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Topic: Deck of Cards (Read 482 times) |
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John_Gaughan
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Deck of Cards
« on: Mar 15th, 2005, 12:26pm » |
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Does cutting a deck of cards after shuffling alter the randomness of the deck?
Common sense tells me it is just a way to foil stacking the deck, but I have never heard a satisfactory explanation either way.
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Grimbal
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Re: Deck of Cards
« Reply #1 on: Mar 15th, 2005, 2:31pm » |
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At least cutting it many times won't make it more "cut". 
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rmsgrey
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Re: Deck of Cards
« Reply #2 on: Mar 15th, 2005, 2:40pm » |
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Certainly for a game like Bridge, where the entire deck is dealt out, all cutting does is rotate the hands between the players - which is obviously only useful in cases of (suspected) foul play.
If only part of the deck is being used, then cutting, which exposes the central portion of the deck may well improve the randomness of the portion in use - assuming that any defective randomness is liable to be localised at the extremes of the deck.
There are a number of magic tricks built on the fact that cutting the deck doesn't change the cyclic order - so if you glimpse the bottom card then place it on top of the chosen card, repeated cutting leaves the same card just above the chosen card.
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Icarus
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Re: Deck of Cards
« Reply #3 on: Mar 15th, 2005, 6:14pm » |
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If you shuffle a lot, you may notice that quite often one card tends to stay at the top. This happens unintentionally because we tend to split the deck in the same way each time we shuffle. Also, when shuffling, cards are likely to start coming from the same side each time, simply by habit. If the person shuffling habitually puts the same part of the deck in the same hand when shuffling, it is likely that the top card, the bottom card, or both will be unchanged after the shuffle (the more practiced shufflers can do this on purpose as well).
The card deck is cut after shuffling to get these static cards off the top and bottom and place them at some random location in the middle of the deck.
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Noke Lieu
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Re: Deck of Cards
« Reply #4 on: Mar 15th, 2005, 7:51pm » |
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I seem to remember that for proper shuffling, there is a limit to the randomness. And that limit is seven. As in if you shuffle an ordered deck of cards seven distinct times, then the level of randomness doesn't increase with subsequent shuffles.
The cut, to my humble brain, might be regarded as a particularly inept shuffle.
Maybe this will help- didn't read thorugh it properly myself.
here
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| « Last Edit: Mar 15th, 2005, 7:52pm by Noke Lieu » |
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John_Gaughan
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Re: Deck of Cards
« Reply #5 on: Mar 16th, 2005, 7:06am » |
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After losing some money playing Blackjack in a casino recently I started to look more closely at the mechanics of decks of cards. Through experimentation, I noticed what Icarus said -- a dealer may have a tendancy to shuffle cards a certain way, leaving some cards at or near one end of the deck.
For example, I split the deck. I put the bottom half in my left hand, top half in my right hand. If I start the shuffle with the cards in my left hand and end it with the ones in my right hand, the cards at the ends of the deck tend to stay at one end of the deck. If I reverse this by starting the shuffle with cards in my right hand and ending with my left, I essentially move cards in a cycle -- cards at the ends move toward the middle, cards in the middle move toward the ends.
My wife and I like to play a solitaire game called Picture Gallery. It is a two-deck game that entails arranging cards in a specific order by suit. The details are irrelevant right now. I also like to play Braid, which in the end leaves cards in a similar arrangement (also a two deck game). I find that it takes a lot of shuffles after playing these games to remove clumps of cards that are in a specific order. I may find clumps of eight or nine face cards in a row, for example.
Anyway, I was curious what effect cutting the deck might have on situation such as these. I think y'all answered this for me about as well as possible.
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| « Last Edit: Mar 16th, 2005, 7:06am by John_Gaughan » |
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rmsgrey
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Re: Deck of Cards
« Reply #6 on: Mar 16th, 2005, 8:26am » |
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As an additional consideration, you could try different types of shuffle - I habitually use an overhand shuffle, mixed with a "pile shuffle" - dealing the pack into some number of piles (typically 5) to forcibly break up runs.
In extreme cases, I also swirl the cards around for a while before picking them up.
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Sjoerd Job Postmus
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Re: Deck of Cards
« Reply #7 on: Mar 16th, 2005, 11:09am » |
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Affecting the randomness depends on where you cut it.
If you cut it inside a sequense, you make the set a bit more random.
