You can become a serverless blackbelt. Enrol to my 4-week online workshop Production-Ready Serverless and gain hands-on experience building something from scratch using serverless technologies. At the end of the workshop, you should have a broader view of the challenges you will face as your serverless architecture matures and expands. You should also have a firm grasp on when serverless is a good fit for your system as well as common pitfalls you need to avoid. Sign up now and get 15% discount with the code yanprs15!
Let p(n) represent the number of different ways in which n coins can be separated into piles. For example, five coins can separated into piles in exactly seven different ways, so p(5)=7.
OOO O O
OO OO O
OO O O O
O O O O O
Find the least value of n for which p(n) is divisible by one million.
// the pentagonal numbers sequence, i.e. 1, 2, 5, 7, 12, 15, 22, ... let pentagonalNumbers = Seq.unfold (fun state -> Some(state, state+1)) 1 |> Seq.collect (fun n -> [n; -1*n]) |> Seq.map (fun n -> int(0.5 * double(n) * double(3 * n - 1))) // the coefficients sequence, i.e. +, +, -, -, +, +, -, -, ... let pCoefficients = Seq.unfold (fun state -> Some(state, -1*state)) 1 |> Seq.collect (fun n -> [n; n]) // cache results to improve performance let mutable cache = Array.init 100000 (fun n -> if n = 0 then 1I else 0I) // define the function p using the pentagonal numbers let rec p k = if cache.[k] <> 0I then cache.[k] else let pSeq = pentagonalNumbers |> Seq.map (fun n -> k - n) |> Seq.takeWhile (fun n -> n >= 0) |> Seq.map p let pk = pCoefficients |> Seq.zip pSeq |> Seq.sumBy (fun (pk, coe) -> pk * bigint(coe)) cache.[k] <- pk pk let answer = Seq.unfold (fun state -> Some(state, state+1)) 1 |> Seq.filter (fun k -> (p k) % 1000000I = 0I) |> Seq.head
Well, this one took some thought and some time to come up with a solution which runs under a minute! A brute force approach simply wouldn’t have worked here as p(k) becomes very large very quickly (p(200) = 3972999029388…). Thankfully, as the wiki page on partition points out, you can build a recurrence for the function p such that:
p(k) = p(k – 1) + p(k – 2) – p(k – 5) – p(k – 7) + p(k – 12) + p(k – 15) – p(k – 22) – …
using pentagonal numbers in the form of
for n running over positive and negative integers (n = 1, –1, 2, –2, 3, –3, …), generating the sequence 1, 2, 5, 7, 12, 15, 22, … The signs in the summation continue to alternate +, +, –, –, +, +, …
Using dynamic programming technique, I’m caching the value of p(k) as they’re generated and in doing so, enabled this solution to run in under 30 seconds.
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.
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!
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.
- 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?
- Many faced threats to Serverless security
- We can do better than percentile latencies
- Yubl’s road to Serverless
- AWS Lambda – should you have few monolithic functions or many single-purposed functions?
- AWS Lambda – compare coldstart time with different languages, memory and code sizes
- Guys, we’re doing pagination wrong
- Top 10 Serverless framework best practices