Advent of Code F# – Day 19

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:

  1
5   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:

  1           1
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  -->
 4          4

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