text:
When exactly do I get to return my Commander to the Command Zone after it goes to the Graveyard?
If my Commander on the field dies, and I send it to the Graveyard, its death becomes a state-based action. My question is, since a Commander has the chance to be returned to the Command Zone from the Graveyard or exile since the last time state-based actions were checked (its death), can I do something that does not involve a stated based action, such as tapping two of my lands to gain mana to activate a creature's ability, and then return my Commander to the Command Zone (after the creature's ability resolves)?

You have a single opportunity to move the commander to the command zone, immediately after it moves to the graveyard. You can't take any actions before making the choice.
The specific rule for moving the commander from the graveyard to the command zone is rule 903.9a:

If a commander is in a graveyard or in exile and that object was put into that zone since the last time state-based actions were checked, its owner may put it into the command zone. This is a state-based action. See rule 704.

Since this is a state-based action, you have exactly one opportunity to make this choice, the next time state-based actions are checked after it moves to the graveyard.
It's important to note that rule 704.3 says, in part,

Whenever a player would get priority (see rule 117, “Timing and Priority”), the game checks for any of the listed conditions for state-based actions, then performs all applicable state-based actions simultaneously as a single event. If any state-based actions are performed as a result of a check, the check is repeated; otherwise all triggered abilities that are waiting to be put on the stack are put on the stack, then the check is repeated...

So, if the commander dies as a result of state-based actions, state-based actions are immediately checked again, and you immediately get the choice of whether to move the commander to the command zone. If the commander instead dies during the resolution of a spell or ability, you make the choice as state-based actions are checked immediately after the spell finishes resolving.

You have it right about when the commander goes back to the command zone; when state based actions are checked, if the commander was put in the graveyard (or exile) since the last time state-based actions are checked, it may be moved to the command zone.
However, mana abilities are activated abilities. So you have to have priority to activate them. It's true that they don't go on the stack, and resolve immediately. But if you're going to move your commander to the command zone, it will go to the command zone before you get priority to activate a mana ability. (The same goes for activating the creature's ability.)


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Bananagrams: dumping rules/etiquette. Is it cheating to keep track of dumped tiles?
From the online rules: “At any time during the game, you may call out “DUMP!” and put one of your letters back, facedown in the BUNCH. However, you MUST take three more tiles in exchange!”
While it is not specifically stated in the rules, they seem to imply that “peeling” from the bunch should be random. However, anyone who’s played bananagrams knows that the peeling isn’t always completely random (unless all the tiles are shuffled every time a tile is discarded, which is somewhat impractical) and that discarding can often be done more or less strategically.
Examples of this are discarding a “hard” letter like “Q” close to where your opponents have been peeling from (to get them to pick it), or discarding an “easier” letter in a specific location for you to use later in the game (like right at the end when picking a bad letter can be fatal). Or simply remembering where you discarded a hard letter to make sure you don’t pick it up again.
While there is nothing in the rules that forbids this, it goes against the idea that peeling should be random. One way to avoid this entirely would be to put the bunch in a large container where discarding would automatically shuffle them (like a large pouch or a hat), even though the rules suggest you should have them facedown on the table. But this “strategic dumping” approach could also be incorporated as part of the game, since it does require some skill to keep track of where tiles are (it becomes analogous to card counting). And also, no one is stopping a player at anytime from shuffling all the tiles to prevent this.
What are people’s thoughts on this? Do you embrace this dumping strategy or do you consider it to be cheating? If so, how do you prevent it?

It is not cheating.
The rules as you state just say peeling is taking a tile from anywhere and dumping in placing a tile in a Bunch and taking three tiles back.
The rules at no point say the Bunch is shuffled.
So if a player chooses to do as you say 'a strategic dump' there is nothing in the rules to prevent this. If in a speed of the game a player can recall what tiles were placed back where they there is nothing to prevent this, although I would suggest its a tricky thing to do.
However you also say "And also, no one is stopping a player at anytime from shuffling all the tiles to prevent this."    I would argue that that IS cheating.  The rules state only peeling and dumping is allowed.   Nothing in the rules permits a player from shuffling the tiles during play.

It might be worth noting that an attentive player could take mental note of other players' strategic dumps in order to avoid them. When I play, I try to take tiles unpredictably around the pile.

