functional programming
2. Recursion

Before we get too deep into the discussion, I want to discuss recursion. If you are already familiar with the concept, feel free to skip over this part.

Recursion

Recursion is simply:

A function calling itself over and over again

It will call itself until one of two things happens:

1. We reach the call stack limit.
2. We define an exit value.

Example for this Problem

Let's say we want to find out the total of all the squared values up to a given value, i.e. 1^2 + 2^2....n^2. We can write a function that calls itself until a condition has been met. I like to call that the "ceiling," because it usually represents the highest value we don't want to exceed.

    // Our function takes in two values: 
    // our limiter (ceiling) and a total that we will return (inititally set at 0)
    function getSumSquares(ceiling, total = 0) {
        // check to see if we have reduced our ceiling to zero. If so...escape!
      if (ceiling === 0) {
        return total;
      }
        // if we still have more work to do, do the work
      let squared = (total += ceiling ** 2);
        // call yourself, but reduce our ceiling by one.
      return getSumSquares(ceiling - 1, total);
    }
    getSumSquares(10)

The function is going to call itself until our condition is met, in this case, ceiling === 0, hence the name recursion.

If you want more detail about recursion, check out my recursion article:

https://dev.to/codenutt/javascript-recursion-explained-in-4-minutes-26oa

Steps

The steps for this problem are fairly simple:

1. Calculate the sum of all squares up to n
2. Calculate the square of the summed values up to n
3. Calculate the difference between the two.

Solution

As I mentioned earlier, we will be composing our solution via functional programming. That means we will be creating three separate functions. The first one we already did in the discussion about recursion!

Sum of all Squares

    function getSumSquares(ceiling, total = 0) {
      if (ceiling === 0) {
        return total;
      }
      total += ceiling ** 2;
      return getSumSquares(ceiling - 1, total);
    }

Square of all Sums

    function getSquareSum(ceiling, total = 0) {
      if (ceiling === 0) {
        return total ** 2;
      }
      total += ceiling;
      return getSquareSum(ceiling - 1, total);
    }

Main Function

    function sumSquareDifference(n) {
      // total for sum of squares of the n natural numbers
      let sumOfSquares = getSumSquares(n);
      // total of square of the sum
      let squareOfSum = getSquareSum(n);
      // get difference between the two
      return squareOfSum - sumOfSquares;
    }

Altogether now

    function getSumSquares(ceiling, total = 0) {
      if (ceiling === 0) {
        return total;
      }
      total += ceiling ** 2;
      return getSumSquares(ceiling - 1, total);
    }

    function getSquareSum(ceiling, total = 0) {
      if (ceiling === 0) {
        return total ** 2;
      }
      total += ceiling;
      return getSquareSum(ceiling - 1, total);
    }

    function sumSquareDifference(n) {
      // total for sum of squares of the n natural numbers
      let sumOfSquares = getSumSquares(n);
      // total of square of the sum
      let squareOfSum = getSquareSum(n);
      // get difference between the two
      return squareOfSum - sumOfSquares;
    }

    let tenSum = sumSquareDifference(10);
    let hundoSum = sumSquareDifference(100);

Final Thoughts

Using those two methods, recursion, and functional programming, we have a nicely composed solution that is highly legible.

Like all things, this can be improved. If you have recommendations or improvements, throw down a comment and let me know!

As always, happy coding!

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Resources

This video is more specific to the event loop, but it covers what happens when the call stack is exceeded around the 7:00 mark.