Originally posted by Dustin De Leeuw:

Originally posted by Kenneth Pletinckx:

a seven card pile … was shown to have a better randomization than a six or five card pile shuffle

This makes me sad.

Me as well. Perhaps Kenneth needs an explanation of why pile counting provides zero randomization, no matter how many piles you do, and no matter how many times you do it? Unsure if Kenneth will return to this thread, but in case he does, I'll provide an explanation for him:

“Randomization”, in the context which is relevant to judges, is defined as “the degree to which a player is able to know the ordering of the contents of their deck” (my own definition, although I think it's pretty good, if any of the ones responsible for documentation would like to steal it for official use, go nuts).

Let's say you start with a non-random deck, in which there exist 1 or more cards whose identity and placement within the deck is known. Because we're discussing a 7-pile could here, let's say it's the 7th card down. Without loss of generality (and because Kenneth's post stated that odd piles are better than even piles) let's compare a 7-pile count with a 6-pile count (which is also a factor of 60 to counter arguments that use that) and an 8-pile count (which is not a factor of 60, to contrast with 6):

If we 7-pile count the deck, then we know that the 7th card down in the deck, which is known, is going to be in whichever pile the 7th card is dealt to (most commonly the 7th pile, but maybe you have some unorthodox pile counting method), and we know how many cards down it is in that pile. Let's say, because it is the most common, that that card is the bottom card in the 7th pile (without loss of generality). Then, we complete the pile count and stack up the piles in some order. Again, without loss of generality, let's say that we stack the piles in order, so pile 1 is on top of your deck and pile 7 is at the bottom. Then we can say that the bottom card of our deck after the pile count is the same as the card that was 7 cards deep before the pile count, and since we knew the identity of the 7th card down, then we know the identity of the bottom card. Therefore, the degree to which our deck is “random” has not changed (as per the definition of “random” above) with respect to this card. If you know, say, the 7th and 14th cards, then under this scheme you will know the bottom 2 cards of your deck. If you know the 7th, 14th, and 21st, then you will know the bottom 3, and so on, and this extends to other piles with other cards as well. To take an extreme example, if you know the complete ordering of the cards before the pile count, then you will likewise know the ordering after the pile count, by applying this argument individually to each of the 60 cards. Which is to say that the degree to which the deck is “random” (by the definition above) has not changed whatsoever.

Now, let's consider the 6-pile count. We know the 7th card down. Again, assuming (without loss of generality) that we individually place each card in a pile, the 7th card down will be the 2nd to bottom card in the 1st pile of our 6-pile count. Next, let's stack up our piles (without loss of generality) so that the 1st pile is on top and the 6th pile is at the bottom. So now we knew our 7th card before, which is now our 9th card down (pile 1 has 10 cards, and the previously 7th card down is the 9th card down of those 10 cards, which are the top 10 cards of our deck). Once again, the deck is not any less “random” then before, and by a similar argument to above we can state the same thing about the deck as a whole.

Considering the 8-pile count, the only thing that changes, again, is the numbers. Our 7th card down is now the bottom card in our 7th (out of 8) piles, and since the last 4 piles have 8 cards in them each (assuming without loss of generality the “standard” method of pile counting), the known card is now the 9th card from the bottom. Again, copy-pasta the argument from above to extend the argument to the whole deck.

Exercise to the reader: Extend this argument to a pile count of n different piles, for any value of n you would like to show does not actually increase the randomness in the deck. I've given you the most common numbers, 6, 7, and 8, but the same argument works for any number of your choosing.

a seven card pile … was shown to have a better randomization than a six or five card piles

As long as one reads “better” as “more advantageously distributed” rather than “more random”, this reads correctly. And since English is such an inconsistent language, it can easily be interpreted that way!

Originally posted by David Poon:

a seven card pile … was shown to have a better randomization than a six or five card piles

As long as one reads “better” as “more advantageously distributed” rather than “more random”, this reads correctly. And since English is such an inconsistent language, it can easily be interpreted that way!

Nice try, but I'm not buying it. “Pile count” and “randomization” have nothing to do with eachother. And that's the message we really want to make clear to players (and judges!): pile counting is counting your deck, it does not randomise, please stop wasting our time.

Originally posted by David Poon:

a seven card pile … was shown to have a better randomization than a six or five card piles

As long as one reads “better” as “more advantageously distributed” rather than “more random”, this reads correctly. And since English is such an inconsistent language, it can easily be interpreted that way!

Even then I would disagree. To wit:

Assume a deck is sufficiently randomized. Note that “randomized” does not imply any pattern of the ordering of the deck; it is just as likely for 7 lands to be stacked in a row as it is for lands to be evenly distributed. If you then pile-count your deck into some number of piles, it is equally likely for you to mana-weave your deck (of course, unintentionally, I don't want to digress into Cheating territory) as it is to mana clump your deck.

For example, take a stack of 14 cards, half land, half nonland, and stack it so that all the land is above all the nonland. This is a valid random configuration of the deck, as any configuration of the deck is acceptable as random. Then pile count into 7 piles. You will necessarily get the distribution land-spell-land-spell. This is “more advantageously distributed”, as claimed.

Next, take the same pile of 14 cards, except this time distribute it 4 spells, then 3 lands, then 3 spells, then 4 lands. Again, this is a possible random orientation. Pile count into 7 piles again. This distribution will be spells on top, lands on bottom, which is obviously not advantageous.

What remains to be proven (exercise to the reader) is to show that the permutations of a pile count are closed; that is, for every possible permutation of the deck, there exists a (possibly different) permutation of the deck such that a pile count starting from the second permutation will lead to the first permutation. I believe this fact to be true but haven't actually verified it.

If it is true, then it is precisely as likely for a pile count of a randomized deck to be advantageous as it is to be disadvantageous, in which case, as Dustin said above, stop wasting our time. And if it is not true, then pile counting should be considered Insufficient Randomization, as it provides a method by which players can produce more advantageous orderings (on average) of a randomized deck, which is not legal.

The issue over piles and randomization is as simple as this:

Give every card in a deck a unique “ID number” from 1 to 40 (for example). If you have perfect recollection of the method taken to create piles, you can always be certain that any given card's ID number is in a specific location once the deck is re-stacked. Even more generally if you don't track all of your actions, you can know with relative certainty what the topmost and/or bottommost card's ID numbers are, simply by virtue of ordered math and the fact that you will almost certainly know where you started or ended your piles.

Randomized means that when you're done, you've got a roughly equal and unpredictably likely chance that the top card is any ID from 1-40. It's not just “lands vs spells” - in terms of discrete objects, each card is unique, even if there are other cards in the deck that have the exact same ink on them, and if you can identify EVEN ONE of those cards you are not randomized.

Wow, that blew up. Did we really need 3 separate explanations as to why pile counting doesn't randomize?

I should clear up the intent of my comment: English is a silly language, and words have multiple meanings. When someone says “shown to have a better randomization” in this thread, it is highly unlikely for that phrase to actually mean “shown to be more random”. What it probably means is “shown to stack the deck with a more favourable distribution of spells and lands when starting from a non-random or poorly shuffled deck”.

Whether or not said someone realises that this is what they actually mean is a different matter.

I agree with David, that blew up :). Personally, I don't find the line of discussion at this point very fruitful and I'm not sure if the one making the comment has been scared off because of it (let's hope not).

I'm not sure there's more going to be said on the original topic of this thread though? If I may nudge the conversation back onto that topic?

This spiraled downward, quickly, from the original topic, so let's wrap it up.

If a player miscounted, then let them correct their mistake. No big deal, nothing to see, move along…

d:^D