----------------------------------------------------------------------------- -- | -- Module : Data.SBV.Examples.Puzzles.Coins -- Copyright : (c) Levent Erkok -- License : BSD3 -- Maintainer : erkokl@gmail.com -- Stability : experimental -- -- Solves the following puzzle: -- -- @ -- You and a friend pass by a standard coin operated vending machine and you decide to get a candy bar. -- The price is US $0.95, but after checking your pockets you only have a dollar (US $1) and the machine -- only takes coins. You turn to your friend and have this conversation: -- you: Hey, do you have change for a dollar? -- friend: Let's see. I have 6 US coins but, although they add up to a US $1.15, I can't break a dollar. -- you: Huh? Can you make change for half a dollar? -- friend: No. -- you: How about a quarter? -- friend: Nope, and before you ask I cant make change for a dime or nickel either. -- you: Really? and these six coins are all US government coins currently in production? -- friend: Yes. -- you: Well can you just put your coins into the vending machine and buy me a candy bar, and I'll pay you back? -- friend: Sorry, I would like to but I cant with the coins I have. -- What coins are your friend holding? -- @ -- -- To be fair, the problem has no solution /mathematically/. But there is a solution when one takes into account that -- vending machines typically do not take the 50 cent coins! -- ----------------------------------------------------------------------------- module Data.SBV.Examples.Puzzles.Coins where import Data.SBV -- | We will represent coins with 16-bit words (more than enough precision for coins). type Coin = SWord16 -- | Create a coin. The argument Int argument just used for naming the coin. Note that -- we constrain the value to be one of the valid U.S. coin values as we create it. mkCoin :: Int -> Symbolic Coin mkCoin i = do c <- exists $ 'c' : show i constrain $ bAny (.== c) [1, 5, 10, 25, 50, 100] return c -- | Return all combinations of a sequence of values. combinations :: [a] -> [[a]] combinations coins = concat [combs i coins | i <- [1 .. length coins]] where combs 0 _ = [[]] combs _ [] = [] combs k (x:xs) = map (x:) (combs (k-1) xs) ++ combs k xs -- | Constraint 1: Cannot make change for a dollar. c1 :: [Coin] -> SBool c1 xs = sum xs ./= 100 -- | Constraint 2: Cannot make change for half a dollar. c2 :: [Coin] -> SBool c2 xs = sum xs ./= 50 -- | Constraint 3: Cannot make change for a quarter. c3 :: [Coin] -> SBool c3 xs = sum xs ./= 25 -- | Constraint 4: Cannot make change for a dime. c4 :: [Coin] -> SBool c4 xs = sum xs ./= 10 -- | Constraint 5: Cannot make change for a nickel c5 :: [Coin] -> SBool c5 xs = sum xs ./= 5 -- | Constraint 6: Cannot buy the candy either. Here's where we need to have the extra knowledge -- that the vending machines do not take 50 cent coins. c6 :: [Coin] -> SBool c6 xs = sum (map val xs) ./= 95 where val x = ite (x .== 50) 0 x -- | Solve the puzzle. We have: -- -- >>> puzzle -- Satisfiable. Model: -- c1 = 50 :: SWord16 -- c2 = 25 :: SWord16 -- c3 = 10 :: SWord16 -- c4 = 10 :: SWord16 -- c5 = 10 :: SWord16 -- c6 = 10 :: SWord16 -- -- i.e., your friend has 4 dimes, a quarter, and a half dollar. puzzle :: IO SatResult puzzle = sat $ do cs <- mapM mkCoin [1..6] -- Assert each of the constraints for all combinations that has -- at least two coins (to make change) mapM_ constrain [c s | s <- combinations cs, length s >= 2, c <- [c1, c2, c3, c4, c5, c6]] -- the following constraint is not necessary for solving the puzzle -- however, it makes sure that the solution comes in decreasing value of coins, -- thus allowing the above test to succeed regardless of the solver used. constrain $ bAnd $ zipWith (.>=) cs (tail cs) -- assert that the sum must be 115 cents. return $ sum cs .== 115