**ps. look out for all my other solutions for Advent of Code challenges here.**

## Day 19

See details of the challenge here.

I initially approached today’s challenge as a dynamic programming exercise, but it quickly transpired that there’s a much better way to do it once I realised that part 1 is in fact the Josephus Problem and there’s a simple solution to it.

To understand the above, watch the YouTube video in the links section below.

### Part 2

Realizing the folly of their present-exchange rules, the Elves agree to instead

steal presents from the Elf directly across the circle. If two Elves are across

the circle, the one on the left (from the perspective of the stealer) is stolen

from. The other rules remain unchanged: Elves with no presents are removed from

the circle entirely, and the other elves move in slightly to keep the circle

evenly spaced.

For example, with five Elves (again numbered 1 to 5):

The Elves sit in a circle; Elf 1 goes first:

15 2 4 3

Elves 3 and 4 are across the circle; Elf 3’s present is stolen, being the one to

the left. Elf 3 leaves the circle, and the rest of the Elves move in:

11 5 2 --> 5 2 4 - 4

Elf 2 steals from the Elf directly across the circle, Elf 5:`1 1 -`

2--> 2 4 4

Next is Elf 4 who, choosing between Elves 1 and 2, steals from Elf 1:`- 2 2 -->`

44

Finally, Elf 2 steals from Elf 4:

2--> 2 -

So, with five Elves, the Elf that sits starting in position 2 gets all the

presents.

With the number of Elves given in your puzzle input, which Elf now gets all the

presents?

I’m not aware of this variation to the Josephus Problem, but I’d wager that there would be some pattern to the results similar to part 1, so I put together a dynamic programming solution to get some outputs. (ps. the solution is not good enough for the input as it’ll take too long to return)

With the help of this I can see a pattern emerging:

n : answer

1 : 1

2 : 2

3 : 3

4 : 1

5 : 2

6 : 3

7 : 5

8 : 7

9 : 9

10 : 1

11 : 2

12 : 3

13 : 4

14 : 5

15 : 6

16 : 7

17 : 8

18 : 9

19 : 11

20 : 13

21 : 15

22 : 17

23 : 19

24 : 21

25 : 23

26 : 25

27 : 27

28 : 1

29 : 2

30 : 3

…

- where
is a power of 3 then the answer is itself**n** - else
can be expressed as**n**where**m****+ l***m*is a power of 3 - where
(eg.**l <= m***n = 5 = 3 + 2*where*m*=*3*and*l*=*2*) then the answer is just**l** - else the answer is
(eg.**m + (l – m) * 2***n = 7 = 3 + 4*where m =*3*and l =*4*and*m + (l – m) * 2 = 5*)

## Links

- Day 19 challenge description
- The Josephus Problem – Numberphile (YouTube)
- Advent of Code 2015
- Solution for Day 18
- All my F# solutions for Advent of Code
- Github repo

**Yan Cui**

I’m an **AWS Serverless Hero** and the author of **Production-Ready Serverless**. I have run production workload at scale in AWS for nearly 10 years and I have been an architect or principal engineer with a variety of industries ranging from banking, e-commerce, sports streaming to mobile gaming. I currently work as an independent consultant focused on AWS and serverless.

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.

- Lambda optimization tip – enable HTTP keep-alive
- You are thinking about serverless costs all wrong
- Many faced threats to Serverless security
- We can do better than percentile latencies
- I’m afraid you’re thinking about AWS Lambda cold starts all wrong
- 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