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Consider the following “magic” 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.
Working clockwise, and starting from the group of three with the numerically lowest external node (4,3,2 in this example), each solution can be described uniquely. For example, the above solution can be described by the set: 4,3,2; 6,2,1; 5,1,3.
It is possible to complete the ring with four different totals: 9, 10, 11, and 12. There are eight solutions in total.
By concatenating each group it is possible to form 9-digit strings; the maximum string for a 3-gon ring is 432621513.
Using the numbers 1 to 10, and depending on arrangements, it is possible to form 16- and 17-digit strings. What is the maximum 16-digit string for a “magic” 5-gon ring?
(see full solution here)
Before we go into details on the solution, let’s first structure the question in a way that’s easy for us to compute.
To construct the magic circle, be it for a 3-gon or a 5-gon ring, we can slice up the numbers into pairs – e.g. A => [1; 2], B => [3; 4], C => [5; 6], D => [7; 8], E => [9; 10] – and the problem becomes finding ways in which the numbers can be permuted such that:
- a0 is the smallest amongst a0, b0, c0, d0, and e0
- the sums of a0 + a1 + b1, b0 + b1 + c1, … are the same
First, we’ll find the different ways the numbers 1 to 10 can be permuted, and for each permutation slice the numbers into pairs:
(this makes use of a permute function defined in the Common.fs source file in the solution).
Then we need a function to sum a pair of number with the last element in the adjacent pair – e.g. a0 + a1 + b1:
For each permutation, we need to check:
- if a0 is the smallest amongst the head elements
- if the sums of the groups of 3 – i.e. a0 + a1 + b1, b0 + b1 + c1, etc. – are the same
This predicate function allows us to find arrangements that meet our criteria. All that’s left is to turn the result groups of 15 numbers into 16/17-digit strings and find the maximum (see full solution below).
Here’s the full solution:
This solution took 26s to execute on my machine.
Hi, I’m Yan. I’m an AWS Serverless Hero and I help companies go faster for less by adopting serverless technologies successfully.
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.
Skill up your serverless game with this hands-on workshop.
My 4-week Production-Ready Serverless online workshop is back!
This course takes you through building a production-ready serverless web application from testing, deployment, security, all the way through to observability. The motivation for this course is to give you hands-on experience building something with serverless technologies while giving you a broader view of the challenges you will face as the architecture matures and expands.
We will start at the basics and give you a firm introduction to Lambda and all the relevant concepts and service features (including the latest announcements in 2020). And then gradually ramping up and cover a wide array of topics such as API security, testing strategies, CI/CD, secret management, and operational best practices for monitoring and troubleshooting.
If you enrol now you can also get 15% OFF with the promo code “yanprs15”.
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!
Here is a complete list of all my posts on serverless and AWS Lambda. In the meantime, here are a few of my most popular blog posts.
- All you need to know about caching for serverless applications
- Lambda optimization tip – enable HTTP keep-alive
- You are wrong about serverless and vendor lock-in
- You are thinking about serverless costs all wrong
- Just how expensive is the full AWS SDK?
- Check-list for going live with API Gateway and Lambda
- How to choose the right API Gateway auth method
- CloudFormation protip: use !Sub instead of !Join
- AWS Lambda – should you have few monolithic functions or many single-purposed functions?
- Guys, we’re doing pagination wrong
- Top 10 Serverless framework best practices
- How to break the “senior engineer” career ceiling
- My advice to junior developers