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Thread: Sorting with Petals of Insight: Shortcutting and Slow Play?

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    Sorting with Petals of Insight: Shortcutting and Slow Play?

    In Carsten Kotter's most recent article, he mentioned the following about Petals of Insight:

    The trick you need to be aware of that I just talked about is simply this: as long as the number of cards in your deck is not divisible by three, casting Petals of Insight again and again in fact changes the order of the cards in your library. You end up slowly moving cards through your deck, pushing a card or two into a different three card block, thereby creating new three card blocks to look at with Petals of Insight every time you move through your complete library. Through this process it is actually possible to create any particular three-card block you want to draw with Petals of Insight at the end of the process.
    SCG commentators Matthias Hunt and Cedric Phillips noted this in commentary to one of the feature matches this weekend, so this "sorting" process has been in everyone's mind.

    However, while finding a particular card (and putting it to the top) doesn't take particularly long--it's an order O(N) operation, where N is the number of cards in your deck--I believe sorting two cards or three cards to the top of your deck becomes O(N^2) or O(N^3), respectively. This can take hundreds to thousands of iterations in some cases, and the player casting Petals may not necessarily "do it right". In other words, this is not trivial for a Magic player (while it might be for a mathematician, dexterity issues aside).

    I have several questions when such a scenario arises:

    1. Is it sufficient for the combo player to say, "cast Petals until I have cards X, Y, and Z in the top pile"?
    2. If no to (1), is it sufficient if he describes his algorithm for performing that? How mathematically rigorous does the description have to be? Hypothetically, there can be more obscure situations where the success of an operation can be mathematically proven, but it would require a judge with a mathematical background to verify the proof. And judges (just like everyone else) can be susceptible to mistakes in this regard.
    3. Would the player have to specify that he can arrive at a particular state after exactly N iterations? In addition to requiring that the player be able to work out the math, this would depend on the original order of the deck, which would require at least one pass through the deck, and the contents of the deck are technically hidden information every time. Furthermore, would this imply that the player not only be able to describe the top 3 cards of the deck after N iterations, but that he needs to be able to describe where all the other cards are in the library?
    4. Assuming the answers to (1)-(3) are negative, the player probably has to perform each iteration manually. If the operation requires hundreds to thousands of moves even when performed optimally, at what point does this become slow play? Technically, the player is advancing the board state. Furthermore, this can be subject to abuse when the combo player has a trivially easy victory in game 1 but uses it to milk the clock before killing the opponent. Granted, the opponent can concede at any point, but now imagine a situation where the combo player is drawing dead* but can repeatedly cast Petals indefinitely. Are judges or opponents allowed to examine the contents of the deck at some point? How long does the combo player have to execute his combo?

    * There are several ways to draw dead. One involves simply not having a combination of cards available in your deck to "get you out" of your situation (for example, you have no more Burning Wishes left in library). Another occurs when the number of cards left in library is a multiple of three and you need a specific combination of cards that is not available in any three-card block. In the latter situation, are judges/opponents allowed to note this before declaring slow play or stalling? What if the condition that leaves an opponent "drawing dead" is more mathematically complicated than it is for Petals?

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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    I watched that match on coverage too and thought the whole Petals thing was nonsense. If you are comboing off and have to quote me some hypothetical equation, then I'm gonna consider you a Deceiver Exarch and make you play it out in the hopes that you suck at math or your fingers can't manipulate your cards correctly in order for your mumbo jumbo mathing to work out. The whole Hypothetical Space Time Continuum shortcut thing is dumb.

    TLDR: I have the same complaint/inquiry but from a stupider person's brain.

    EDIT: sry. I thought this was in the Community Thread and was an open space for ranting. I'll just quietly leave now...
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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    I also don't think the whole scenario is as trivial as people always used to make it when Petals of Insight was involved. When OmniTell with Burning Wish first became a deck in 2012, it didn't matter too much because you would usually only look for a single card. With the new versions often trying to create piles with 2+3 specific cards, I assume it's possible to also manipulate the entire rest of the library to just leave it in any arbitrary state....but not yet fully convinced. I'm siding with #TeamStupid and our glorious leader Dan for now.
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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Quote Originally Posted by lordofthepit View Post
    ...
    However, while finding a particular card (and putting it to the top) doesn't take particularly long--it's an order O(N) operation, where N is the number of cards in your deck--I believe sorting two cards or three cards to the top of your deck becomes O(N^2) or O(N^3), respectively. This can take hundreds to thousands of iterations in some cases, and the player casting Petals may not necessarily "do it right". In other words, this is not trivial for a Magic player (while it might be for a mathematician, dexterity issues aside).
    It's effectively a bubble sort so you can get an arbitrary order in O(N^2) steps as long as the number of cards in the library isn't divisible by 3.

