Sunday, September 20, 2009

On solving Project Euler - Part 1

For those of you who have never heard about it, Project Euler is a website publishing almost every week new programming and mathematical challenges. It has been almost one year since I discovered it, and I must tell that I have really become addicted to it.
I would recommend it to any programmer who do not hate mathematics, as it can be used for several purposes: challenging your mathematical and programming skills, learning new mathematics concepts, and it is especially a good practice for learning a new programming language. Of course, it won't help you to become better at designing software, but it might help help you to improve your algorithm writing skills.

As I said, both mathematical and programming skills are needed to solve Project Euler problems. While some of them might be solved with just a simple brute force algorithm, or on the opposite, with a clever formula, most of the problems will require to use both your mathematical and algorithm implementation skills.

On the listing page, the problems are ordered by publishing date. Usually, trying to solve them in this order is better, as concepts introduced in older problems are often reused in newer problems, and it will make them easier to solve than just starting from zero.
The listing can also be sorted by the difficulty of the problems (you will need to register), which is determined by the number of people who solved them: indeed, once you solved a problem, you can input your answer in a form and verify if it is correct. In the case it is, you will then be able to access the forum thread of this problem, where people usually post their solution. This is a very interesting feature of Project Euler, because you can see what are the other algorithms that can be used, in several programming languages.

We could also classify the problems in different categories based on how they can be solved, and what are the most efficient languages to solved them.

A first category could be the one of problems which can be solved by implementing the obvious algorithm transcribed from the problem itself. An important point that I didn't mention before is that all the problems have been designed according to a "one-minute rule", meaning that there always exist an algorithm that can solve the problem in less that one minute even on modest hardware. Although this rule will usually lead you to rethink your first (slow) brute-force algorithm in order to make your solution comply to it, some problems can just be brute-forced as I said before.

The problem 1 is a nice and simple example from this category:

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.

Find the sum of all the multiples of 3 or 5 below 1000.

An obvious brute-force algorithm in Ruby could be:

This simple algorithm comply with the one-minute rule, but might not scale well for bigger values, for which we might need a better algorithm or formula. In same same naive way, the following Clojure code gives the answer to the problem 6:

This problem can also be solved with the following Factor code:

We will see in the next posts the other categories of challenges available on Project Euler, and the way to solve them efficiently. Indeed, implementing brute-force algorithms is not the real fun of Project Euler, and is really not what you need to improve your problem solving skills!

As an exercise, you can try to solve the problem 187, which seem simple to solve with a naive brute-force algorithm, but there is little chance that it complies with the one-minute rule!
A more clever algorithm can solve this problem in seconds, and a more advanced one can solve it in an instant...

No comments:

Post a Comment