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Settlers of Catan: does the "knight card steal" have to be random?
When using a knight card, can you negotiate with your opponent to agree on a specific resource card that may be mutually beneficial?
For example, let's say I desperately need a sheep and use a knight card (or roll a 7). My opponent has one or multiple sheep cards they don't particularly need. However, they have a brick they really need for something else and woudn't want to have it stolen. Can both players agree to handing over a sheep?

Oops didn't notice the duplicate before.

The steal itself must be random, but you can easily "fix" this by doing a trade afterwards.
From the rules:

(2) Then you steal 1 (random) resource card from an
opponent who has a settlement or city adjacent to the target
terrain hex. The player who is robbed holds his resource
card hand face down. You then take 1 card at random.

If you end up stealing a brick, then you and your opponent can now simply agree to trade that brick for a sheep; and you both end up with what you want. It's the same end result; but you have the ability to decide whether or not to make that trade only after seeing what you steal. It also gives you more information; if you had stolen a sheep right away you may have not known or remembered if your opponent had any brick.

It follows that one can make it a house rule allowing to ask for voluntarily giving up a sheep without violating the intent of the official rule. The asker would then voluntarily forfeit the extra information mentioned in the answer and tell the whole table what resource exactly they need, but it would improve the playing speed.

I would argue that when you shuffle and fan your cards, for one to be taken at random, then it would be legal to say my sheep is on the right hand side.
The other player would obviously have to trust you and this trust would be based off the previous conversation about who wanted to gain/lose what.

Interesting idea!

Some house rules might require that the player shuffle their hand face-down so that even they don't know what cards are where before the steal, which would defeat that strategy.

This seems like spurious semantics. If you tell the opponent something about your hand to make the steal not random, then the opponent would have to resort to a die or something to follow the rule: "take 1 card at random".

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Settlers of Catan: is it better to go first or last?
In the preliminary stage of the game, going first/last affects (and limits) your ability to pick the location for your two initial settlements. Either scenario seems to have advantages/disadvantages. If you go first, you get to put your first settlement on the best position (there may not always be a clear best position, but let's assume there is). However, for your second settlement, most good positions may have been already taken. If you go last, you don't necessarily get the best positions, but you get to choose your two positions back to back, which allows you to develop a more coherent strategy.
Obviously there are many parameters that come into play when determining who wins the game, but I've always wondered how significant the order of this initial turn might be. Has anyone looked into this?

Related, but not an answer. https://boardgames.stackexchange.com/questions/2877/settlers-of-catan-data

In addition to the question at hand, I think it'd be interesting to compare the two possibilities with neither going first nor last. The question seems to suggest that either of these possibilities will be optimal, but it would be nice to see a reason for that, if it is the case at all.

Yes, I may not have expressed it clearly enough in my question, but I'm also considering that scenario as a potential optimal initial situation. I also didn't mention whether I was referring to a 3 or 4 person game, in which case the dynamic might be different. For example, going last in a 4 person game is pretty rough, and the back to back placement doesn't even really make up for it because all the best spots have usually been taken.

You also need to factor in the group you play with as going first with one group could be important while going last with another group is better.

If by "group" you mean the players, I agree. Here I'm assuming all players are good and will pick close-to-optimal positions on their turn.
Another important variable is the board itself too. But overall, I'm curious to hear folks' thoughts on the effects of starting turn and if there are any underlying truths we can come up with.

Also related, but probably not an answer https://boardgames.stackexchange.com/questions/54080/are-there-different-imperatives-for-the-four-players-when-choosing-starting-se/54083

After having played the video game a lot for many years, I'd have to say that yes, going first is somewhat of an advantage.

Keep in mind, though, that there are many strategies for choosing resources toward your gameplay strategy.

Optimal places are usually those in the middle of three areas, with an 8 or 6, a 5 or 9, and a 5 or 9. Usually there's only one of those on the board, if that. If you can't get a 5 or a 9, go for a 4 or 10. Also, where it is on the board matters. If you can get to two shorelines from a single 6 or 8, each with two roads from a different direction, then you can get two additional 6 or 8s pretty quickly, and deter others from wanting to build in places you don't want them (since they're no longer optimal). Harbor proximity matters. …

Double numbers can be helpful since it ensures you get a lot of resources at once (making them usable at once, and less vulnerable to robbing in some circumstances). So, if you can get a city on an area with two 10s, it's not the end of the world. But it really depends.

