Mind-Boggling Card Trick

September 4, 2018

Today’s exercise is a mind-boggling card trick:

Create a pack of 52 cards, half red, half black, and shuffle them. With the cards face down, turn over the top card. If it is black, add the next card, unseen, to the black stack; if it is red, add the next card, unseen, to the red stack. Add the original top card to the discard stack. Repeat these steps for all the cards in the pack. Now, randomly choose a number less than the size of the smaller of the black and red stacks, choose that many cards randomly from each of the two stacks, and swap those randomly-chosen cards from one stack to the other. The number of black cards in the black stack will equal the number of red cards in the red stack.

Your taks is to write a program to simulate the card trick and confirm that the number of black cards in the black stack equals the number of red cards in the red stack. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.


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4 Responses to “Mind-Boggling Card Trick”

  1. Informatmiago said

    This is indeed a magic trick:

    The first phase is not random at all, because the operation ensures that there are the same number of black in the black stack as there are reds in the red stack. Discading the seen cards, ensures that there will remain the exact same number of cards (either of the same color, or the opposite color) to be moved to the other stack.

    The second part is just for the spectacle, since it’s an operation that does not change that property (if two cards of the same color are exchanged, then no change; if cards of different colors are exchanged, then either it increments both counts (if the right color leaves each stack), or it decrements both counts (if red leaves the red stack and black leaves the black stack). Randomness does not matter, you could choose the cards you want, as long as you exchange the same number from both stacks :-)

  2. Rutger said

    This code almost reads like the assignment to me ;)

    from random import shuffle
    from random import randint
    deck = 26*['black'] + 26*['red']
    red_stack = []
    black_stack = []
    while deck:
    	item = deck.pop()
    	if item == 'black':
    swap = randint(1, min(len(black_stack), len(red_stack)))
    black_stack, red_stack = red_stack[:swap]+black_stack[swap:], black_stack[:swap]+red_stack[swap:]
  3. Globules said

    A Haskell version.

    import Control.Monad.Random (RandT, evalRandT, getRandomR, liftIO)
    import Data.Tuple.Extra (both)
    import Data.List (partition)
    import System.Random.Shuffle (shuffleM)
    import System.Random.TF.Gen (TFGen)
    import System.Random.TF.Init (initTFGen)
    import Text.Printf (PrintfArg, formatArg, formatString, printf)
    data Card = Black | Red deriving (Eq, Show)
    orderedDeck :: [Card]
    orderedDeck = replicate 26 Black ++ replicate 26 Red
    -- Look at the first card of the list.  If it's black add the next card to the
    -- black stack, otherwise add it to the red stack.  Discard the first card.
    -- Repeat this until the list has been exhausted.  Return the black and red
    -- stacks.  If the list contains an odd number of cards the last one will simply
    -- be discarded.
    makeStacks :: [Card] -> ([Card], [Card])
    makeStacks = both (map snd) . partition ((== Black) . fst) . pairs
      where pairs (x:y:zs) = (x, y) : pairs zs
            pairs _        = []
    -- The result of randomly swapping elements between two lists.  (We don't bother
    -- retaining the relative order of the original elements.)
    randomSwap :: ([a], [a]) -> RandC ([a], [a])
    randomSwap (xs, ys) = do n <- getRandomR (0, (length xs `min` length ys) - 1)
                             (xs1, xs2) <- splitShuffle n xs
                             (ys1, ys2) <- splitShuffle n ys
                             return (ys1 ++ xs2, xs1 ++ ys2)
      where splitShuffle n zs = splitAt n <$> shuffleM zs
    -- Simulate the card trick described in the exercise.
    cardTrick :: RandC ()
    cardTrick = do deck <- shuffleM orderedDeck
                   (blackStack, redStack) <- randomSwap (makeStacks deck)
                   printNumberOf Black blackStack
                   printNumberOf Red   redStack
    main :: IO ()
    main = initTFGen >>= evalRandT cardTrick
    ---------------------------------- Utilities ----------------------------------
    type RandC a = RandT TFGen IO a
    instance PrintfArg Card where
      formatArg c = formatString (show c)
    printNumberOf :: Card -> [Card] -> RandC ()
    printNumberOf c = liftIO . printf "%s cards in %s stack: %d\n" c c
                             . length . filter (== c)
    $ for i in $(seq 1 10); do ./cardtrick; done
    Black cards in Black stack: 4
    Red cards in Red stack: 4
    Black cards in Black stack: 6
    Red cards in Red stack: 6
    Black cards in Black stack: 8
    Red cards in Red stack: 8
    Black cards in Black stack: 5
    Red cards in Red stack: 5
    Black cards in Black stack: 9
    Red cards in Red stack: 9
    Black cards in Black stack: 10
    Red cards in Red stack: 10
    Black cards in Black stack: 6
    Red cards in Red stack: 6
    Black cards in Black stack: 7
    Red cards in Red stack: 7
    Black cards in Black stack: 6
    Red cards in Red stack: 6
    Black cards in Black stack: 6
    Red cards in Red stack: 6
  4. Daniel said

    Here’s a solution in Python.

    from collections import deque
    import random
    deck = [0] * 26 + [1] * 26
    stacks = [deque(), deque()]
    for top, bottom in zip(deck[0::2], deck[1::2]):
    for stack in stacks:
    min_len = min(len(stack) for stack in stacks)
    num_swaps = random.randint(0, min_len - 1)
    for idx in range(num_swaps):
    assert stacks[0].count(0) == stacks[1].count(1)

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