I do not know if this is true or not but yet another rumor came around to me one time that if you have a deck of cards and you were to shuffle it perfectly 8 times, the outcome would be how the deck started in the first place. Is this true?
no. but like all great lies it’s got a great little booger of truth to it.
for one thing, there’s no such thing as a “Perfect Shuffle”. but enough of details; let’s go straight away to the big picture. the fact of the matter is that if you have any set of data (pixels in a picture or cards in a deck), if you randomly mix it up, you will eventually start seeing some patterns repeat.
but, and this is the big but, the whole set of data wouldn’t magically reform itself.
Your right if you took 6.5 cards everytime you shuffled you’d end up where you stared . Good trick , I’d like to see it done :rolleyes:
Oh and also from dictionary.com
shuf·fle
5 :Games. To mix together (playing cards, tiles, or dominoes) so as to make a random order of arrangement.
So if you are moving the cards in a non-random manner you’re not really shuffling them .
There is such a thing as a perfect shuffle … many magicians can do it at will.
A perfect shuffle is far from random. However, there are two types: for an “in-shuffle” the original top card is the second card after the shuffle, while for an “out-shuffle” the original top card is stil the top card after the shuffle.
A 52-card deck returns to its original order after eight out-shuffles.
Well I’ll be a simians fathers brother .
Areyou sure about that, JonF? Last time I tried it, it took exactly thirteen perfect out-shuffles to put the deck in reverse order, and then (of course) another 13 to restore it.
No, I’m not a magician, I just did it one card at a time. I was bored.
Nope, that’s just what it said at one of the sites I found. haven’t done it myself.
Martin Gardner wrote about it in Scientific American many years ago. I don’t remember what he said.
Chronos, what amazes me is not that you had the time to “shuffle” a deck one card at a time. What amazes me is that you were able to verify when the deck was reassembled!
Wow, Chronos – much like a clock, you have too much time on your hands.
Perhaps what they meant was that, if shuffled ‘properly’, no more than seven shuffles are required to lead to maximum randomness. In other words, even if the cards are in perfect order at first, after seven shuffles they will be completely mixed. Further shuffling will not mix them any better.
This was recently proved and there is a link to it on the Web, but I can’t find it just yet.
I would think that verification would be simple. All you do is start out with the deck in order.
In other news, I’ve heard something entirely different regarding the words “shuffling” and “eight”. I’ve heard that if you shuffle a deck 8 times (in a ‘random’ manner), you nearly insure randomizing the deck - which would be the exact opposite of the OP’s question.
Any suggestions/comments?
As an out-shuffle maintains the top and bottom cards in their original places. If the cards reversed order, you weren’t doing out-shuffles. My recollection is that 8 out-shuffles do, indeed, return the cards to their original places. Plugging in some test numbers to the equation
x[sub]i[/sub] = { x[sub]i-1[/sub] <= 26 ? 2x[sub]i-1[/sub] - 1 : 2x[sub]i-1[/sub] - 52 }
lends credence to the 8 out-shuffle statement.
1 -> 1
2 -> 3 -> 5 -> 9 -> 17 -> 33 -> 14 -> 27 -> 2
4 -> 7 -> 13 -> 25 -> 49 -> 46 -> 40 -> 28 -> 4
6 -> 11 -> 21 -> 41 -> 30 -> 8 -> 15 -> 29 -> 6
The other four rings should be just mirror-images of the above four.
I think Chronos may have made an error in his shuffling. I spent a little time playing ‘perfect shuffle solitaire’ myself once, just to figure out the pattern. I’m writing this from memory, so my formulas might be a little off, but the general picture should come out.
To make it simple, put the deck in suit & rank order : from bottom to top, A-K of Hearts, A-K of Spades, A-K of Diamonds, A-K of Clubs.
So with perfect-out shuffles, it’s always the A-Hearts followed by the 27th card (currently A-Diamonds).
Shuffle twice. What you’ll get is each rank right next to each other (all the aces, then the twos, etc.) in the suit order : Hearts Spades Diamonds Clubs.
To confuse things, we’ll label the cards after each shuffle as 0 to 51 from bottom to top (A-Hearts is 0, K-Clubs currently 51).
At each shuffle from here on out, the ranks will be displaced twice as much as in the last shuffle. One shuffle moves the second ace to position 2, the next moves it to position 4, etc. Each successive rank follows, spread out by the same amount.
The ‘magic’ of it is of course that the offset ends up being mod 51, i.e. it cycles around (mod(-ulo) 51 == remainder when divided by 51).
So, after 6 shuffles we’re offset
2^6 = 64 and 64 mod 51 = 13.
So the A-Spades that was at pos. 1 (after 2 shuffles) moves to pos. 13, where it started way back at the beginning.
I’m going to call that QED for now. Although it isn’t quite completely stated, I’m hoping it’s apparent.
panama jack
http://www.coolsig.com - they’re all mine today.
Whoops, you’re right, I was doing in-shuffles. I misread your definitions of “out” and “in”.
And yes, of course I started with the deck in order. I don’t think that I made any mistakes, because most mistakes I could conceivably have made would make it nearly impossible to ever reach the original order again. On the other hand, it was five years ago that I did this, so I may well be misremembering.
I’m losing it folks! What’s happening to me?
jb_farley, all I can say is, don’t even think about what happens if you fall through the earth.
I think blitzzen is having a strange effect on you. Don’t go outside in a storm, especially in open spaces.
panama jack
Yeah, I read that Gardner essay, its in one of his collections. I used to know how to do one of Gardner’s card tricks, it was cool. You divide the deck into red and black cards. Then you have someone “riffle shuffle” the 2 decks together a number of times, any way they like (no “perfect shuffle” required). The cards are now randomized, but if you deal the cards, they come out in red-black pairs. Every single pair is a set of black/red cards.
There are whole mystical systems based on how you shuffle cards. One of my favorites is “The Mystic Test Book” by Olney Richmond, who used have a “temple” in Chicago during the late 19th century. He devised a system of shuffling cards that he used as divination. Its extremely complex, you shuffle the cards in a certain way that the cards repeat after 96 shuffles. He said this represents 96 years of a life, and the positions of the cards represent the influences on your life during that year.
It may be cool but it doesn’t work as you have described it.
In my shuffling I alternate 2 red cards with 2 black cards. After any number of shuffles the combined deck is RRBBRRBB…
For a 52-card deck, a shuffle of the top 26 and bottom 26 cards will take a full 52 shuffles to reassemble, conditioned that the top 26 become the 2nd, 4th, … 52nd cards and the bottom 26 become the 1st, 3rd, … 51st. Each card will take a turn being the nth card of the deck, 1 <= n <= 52.
Here is the order a card will go through:
1 27 40 20 10 5 29 41 47 50 25 39 46 23 38 19 36 18 9 31 42 21 37 45 49 51 52 26 13 33 43 48 24 12 6 3 28 14 7 30 15 34 17 35 44 22 11 32 16 8 4 2 1.