And road building is more than about getting the longest road; it's helpful to cut off your opponents, so they can't go anywhere (and so you reserve more area for you to build on). So, getting wood and bricks at first is helpful if you're doing that strategy. However, just going all out for development cards is surprisingly effective, if the available resources are there for that.

The authors of the game have derived the system to make it as levelled as possible, and in that sense, the odds should be the same irrespectively of the order one starts with.
Having played the game a few hundred times, I have never really experienced any difference in the outcome based on starting order, so if an effect exists, it is probably small.
But then again, it might be impossible to estimate such an effect, because there are too many confounding factors that you cannot realistically control for.
I believe it is really hard to isolate an effect because even if you were able to conduct a proper experiment, where each player would play the game many times with 3 or 5 other players, and then evaluate how many times they won given the starting order, I suspect that such results would only be valid for that set of players.
Depending on the set of players that you might be playing with, say players with an aggressive versus a defensive strategy, or players that are fond of progress cards vs players that are fond of ports, or whatever, you would end up with a different result.
Worse, even for a given set of players, over time folks will adjust play based on tactics other players tend to use, and so any advantage a given strategy could have in the beginning might change over time, and then the effect of starting order would change over time too.
Without really knowing of a proper study that looks into it, I’m willing to bet that this will not have been looked at in an experimental setup, simply because it would be really hard to so in practice. It seems like this would be possible to evaluate this via a simulation where each player has clearly defined constant strategies, but whether that would be of any use in a real-life scenario is a different question. Maybe it would be a fun project for an MSc in Operational Research or some such!

I will add that the developers intending and trying to make it balanced neither means that they necessarily succeeded, nor that it's possible to do so. It isn't uncommon for systems to appear sound at first glance (to developers and players alike), but turn out to be broken at closer inspection. Perhaps this game has seen enough play to conclude that there's only insignificant differences - but I still have to remark the logical fallacy.

"too many confounding factors" - This is the key point. There are absolutely map set ups where there is a single clear fantastic starting spot and absent awful dice rolls the first player will ride this to victory. Equally there are maps with lots of decent spots that being able to pick two together is so advantageous that the last player romps home. And there's so much in between that relies on good/odd picks by players 2 and 3. And then there's Dice. You can make all the right choices, and then never see a 6 rolled all game.

I like first or last the most.
As a first you can get the best triple of ressources, but need to have less decision on the second triple.
As the last player you have less choice but you can instantly set two settlements.
But as others said, the system of the starting in Catan is perfectly balanced.

"the system of the starting in Catan is perfectly balanced." Do you have evidence for this statement - e.g. the distribution of wins vs starting position in high-level play?

I never had any problems starting with other positions. The only important part is to take the correct ressources. Even had once that last player only had shit but he could rule the world with sheep monopoly.
If you really want to dig statistics, you can look at this excel, but I didn't studied it
https://docs.google.com/spreadsheets/d/1SRp8KSTnyk_kLMJ4HjIQhd1g4-bT4TVSFmWqAH8IUK0/edit#gid=201632411

"I never had any problems" is not the same as "the system is perfectly balanced"; or in other words, the plural of "ancedote" is not "data".


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Does Konami have this?
Does Konami have e-mail or whatever to fan-cards for Yu-gi-oh? I'm know they not take fan-cards from net, but maybe they have the special e-mail for cards or some contests. I just saw on Yu-gi-oh wiki that was some kind of competition or something like that, where fans drew their cards and some of them were even added to the anime later

I’m voting to close this question because it should be addressed to Konami customer support.

I'm be glad if you say me how I can contact with them

What happens if you Google "Konami customer support"?

From a very general standpoint, most creators (including game publishers, authors, etc) do not accept unsolicited fan submissions. The main reason for this is liability - if I send Konami a fan card, and some time later Konami releases a card that is very similar, then I might think they stole my idea and try to get them to give me credit. In reality it's likely that nobody involved in designing that card even saw my email, but proving it could result in a lot of painful legal proceedings where no-one benefits other than the lawyers.
In fact, Konami has a very similar answer to this in their FAQ: Can I submit a new game idea?
It's possible that they could hold a competition to design a card, but if they choose to then it will (a) be heavily promoted, and (b) have very clear guidelines on entries that boil down to "once you send us an idea, we own everything about it and can do whatever we want with it". On the one hand, this can be seen as a nasty corporate move to get free access to people's creative work, but on the other hand it ensures that they don't have to worry about those legal proceedings.


