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#### Problem

Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae:

The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties.

- The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).
- Each polygonal type: triangle (P
_{3,127}=8128), square (P_{4,91}=8281), and pentagonal (P_{5,44}=2882), is represented by a different number in the set.- This is the only set of 4-digit numbers with this property.
Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set.

#### Solution

(see full solution here),

The tricky thing here (at least for me) was to remember that the six 4-digit numbers have to come from different sets, but not necessarily in the order of P_{3}, P_{4}, … P_{8}. Once that is cleared up, the rest is fairly straight forward. In the solution linked above, I first created a set of functions for generating triangle, square, pentagonal, … octagonal numbers:

Since the question concerns only 4-digit numbers, so for efficiency sake let’s generate the desired 4 digit numbers ahead of time and safe them for later use:

The *is4digit *predicate function is self-explanatory. *naturalNumbers *is an infinite sequence of integers starting from 1, we use this sequence to generate the figurate numbers we need, but only keep those that are actually 4 digits.

So far so good, we have all the figurate numbers in an array where [0] => P_{3}, [1] => P_{4}, and so on.

Next, create permutations of the figurate numbers such that we exhaust all possible sequence of figurate numbers:

P_{3} => P_{4} => P_{5} => P_{6} => P_{7} => P_{8}

P_{3} => P_{4} => P_{6} => P_{7} => P_{8} => P_{5}

P_{4} => P_{3} => P_{5} => P_{6} => P_{7} => P_{8}

…

(P.S. the permute function here comes from the *Common.fs* source file in the solution)

To find the answer to the problem, we process each permutation to find our answer, take a moment to understand this code:

The *processPermutation *function processes one permutation of the figurate numbers and the termination conditions for the inner *loop* function are:

1. we have one number from each figurate number set and that the last 2 digits of the last number = first 2 digits of first number

2. we have one number from each figurate number set but last 2 digits of last number <> first 2 digits of first number (so close!)

3. one of the figurate number set in the sequence doesn’t contain a number whose first 2 digits = the last 2 digits of the number selected from the last figurate number set (short-circuited)

For each number in the current set of figurate numbers we build up a new predicate function – e.g. if x = 1282 then the predicate function would find 4-digit numbers whose first two digit = 82 – and use it to process the next set of figurate numbers in the sequence.

The *loop* function returns *int list option *where the *int list* represents the cyclic figurate numbers we’re looking for, so all that’s left is to unpack the option type and then sum the list.

This solution took 17ms to find the solution 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.

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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.

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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.

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Further reading

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