Riffle stacking

[John Bodine]

I've recently been playing around with riffle stacking and was curious about a few things.

The method I've been using is to control the cards I want (currently only 2) to the top and bottom. Depending on the number of hands I want, I stack either 1, 2, or 3 cards on top of the top card and repeat as necessary. I then perform a series of cuts to shift the bottom card to the top, repeat step 1 of riffle shuffling cards onto the new top card, and I'm ready. With this method I can stack 2 cards into any hand up to 7 players in no more than 6 shuffles and 1 set of cuts.

My question is the importance of being able to hold back cards with either the left or right hands, and whether or not the cuts in the middle create a problem. I've watched friends shuffle/cut while we play poker (not the reason I'm working on this. :) ) and it's not uncommon for them to shuffle that many times and cut in the middle of their shuffle.

John Bodine

[Guest]

Hi John,

If it's taking up to six shuffles, with cuts in between to stack two cards, here's a challenge for you.

It should take no more than two shuffles and a cut after the shuffles to stack those two cards.

Happy shuffling,
Joe

[John Bodine]

Thanks for the respone James. I realize that there are many different ways I could approach stacking just 2 cards, and that I could just cut the bottom card and the correct number of cards above it to the top to achieve the last stack thereby dramatically reducing the number of shuffles required. Additionally, I could start with everything on top and stack between the desired cards. Albeit a bit more difficult, it's certainly within the grasp given some serious practice.

My question is more accurately this: to those of you who riffle stack, do you try to stack in as few shuffles/cuts as possible, or do you make an attempt to mimic how the average player would shuffle and cut. I'm inclined to favor the latter, and throw in the bit of knowledge that Mr. Persi Diaconis has found that 7 shuffles is necessary to randomize a deck.

john bodine

[Guest]

Hello again John,

I am really curiopus about the seven shuffles to "randomize" a deck.

First, what is a "randomized" deck?

And, what does seven shuffles do to a deck that six or eight shuffles does not.

Is a deck that has been shuufles seven times more mixed up than one that was shuffled only five times?

What was the condition of the deck before the shuffles?

I hope this is not coming across as argumentative but I think the seven shuffle bit is very interesting.

Joe

[Bill Mullins]

See THIS article for details on how much to shuffle a deck. For other info, do a yahoo/google on (Diaconis Shuffle) and other information will come up.

[DChung]

John, the average schmo is very unlikely to be shuffling the deck 7 times. And he is even less likely to know (or care) that it takes that many shuffles for a deck to be "random".

I'm a relative newbie to riffle stacking myself, but if you're taking that much time to stack two cards, good luck with four or five.

My stance is the fewer the better. You can always false shuffle if you need to.

[Pete McCabe]

This is based on reading a NY Times article on Diaconis's work about 20 years ago, so it may not be perfectly accurate. But the basics are correct.

There is no such thing as a "randomized" deck. There are many different definitions of random -- or to be more precise, many different ways to measure how random a deck (or shuffling procedure) is.

Point is, it's somewhat arbitrary. Different definitions make more or less sense to different people. Judgement is involved.

Diaconis chose as his measure the probability that any given card ends up in the same position after N shuffles as it was before the shuffles. In other words, start with the deck in new deck order. Shuffle three (or however many) times. Now see what percentage of cards are still in their original position. The less likely it is that any given card is in the same place, the more "randomized" the deck is.

Now Diaconis used another standard (but also arbitrary) convention. In any experiment there is always a possibility that the results occurred by chance. If I flip a coin two times and both come up heads, that's not a very convincing demonstration that it's a two-headed coin. Ten heads in a row, much more convincing. A hundred times, all heads, you're quite confident it's a two-headed coin, even though this can, theoretically, happen randomly.

So a convention has been established that if there's a less than 5% chance the results of an experiment could occur randomly, the results are said to be significant. Less than a 1% chance is said to by highly significant.

These figures are arbitrary. If you use different percentages, you will get different results than Mr. Diaconis. You will not be wrong.

After seven shuffles, fewer than 1% of cards are still in its same position. Thus, seven shuffles are "necessary" to produce a "random" deck.

It's worth noting that most of the "randomness" (by this scale) comes from the very first shuffle. Each additional shuffle adds less and less randomness, until the probability creeps below 1% with the seventh shuffle. I have seen many people summarize Diaconis' results by saying "The deck doesn't really start getting random until the seventh shuffle." But by this measure of randomness, each succeeding shuffle adds less and less randomness.

If anyone can alert me to any errors in the above analysis I'll appreciate it. (Update: thanks Bill Mullins)


Back to the subject at hand:

When I play poker, as soon as the dealer of the next hand folds, he starts shuffling (and occasionally cutting) the deck. This can easily add up to twenty shuffles, all done on autopilot while watching the game.

If you want to mimic this behavior, write a script about the presentation which you say while you are shuffling. I would recommend that you not talk about the shuffling, by the way. The less the script has to do with the shuffling, the less the audience will pay attention to the shuffling.

[Bill Mullins]

Originally posted by Pete McCabe:

Diaconis chose as his measure the probability that any given card ends up in the same position after N shuffles as it was before the shuffles. In other words, start with the deck in new deck order. Shuffle three (or however many) times. Now see what percentage of cards are still in their original position. The less likely it is that any given card is in the same place, the more "randomized" the deck is.

Another definition of randomness that he has used is the probability that a gambler can guess what the next card flipped over is. THIS LINK goes through some of his work on that subject. Since some of his research has been paid for by casinos, this is a very relevant measure of randomness -- how much shuffling should occur, before an intelligent gambler can no longer take advantage of residual order in a shuffled deck?

It's worth noting that most of the "randomness" (by this scale) comes from the very first shuffle. Each additional shuffle adds less and less randomness, until the probability creeps below 1% with the seventh shuffle. I have seen many people summarize Diaconis' results by saying "The deck doesn't really start getting random until the seventh shuffle." This is entirely backwards. Each succeeding shuffle adds less and less randomness.

This isn't how I read the situation. There is still a great deal of order left in the deck after five shuffles. (example: Take the order of a new deck. The two of spades immediately follows the ace. After one riffle shuffle, it still follows it somewhere in the deck. After two, it still follows it. After three, it still follows it. ) This sort of order stays in the deck in a measurable fashion, until seven shuffles occur, at which point it drops rapidly. For eight and more shuffles, very little randomness is added.

see HERE for anther good article.

[Pete McCabe]

Bill:

I read the article you linked to and found it excellent. Interesting to see how the different measures of randomness produced different outcomes. The "detectable order" measure is obviously much more important for card playing than the "same position" test in the article I read.

I think this difference accounts for our different takes on the speed of randomization.