2023-24 part 2 in haskell

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diff --git a/2023/24-Never_Tell_Me_The_Odds/second.hs b/2023/24-Never_Tell_Me_The_Odds/second.hs
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+-- requires cabal install --lib megaparsec parser-combinators heap vector matrix
+module Main (main) where
+
+import           Control.Applicative.Permutations
+import           Control.Monad                    (void, when)
+import qualified Data.Char                        as C
+import           Data.Either
+import           Data.Functor
+import qualified Data.Heap                        as H
+import qualified Data.List                        as L
+import qualified Data.Map                         as M
+import qualified Data.Matrix                      as MTX
+import           Data.Maybe
+import           Data.Ratio
+import qualified Data.Set                         as S
+import qualified Data.Vector                      as V
+import qualified Data.Vector.Unboxed              as VU
+import           Data.Void                        (Void)
+import           Text.Megaparsec
+import           Text.Megaparsec.Char
+
+import           Debug.Trace
+
+exampleExpectedOutput = 47
+
+type Hail = (Rational, Rational, Rational, Rational, Rational, Rational)
+type Input = [Hail]
+
+type Parser = Parsec Void String
+
+parseNumber :: Parser Integer
+parseNumber = read <$> some (char '-' <|> digitChar) <* optional (char ',') <* optional hspace <* optional (char '@' <* hspace)
+
+parseHail :: Parser Hail
+parseHail = (,,,,,) <$> (fromInteger <$> parseNumber)
+                    <*> (fromInteger <$> parseNumber)
+                    <*> (fromInteger <$> parseNumber)
+                    <*> (fromInteger <$> parseNumber)
+                    <*> (fromInteger <$> parseNumber)
+                    <*> (fromInteger <$> parseNumber)
+
+parseInput' :: Parser Input
+parseInput' = some (parseHail <* eol) <* eof
+
+parseInput :: String -> IO Input
+parseInput filename = do
+  input <- readFile filename
+  case runParser parseInput' filename input of
+    Left bundle  -> error $ errorBundlePretty bundle
+    Right input' -> return input'
+
+-- rock in (x, y, z, a, b, c)
+-- with a known hail (x1, y1, z1, a1, b1, x1) we have
+-- x + a * t1 = x1 + a1 * t1
+-- => t1 = (x - x1) / (a1 - a) = (y - y1) / (b1 - b)
+-- => (x - x1)(b1 - b) = (y - y1)(a1 - a)
+-- => xb1 - xb - x1b1 + x1b = ya1 - ya - y1a1 + y1a
+-- => ya - xb = ya1 - y1a1 + y1a - xb1 + x1b1 - x1b
+-- With a second hail we also get:
+--    ya - xb = ya2 - y2a2 + y2a - xb2 + x2b2 - x2b
+-- By substracting the two equations, we get:
+-- => y(a1-a2) + a(y1-y2) - x(b1-b2) - b(x1-x2) = y1a1 - y2a2 - x1b1 + x2b2
+-- => a(y1-y2) + b(x2-x1) + x(b2-b1) + y(a1-a2) = y1a1 - y2a2 - x1b1 + x2b2
+-- We build 4 equations on this template and populate a matrix to triangularise
+-- and solve for x and y, then repeat with z.
+compute :: Input -> Rational
+compute input = x + y + z
+  where
+    (x1, y1, z1, a1, b1, c1)
+      :(x2, y2, z2, a2, b2, c2)
+      :(x3, y3, z3, a3, b3, c3)
+      :(x4, y4, z4, a4, b4, c4)
+      :(x5, y5, z5, a5, b5, c5)
+      :_ = input
+    (Right mxy) = MTX.rref $ MTX.fromList 4 5 [ y1-y2, x2-x1, b2-b1, a1-a2, y1*a1-y2*a2-x1*b1+x2*b2
+                                              , y1-y3, x3-x1, b3-b1, a1-a3, y1*a1-y3*a3-x1*b1+x3*b3
+                                              , y1-y4, x4-x1, b4-b1, a1-a4, y1*a1-y4*a4-x1*b1+x4*b4
+                                              , y1-y5, x5-x1, b5-b1, a1-a5, y1*a1-y5*a5-x1*b1+x5*b5 ]
+    (Right mxz) = MTX.rref $ MTX.fromList 4 5 [ z1-z2, x2-x1, c2-c1, a1-a2, z1*a1-z2*a2-x1*c1+x2*c2
+                                              , z1-z3, x3-x1, c3-c1, a1-a3, z1*a1-z3*a3-x1*c1+x3*c3
+                                              , z1-z4, x4-x1, c4-c1, a1-a4, z1*a1-z4*a4-x1*c1+x4*c4
+                                              , z1-z5, x5-x1, c5-c1, a1-a5, z1*a1-z5*a5-x1*c1+x5*c5 ]
+    y = mxy MTX.! (4, 5)
+    x = mxy MTX.! (3, 5)
+    z = mxz MTX.! (4, 5)
+
+main :: IO ()
+main = do
+  example <- parseInput "example"
+  let exampleOutput = compute example
+  when  (exampleOutput /= exampleExpectedOutput)  (error $ "example failed: got " ++ show exampleOutput ++ " instead of " ++ show exampleExpectedOutput)
+  input <- parseInput "input"
+  print $ compute input