    Quote Originally Posted by Julian23
    I assume it's possible to also manipulate the entire rest of the library to just leave it in any arbitrary state....but not yet fully convinced.
    Let's say that there are L cards in the library, and we write a number from 1 through L on the cards. Let's also assume that the number of cards is *not* divisible by 3.

    Then, whenever we pick up and look at 3 cards with Petals, if the group of cards contains card 1, then we leave it in order and put it on bottom. If the group does not contain 1 then put it into numerical order, and put it back on bottom.

    As long as the cards are not in order, this procedure is guaranteed to change the library order at least once every L castings of petals.

    This procedure will always move low numbered cards 'forward' and high numbered cards backward so it will never reverse the changes it makes.

    There is a finite number of possible library orderings, so it will get to the desired order in finite time.

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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    It has always been possible to shortcut a loop with a mathematical model, but it's (head) judge's discretion as to whether the model is accepted.

    If performed manually, it's absolutely subject to slow play rules - the game state is not being sufficiently advanced. Loops are specifically covered under Slow Play in the IPG:

    It is also slow play if a player continues to execute a loop without being able to provide an exact number of iterations and the expected resulting game state.
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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Petals is fine. You can describe the entire process deterministically (in fact, I'm pretty sure you can do it regardless of whether the number of cards in library is a multiple of three), and that's all we care about.

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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Quote Originally Posted by ubernostrum View Post
    ...(in fact, I'm pretty sure you can do it regardless of whether the number of cards in library is a multiple of three)...
    If the number of cards in the library is divisible by three, you keep reordering the same groups of three.

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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Quote Originally Posted by cdr View Post
    It has always been possible to shortcut a loop with a mathematical model, but it's (head) judge's discretion as to whether the model is accepted.

    If performed manually, it's absolutely subject to slow play rules - the game state is not being sufficiently advanced. Loops are specifically covered under Slow Play in the IPG:

    It is also slow play if a player continues to execute a loop without being able to provide an exact number of iterations and the expected resulting game state.
    Can you define what constitutes "expected resulting game state"? Is it sufficient to specify the top 3 cards, or would you have to be able to specify the positions of all cards in the library?

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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Quote Originally Posted by rufus View Post
    If the number of cards in the library is divisible by three, you keep reordering the same groups of three.
    For some reason I was thinking there was a second Petals in the sideboard of the current lists (which does put it back into purely deterministic territory regardless of the starting count of the library). In that case, modify to "cantrip first, then Petals" :)

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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Quote Originally Posted by rufus View Post
    There is a finite number of possible library orderings, so it will get to the desired order in finite time.
    I understand that. What I'm wondering is the step you go about without deeper explanation:

    This procedure will always move low numbered cards 'forward' and high numbered cards backward so it will never reverse the changes it makes.
    How do we know that at some point we won't just end up in an infinitely repeating pattern where, no matter how often we go over it, certain cards won't ever be in the same pile?
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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Quote Originally Posted by Julian23 View Post
    I understand that. What I'm wondering is the step you go about without deeper explanation:
    How do we know that at some point we won't just end up in an infinitely repeating pattern where, no matter how often we go over it, certain cards won't ever be in the same pile?
    Doesn't what you describe get dangerously close to Four Horseman in a "eventuality" combo, but still illegal under shortcut rules?
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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    No. Because Four Horseman is not deterministic. You have no idea whether the constellation you are looking for will ever come up, it's just very likely. And once it (99,999..% likely) does, you don't know what the rest of the library will look like.
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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Quote Originally Posted by Julian23 View Post
    ...

    How do we know that at some point we won't just end up in an infinitely repeating pattern where, no matter how often we go over it, certain cards won't ever be in the same pile?
    In the process that I described, card 1 never gets "reordered" so if there are L cards in the library, and you start with card 1 on top of the library, and you cast petals of insight L times you end up with card 1 on top of the library again.

    Now, consider what happens to card 2: Until card 2 is directly after card 1 it will get "sorted up" at least one slot every time petals of insight gets cast L times. So card 2 will eventually get sorted behind card 1.

    Now, consider what happens to card 3, once cards 1 and 2 are sorted: Until card 3 is directly after cards 1 and 2, it will get "sorted up" at least one slot every time petals gets cast L times. So card 3 will eventually get sorted.

