Euler Problem 6: Sum Square Difference

January 5, 2017
By

(This article was first published on The Devil is in the Data, and kindly contributed to R-bloggers)

Euler Problem 6 Definition

The sum of the squares of the first ten natural numbers is:

1^2 + 2^2 + \ldots + 10^2 = 385

The square of the sum of the first ten natural numbers is:

(1 + 2 + \ldots + 10)^2 = 552 = 3025

The difference between the sum of the squares of the first ten natural numbers and the square of the sum is 3025 - 385 = 2640. Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

Solution

This is a straightforward problem for vector processing capabilities in R.

answer <- sum(1:100)^2-sum((1:100)^2)

This problem can also be solved arithmetically. When Carl Friedrich Gauss (1777-1855) when he was a child his teacher challenged his students to add all numbers from 1 to 100. All of his friends struggled to add all the 100 numbers one by one but Carl completed the task in a few seconds.

The same principle applies to computing. The first solution is like Carl’s classmates who slavishly add all numbers. This solution is based on arithmetic progressions.

The sum of natural numbers can be expressed as:

\frac{n(n + 1)}{2}
The sum of the squares of the first n natural numbers is:

\frac{n(n+1)(2n+1)}{6}
With these two formulas, a fast solution without having to use loops can be written.

n <- 100
answer <- ((n*(n+1))/2)^2 - (n*(n+1)*(2*n+1))/6

The post Euler Problem 6: Sum Square Difference appeared first on The Devil is in the Data.

To leave a comment for the author, please follow the link and comment on their blog: The Devil is in the Data.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...



If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Comments are closed.

Sponsors

Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)