Project Euler – Problem 18 Solution

You can become a serverless blackbelt. Enrol in my course Learn you some Lambda best practice for great good! and learn best practices for performance, cost, security, resilience, observability and scalability. By the end of this course, you should be able to make informed decisions on which AWS service to use with Lambda and how to build highly scalable, resilient and cost efficient serverless applications.

Problem

By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top to bottom is 23.

image

 

That is, 3 + 7 + 4 + 9 = 23.

Find the maximum total from top to bottom of the triangle below:

image

 

NOTE: As there are only 16384 routes, it is possible to solve this problem by trying every route. However, Problem 67, is the same challenge with a triangle containing one-hundred rows; it cannot be solved by brute force, and requires a clever method! ;o)

Solution

With the warning of problem 67 in mind, I set out to tackle this problem with a somewhat efficient solution which will also work for problem 67. To start off, let’s look at the simple example, as you traverse down the triangle from the top you have two options at every turn – go left, or go right. From the third row onwards, some of the paths will end up in the same place in the triangle:

image

From the above image you can see that the path 3-7-4 and 3-4-4 both exit the mini triangle at the same location but carrying two different totals – green = 14, red = 11. If you were to continue down the triangle from this point on, regardless of whether you’re going to turn left or right in the next move, the max total will surely have to come from the path which has generated the highest total so far.

If we apply the same logic to the entire triangle we can build up a triangle of totals for the raw triangle where every element represents the max total achievable by any path that has lead to it:

image

Let’s call this triangle of totals T, and the original triangle R, we have:

image image

You can see that T(2,1) differed from T(2,0) and T(2,2) in that it required us to find the greater between T(1,0) and T(1,1). For simplicity sake, let’s say that for a given row n, in T (where n >= 2) we have:

image

The max total from top to bottom of the triangle R is equal to the max total in the last row of the triangle T. Seeing as we’re only interested in the max total rather than the triangle T itself, my solution below only returns the last row of T which is all we need to solve the problem:


Liked this article? Support me on Patreon and get direct help from me via a private Slack channel or 1-2-1 mentoring.
Subscribe to my newsletter


Hi, I’m Yan. I’m an AWS Serverless Hero and the author of Production-Ready Serverless.

I specialise in rapidly transitioning teams to serverless and building production-ready services on AWS.

Are you struggling with serverless or need guidance on best practices? Do you want someone to review your architecture and help you avoid costly mistakes down the line? Whatever the case, I’m here to help.

Hire me.


Check out my new podcast Real-World Serverless where I talk with engineers who are building amazing things with serverless technologies and discuss the real-world use cases and challenges they face. If you’re interested in what people are actually doing with serverless and what it’s really like to be working with serverless day-to-day, then this is the podcast for you.


Check out my new course, Learn you some Lambda best practice for great good! In this course, you will learn best practices for working with AWS Lambda in terms of performance, cost, security, scalability, resilience and observability. We will also cover latest features from re:Invent 2019 such as Provisioned Concurrency and Lambda Destinations. Enrol now and start learning!


Check out my video course, Complete Guide to AWS Step Functions. In this course, we’ll cover everything you need to know to use AWS Step Functions service effectively. There is something for everyone from beginners to more advanced users looking for design patterns and best practices. Enrol now and start learning!


Are you working with Serverless and looking for expert training to level-up your skills? Or are you looking for a solid foundation to start from? Look no further, register for my Production-Ready Serverless workshop to learn how to build production-grade Serverless applications!

Find a workshop near you