    ...

    Now consider what happens to card N once cards 1 through N-1 are sorted...

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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    If you determine a number of iterations(N^2) and tell how the library is stacked, should be enough no? Not sure if you can do a shortcut if you only tell how the top 3 are stacked.

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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Quote Originally Posted by rufus View Post
    In the process that I described, card 1 never gets "reordered" so if there are L cards in the library, and you start with card 1 on top of the library, and you cast petals of insight L times you end up with card 1 on top of the library again.

    Now, consider what happens to card 2: Until card 2 is directly after card 1 it will get "sorted up" at least one slot every time petals of insight gets cast L times. So card 2 will eventually get sorted behind card 1.

    Now, consider what happens to card 3, once cards 1 and 2 are sorted: Until card 3 is directly after cards 1 and 2, it will get "sorted up" at least one slot every time petals gets cast L times. So card 3 will eventually get sorted.

    ...

    Now consider what happens to card N once cards 1 through N-1 are sorted...
    I agree to that, but you didn't address what I was asking: how we can 100% rule out that halfway through the process the decks ends up in a state that through a period of N castins of Petals of Insight will not repeat itself?

    I did the process in my head all the way through for decksizes of up to 10 and it always works. But I know that that is just anecdotical evidence. I was wondering about the actual mathematical proof.
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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Quote Originally Posted by Julian23 View Post
    I agree to that, but you didn't address what I was asking: how we can 100% rule out that halfway through the process the decks ends up in a state that through a period of N castings of Petals of Insight will not repeat itself?
    I'm not sure I understand. Do you mean, "how can we be sure it won't repeat until it's sorted?", or do you mean "how can we calculate the number of castings so that we can specify N so that we don't keep cycling the deck once it's sorted"?

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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Quote Originally Posted by rufus View Post
    I'm not sure I understand. Do you mean, "how can we be sure it won't repeat until it's sorted?", or do you mean "how can we calculate the number of castings so that we can specify N so that we don't keep cycling the deck once it's sorted"?
    Initially I was thinking about the former question.

    But now that you mention it, the second one is also interesting. In general choosing too high of a value for N wouldn't matter because once it's sorted you would just put back things in the same order. The part that interests me is that - unlike with other loops - you can't tell your opponent you are performing the loop x times or until x condition is met, because the cards and order in your library is unknown. Which makes me wonder: everyone agrees that choosing N = 1 million is easily enough...because we have a pretty good feeling it will easily be. But that's not how the rules work; what's the actual proof that it will work...which is basically the question you mentioned: what value of N guarantees the desired outcome?
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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Quote Originally Posted by Julian23 View Post
    Initially I was thinking about the former question.

    But now that you mention it, the second one is also interesting. In general choosing too high of a value for N wouldn't matter because once it's sorted you would just put back things in the same order. The part that interests me is that - unlike with other loops - you can't tell your opponent you are performing the loop x times or until x condition is met, because the cards and order in your library is unknown. Which makes me wonder: everyone agrees that choosing N = 1 million is easily enough...because we have a pretty good feeling it will easily be. But that's not how the rules work; what's the actual proof that it will work...which is basically the question you mentioned: what value of N guarantees the desired outcome?
    It's not hard to show if there are L cards in the library, L>3 and L is not divisible by 3, then casting petals at most L^2+L times will get you there.

    Phase 1 (casting petals at most L times):

    Cast petals.
    Put the top 3 cards in numerical order and put them on the bottom of the library.

    If you saw card L (the card that will be on bottom when you're done) continue to phase 2, otherwise repeat.

    Phase 2

    Cast petals L^2 times. Each time...

    If the top three cards of the library contain card L, don't change the order and put them on bottom.
    Otherwise, put them in numerical order, and put them on bottom.

    This is a sort of bubble sort:
    https://en.wikipedia.org/wiki/Bubble_sort

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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    Its definitely an awkward form of Bubble Sort, which is deterministic. I'm not so sure that means executing it isn't slow play (And if it isn't, does your opponent asking you to execute it manually slow play a-la Mind's Desire?)

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    Re: Sorting with Petals of Insight: Shortcutting and Slow Play?

    All this math talk is confusing. I am a simpleton and it's weird to me that an opponent can show me a math equation in order to avoid actually playing Magic. If an opponent starts going into this combo, is it asinine of me to make them play it out and then call a judge over to watch for slow play? With the vague hope that my opponent makes a mistake somewhere? Is that a dick move or totally reasonable?
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