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# Day 1

all / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15 / 16 / 17 / 18 / 19 / 20 / 21 / 22 / 23 / 24 / 25

So there's a simple-ish Haskell solution for these problems,

`tails` lets you separate out each item in a list with the list of items after it:

```ghci> tails [1,2,3,4]
[1:[2,3,4], 2:[3,4], 3:[4], 4:[]]```
```findPair :: [Int] -> Maybe Int
findPair xs = listToMaybe \$ do
x:ys <- tails xs
y    <- ys
guard (x + y == 2020)
pure (x*y)

findTriple :: [Int] -> Maybe Int
findTriple xs = listToMaybe \$ do
x:ys <- tails xs
y:zs <- tails ys
z    <- zs
guard (x + y + z == 2020)
pure (x*y*z)```

But this method is a little bit "extra", since we actually don't need to search all of `ys` for the proper sum...if we pick `x` as `500`, then we really only need to check if `1520` is a part of `ys`.

So we really only need to check for set inclusion:

```import qualified Data.IntSet as IS

findPair :: Int -> IS.IntSet -> Maybe Int
findPair goal xs = listToMaybe \$ do
x <- IS.toList xs
let y = goal - x
guard (y `IS.member` xs)
pure (x * y)```

And our first part will be `findPair 2020`!

You could even implement `findTriple` in terms of `findPair`, using `IS.split` to partition a set into all items smaller than and larger than a number. Splitting is a very efficient operation on a binary search tree like `IntSet`:

```findTriple :: Int -> IS.IntSet -> Maybe Int
findTriple goal xs = listToMaybe \$ do
x <- IS.toList xs
let (_, ys) = IS.split x xs
goal'   = goal - x
case findPair goal' ys of
Nothing -> empty
Just yz -> pure (x*yz)```

But hey...this recursive descent is kind of neat. We could write a general function to find any goal in any number of items!

```-- | Given a number n of items and a goal sum and a set of numbers to
-- pick from, finds the n numbers in the set that add to the goal sum.
knapsack
:: Int              -- ^ number of items n to pick
-> Int              -- ^ goal sum
-> IS.IntSet        -- ^ set of options
-> Maybe [Int]      -- ^ resulting n items that sum to the goal
knapsack 0 _    _  = Nothing
knapsack 1 goal xs
| goal `IS.member` xs = Just [goal]
| otherwise           = Nothing
knapsack n goal xs = listToMaybe \$ do
x <- IS.toList xs
let goal'   = goal - x
(_, ys) = IS.split x xs
case knapsack (n - 1) goal' ys of
Nothing -> empty
Just rs -> pure (x:rs)```

And so we have:

```part1 :: [Int] -> Maybe Int
part1 = knapsack 2 2020 . IS.fromList

part2 :: [Int] -> Maybe Int
part2 = knapsack 3 2020 . IS.fromList```

And we could go on, and on, and on!

Definitely very unnecessary, but it does shave my time on Part 2 down from around 2ms to around 20μs :)

Back to all reflections for 2020

## Day 1 Benchmarks

``````>> Day 01a
benchmarking...
time                 5.564 μs   (5.347 μs .. 5.859 μs)
0.987 R²   (0.979 R² .. 1.000 R²)
mean                 5.499 μs   (5.390 μs .. 5.783 μs)
std dev              546.8 ns   (238.7 ns .. 928.6 ns)
variance introduced by outliers: 87% (severely inflated)

* parsing and formatting times excluded

>> Day 01b
benchmarking...
time                 51.91 μs   (51.03 μs .. 53.43 μs)
0.988 R²   (0.978 R² .. 0.995 R²)
mean                 58.57 μs   (56.07 μs .. 61.01 μs)
std dev              9.320 μs   (8.111 μs .. 10.06 μs)
variance introduced by outliers: 93% (severely inflated)

* parsing and formatting times excluded
``````