handle colinear case

solutions/src/2024/13.hs

@@ -1,4 +1,4 @@
-{-# Language QuasiQuotes #-}
+{-# Language QuasiQuotes, NumDecimals #-}
 {-|
 Module      : Main
 Description : Day 13 solution
@@ -37,32 +37,49 @@ Prize: X=18641, Y=10279
 module Main (main) where
 
 import Advent (format)
+import GHC.Num (integerGcde)
+import Debug.Trace
 
 -- | >>> :main
 -- 29877
 -- 99423413811305
 main :: IO ()
 main =
  do input <- [format|2024 13
-      (Button A: X%+%u, Y%+%u%n
-      Button B: X%+%u, Y%+%u%n
-      Prize: X=%u, Y=%u%n)&%n|]
-    print (sum (map (cost              0) input))
-    print (sum (map (cost 10000000000000) input))
+      (Button A: X%+%lu, Y%+%lu%n
+      Button B: X%+%lu, Y%+%lu%n
+      Prize: X=%lu, Y=%lu%n)&%n|]
+    print (sum (map (cost    0) input))
+    print (sum (map (cost 1e13) input))
 
-cost :: Int -> (Int, Int, Int, Int, Int, Int) -> Integer
+cost :: Integer -> (Integer, Integer, Integer, Integer, Integer, Integer) -> Integer
 cost extra (ax, ay, bx, by, x, y)
-  | det == 0 = error "colinearity not supported"
-  | a >= 0, b >= 0, ar == 0, br == 0 = 3 * a + b
+  | det == 0 = colinear ax ay bx by x y
+  | (a, 0) <- (by * x' - bx * y') `quotRem` det, a >= 0
+  , (b, 0) <- (ax * y' - ay * x') `quotRem` det, b >= 0
+  = 3 * a + b
   | otherwise = 0
   where
-    ax' = toInteger ax
-    ay' = toInteger ay
-    bx' = toInteger bx
-    by' = toInteger by
-    x'  = toInteger (x + extra)
-    y'  = toInteger (y + extra)
-
-    det = ax' * by' - ay' * bx'
-    (a, ar) = (by' * x' - bx' * y') `quotRem` det
-    (b, br) = (ax' * y' - ay' * x') `quotRem` det
+    det = ax * by - ay * bx
+    x'  = x + extra
+    y'  = y + extra
+
+colinear :: Integer -> Integer -> Integer -> Integer -> Integer -> Integer -> Integer
+colinear ax ay bx by x y
+  | ax*by == bx*ay          -- buttons are colinear with each other
+  , ax*y  == x*ay           -- output is colinear with buttons
+  , let (d, e, f) = integerGcde ax bx
+  , (h, 0) <- x `quotRem` d -- target x is a multiple of the gcd of the two button +x
+  , let a = e * h -- prototypical (but potentially negative) solution
+  , let b = f * h -- prototypical (but potentially negative) solution
+  , let u = ax `quot` d
+  , let v = - (bx `quot` d)
+  , let klo = a `divCeil` v
+  , let khi = b `div` u
+  , klo <= khi
+  = minimum [3 * (a - k * v) + (b - k * u) | k <- [klo, khi]] -- the relation is linear so one of the extremes is the answer
+  | otherwise = 0
+
+-- Division that rounds up
+divCeil :: Integral a => a -> a -> a
+x `divCeil` y = (x + y - 1) `div` y