import Data.Packed
import Data.Maybe

data Car = Car { airparts :: Int, chambers :: [Chamber] }
data Chamber = MainChamber { upperPipe, lowerPipe :: Pipe } | AuxChamber { upperPipe, lowerPipe :: Pipe }
data Pipe = Pipe { sections :: [Section] }
data Section = Section { tank :: Int }

data Fuel = Matrix Int

-- Cars
-- 2211 220 0 0 1 0 2210 0 1 0 0 2200 1 1 0 2210 0 1 0 1 1 2210 0 1 0 1 1 1 0 2211 0 1 0 1 0 1 1 2210 1 0 1 0 0 1 0 220 0 1 0
-- 22 111001101111010 22 010010111 22 011010 22 1001011
-- 2211 1 0 0 220 1 0 1 1 220 0 1 0 1 0 220 1 0 0 220 0 1 1 1 0 220 1 1 0 220 1 21110 220 1 20
-- 2210 220 0 0 0 220 1 0 1 1 220 1 0 1 0 0 220 0 1 1 220 1 1 1 1 0 220 0 1 0

-- 220 220 0 0 0 1 1 0 220 1 0 1 0 0 1 0
-- [ ([0,0],0,[1]), ([1,1],0,[0]) ]

-- 1 220 0 0 0 1 0 : 1 chamber, 1 tank
-- [([0,0],0,[0])]

-- 1 220 0 0 0 1 1 0: In the car, tank 1 is not properly connected to tank 0
-- [([0,0],0,[1])]

-- 1 220 0 0 0 1 1 1: In the car, tank 2 is not properly connected to tank 0
-- 1 220 0 0 0 1 1 2: In the car, tank 3 is not properly connected to tank 0
-- 1 220 0 0 0 1 22000: In the car, tank 4 is not properly connected to tank 0

-- correct fuels:
--  1 1 1 220: for 1 tanks, using 1 ingredients of air, checking reaction chamber 0
--  220 1 1 1 1 1 1 1: for 2 tanks, using 1 ingredients of air, checking reaction chamber 0, checking reaction chamber 1, surplus of ingredient 1 in lower pipe when air consists of ingredient 1 only
--    [[2],[1]]
-- 1111: for 1 tanks, using 1 ingredients of air, checking reaction chamber 0, first fuel component must increase

-- 1110: for 1 tanks, using 1 ingredients of air, check fuel for tank 0, c_{1,1} must be >= 1
-- 220111011: for 2 tanks, using 1 ingredients of air, check fuel for tank 1, c_{1,1} must be >= 1
-- 220111111: for 2 tanks, using 1 ingredients of air, check fuel for tank 1, c_{1,1} must be >= 1
-- 2201 1111: for 2 tanks, using 1 ingredients of air, dimension mismatch
-- 10: for 1 tanks, using 0 ingredients of air, dimension mismatch
-- 11: for 1 tanks, using 1 ingredients of air, dimension mismatch
-- 11011000000: for 1 tanks, using 1 ingredients of air, dimension mismatch
-- 111220000: for 1 tanks, using 1 ingredients of air, fuel coefficients missing for tank 1
-- 1111220000: for 1 tanks, using 1 ingredients of air, fuel coefficients missing for tank 1
-- 1220: for 1 tanks, using 2 ingredients of air, dimension mismatch
-- 12210: for 1 tanks, using 3 ingredients of air, dimension mismatch
-- 12211: for 1 tanks, using 4 ingredients of air, dimension mismatch
-- 12222000: for 1 tanks, using 6 ingredients of air, dimension mismatch
-- 12222001: for 1 tanks, using 7 ingredients of air, dimension mismatch
-- 1222201: for 1 tanks, using 9 ingredients of air, dimension mismatch
-- 122221: for 1 tanks, using 15 ingredients of air, dimension mismatch
-- 2200: for 2 tanks, using 0 ingredients of air, dimension mismatch
-- 2201: for 2 tanks, using 1 ingredients of air, dimension mismatch
-- 220220: for 2 tanks, using 2 ingredients of air, dimension mismatch
-- 2202222022: for 2 tanks, using 14 ingredients of air, dimension mismatch
-- 22100: for 3 tanks, using 0 ingredients of air, dimension mismatch
-- 22101: for 3 tanks, using 1 ingredients of air, dimension mismatch
-- 2210220: for 3 tanks, using 2 ingredients of air, dimension mismatch
-- 22102210: for 3 tanks, using 3 ingredients of air, dimension mismatch
-- 22110: for 4 tanks, using 0 ingredients of air, dimension mismatch
-- 22111: for 4 tanks, using 1 ingredients of air, dimension mismatch
-- 22120: for 5 tanks, using 0 ingredients of air, dimension mismatch
-- 22121: for 5 tanks, using 1 ingredients of air, dimension mismatch
-- 22220001: for 6 tanks, using 1 ingredients of air, dimension mismatch
-- 22220011: for 7 tanks, using 1 ingredients of air, dimension mismatch
-- 22220101: for 9 tanks, using 1 ingredients of air, dimension mismatch
-- 22220111: for 10 tanks, using 1 ingredients of air, dimension mismatch
-- 222210001: for 16 tanks, using 0 ingredients of air, dimension mismatch
-- 2222100011: for 16 tanks, using 1 ingredients of air, dimension mismatch
-- 222210010: for 18 tanks, using 0 ingredients of air, dimension mismatch
-- 2222100101: for 18 tanks, using 1 ingredients of air, dimension mismatch
-- 222210011: for 19 tanks, using 0 ingredients of air, dimension mismatch
-- 2222100111: for 19 tanks, using 1 ingredients of air, dimension mismatch


-- Numbers
--  0: 0
--  1: 10
--  2: 11
--  3: 12
--  4: 22000
--  5: 22001
--  6: 22002
--  7: 22010
--  8: 22011
--  9: 22012
-- 10: 22020
-- 11: 22021
-- 12: 22022

-- List Lengths:
--  0: 0
--  1: 1
--  2: 220
--  3: 2210
--  4: 2211
--  5: 2212
--  6: 2222000
--  7: 2222001
--  8: 2222002
--  9: 2222010
-- 10: 2222011
-- 11: 2222012
-- 12: 2222020
-- 13: 2222021
-- 14: 2222022
-- 15: 222210000
-- 16: 222210001
-- 17: 222210002
-- 18: 222210010
-- 19: 222210011
-- 20: 222210012
-- 21: 222210020
-- 22: 222210021
-- 23: 222210022
-- 24: 222210100
-- 25: 222210101
-- 26: 222210102
-- 27: 222210110
-- 28: 222210111
-- 29: 222210112
-- 30: 222210120
-- 31: 222210121
-- 32: 222210122
-- 33: 222210200
-- 34: 222210201
-- 35: 222210202
-- 36: 222210210
-- 37: 222210211
-- 38: 222210212
-- 39: 222210220
-- 40: 222210221
-- 41: 222210222