If you cut it inside a random part, you leave the set as random as it is, or you make it a bit more ordered.
But, when you talk about shuffling, there are several methods.
1/ The 'inexperienced' way. You grab the deck with your righthand, and have your lefthand open. You then move your right hand towards your left hand, and slightly release the deck, so only the top few cards slide off. Then, you repeat this, until your right hand is empty. Sometimes you 'throw' groups of 8 or so at a time. Not good. Even some experienced players shuffle like that.
2/ The really old way. You throw the cards at a table, at random, and you just try to cover as much of the table with that, effictively splitting/merging cards as you go. Works real well, but a pain to get the cards back together.
3/ The experienced way(need proper cards). You split the deck in two. (not accurately), then you just try to push these two decks into eachother. Works real well, but hard to do.
4/ The other experienced way. You split the deck in two, and put half a deck in each hand. You then bend/release them, so they fall approx one-to-one inside eachother. If you do this perfectly without fail, you'd better shuffle approx 5 or 6 times at most. When done perfectly, ater 8 times you have the deck you started with. Not desirable. Some prefer to do this in the air, others prefer to do it so it stacks up on the table.
What other ways are there?
By the way, what's the proper blackjack set? All cards ( minus jokers ), or 7 through Ace?
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Icarus
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Re: Deck of Cards
« Reply #8 on: Mar 16th, 2005, 3:56pm » |
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Sjoerd - Your 4th method is called a "Riffle Shuffle". Only the very talented and practiced are able to do a "perfect" riffle shuffle, where the cards coming from each side interweave 1 by 1. How many cycles it takes to return the deck to its original state depends on whether you do an "in-shuffle", where the top card is unchanged by shuffling (much as I described in my previous post), or an out-shuffle (where the top card becomes the 2nd card after the shuffle, because the card from the other half ends up on top). An in-shuffle takes only 8 shuffles to restore the order of the deck. An out-shuffle takes longer (13 shuffles, if I recall correctly). A few magicians have managed to learn to do a perfect riffle exactly for this reason. Restoring the deck to its original order by shuffling is unexpected by the audience, and can be used in a number of tricks.
However, most people are not capable of doing a perfect riffle shuffle, nor are they desirous of doing one, exactly for the reason that perfect riffle shuffles are not random. Instead, two or three cards will sometimes fall from one side between adjacent cards from the other. It is this randomness that riffle shuffles depend on. According to the analysis Noke linked to, it takes about 7 shuffles by an average dealer to produce randomness.
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rmsgrey
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Re: Deck of Cards
« Reply #9 on: Mar 17th, 2005, 7:15am » |
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For Sjoerd Job Postmus' types of shuffling:
1) is the Overhand shuffle I favour - it tends to blur small-scale structures in the deck, and can reverse the relative order of neighbouring cards much faster than repeated riffling - since I do it by taking one large-ish chunk (1/3 to 1/2 the deck) then a large number of small packets (typically 1-4 cards) it also produces a large scale cycling of ends to middles.
2) I don't know of any official name for - though it's closely related to the "game" "52-card pickup" (what you play when the local bully throws your cards on the floor)
3) this closely resembles the riffle shuffle in effect.
4) As Icarus noted, this is the riffle shuffle. A "perfect" riffle shuffle - interleaving the two exact halves of the deck so every second card comes from the same half - is also known as a "faro shuffle". According to Martin Gardner (Mathematical Carnival, chapter 10) it takes a massive 52 in-faroes to restore a deck (or 8 out-faroes). A riffle shuffle where you bend the cards significantly (as usual when performing it in midair) is called a bridge shuffle.
Another model for "perfection" in riffle shuffling (the one used in the analysis Noke linked to) is that you cut the deck approximately in half (actual cut point determined by a binomial distribution - so possible to get 52-0 split, but (very) unlikely) and then drop the cards so that the probability of dropping from each pile is proportional to the current size of the pile. Using that model, the analysis concluded that one measure of shuffling gave "good enough" randomness after somewhere between 5 and 11 shuffles (so 7 is a reasonable compromise) while a second (expected time to separate all histories - so for each pair of cards there has been at least one shuffle where the two cards weren't in the same "half" of the pack as each other) gives a number of shuffles around 11.7 - or 12 if you are actually doing it...
Also according to the linked analysis, riffle shuffles are never truly randomising in the sense of producing a uniform distribution over possible card orders beause they always have a better than average chance of preserving the original order.
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