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Identifying barricade-looking game
When my parents moved, they gave
my old games from their place and there was this game (pieces are just putted like this, no idea if it's supposed to be like that, neither if the orientation is right):

When searching with Google Image, "barricade" is suggested but it's not the same holes.
I think there was a paper with name and rules but it's lost. No memories of playing with it.
No idea if there are missing sticks, neither if some of the one I have are spares.
In France but don't know about the origin of the game.

I would say google is right. Orientation is correct, first player to reach the top with and you can block other people with your pins.

That's barricade. Happy to help.

Unsurprisingly there has been more than one version: https://boardgamegeek.com/thread/2121481/various-versions-malefiz-barricade

The game is known under Barricade or Malefiz. Your oriantation is correct, the goal is to reach the top.
I would say you are missing a bunch of green pins, in the version I know you have a certain color (usually white) that adds as barricade stones, that are blocking your way to the goal.
As for way you can't find the exact layout on google, there are a lot of different board layouts to this game. In some of them the goal is even in the middle and players are racing from both sides.
You can play this game as well on just a paper with a few coins (since you need to move the barricades around, they are not suited to be just marked on the paper)
I found on google a version that matches your image somehwat

Thanks for the answer, so it's really a variation of barricade. Now I just have to redo 1 missing blue stick, 2 missing green and I guess the barricades are the black ones, so I need more of them.

@zigarn as you can see in the Mario version (since it uses the same layout), it has at least 10 blue circles (those are the barricade positions). But they have as well only 6 orange blocks. Might be the smaller game uses less or it is just a demo picture for demonstartions


text:
How do I calculate Mulligan probability?
I am playing a game of Commander and I have a deck of 99 cards. In this deck are 37 lands. Using a hypergeometric calculator it returned a 29.12% chance of drawing exactly 3 Lands on a 7 card draw using the probability density function. My question is how do I calculate the probability for drawing 3 lands with one or more mulligans?

Which mulligan rule are you using?

@LeppyR64 I think we should assume the default Commander rules as specified in the CR unless a poster specifies otherwise.

The probability to draw exactly 3 lands after n attempts (including your first non-mulligan draw) is 1-(1-0.2912)^n, with n<6. With the maximum of 4 mulligans, you have a ~82.1% chance to get a 3-lander.
On each mulligan, you draw a new hand of 7 cards and then put as many cards on the bottom of the library as you have taken mulligans.

103.4. Each player draws a number of cards equal to their starting hand size, which is normally seven. (Some effects can modify a player’s starting hand size.) A player who is dissatisfied with their initial hand may take a mulligan. First, the starting player declares whether they will take a mulligan. Then each other player in turn order does the same. Once each player has made a declaration, all players who decided to take mulligans do so at the same time. To take a mulligan, a player shuffles the cards in their hand back into their library, draws a new hand of cards equal to their starting hand size, then puts a number of those cards equal to the number of times that player has taken a mulligan on the bottom of their library in any order. [..]

This limits you to a maximum of 4 mulligans because if you take 5 or more, you will have fewer than 3 cards left in your starting hand and you can never have a 3-lander.
Also, since you shuffle the library between mulligans, the attempts are independent of each other.
If you have an event A with probability P(A), the probability of it failing is its inverse, or

1-P(A)

If you want to calculate the odds of n independent instances of A having at least 1 success, you first calculate the odds of all of them failing. Since the attempts are independent of each other in your case, you can simply multiply their probabilities with each other to get the result:

(1-P(A))^n

The inverse of that is the probability that at least one of them does not fail:

1-(1-P(A))^n

That means that if you have seen n=5 hands (after 4 mulligans), the probability of having at least one of them being a 3-lander is

1 - (1-0.2912)^5 = 1 - 0.7088^5 = 1 - 0.1789 = 0.8211, or 82.1%

Does this analysis include the fact that the first mulligan in Commander is free? ("103.4c  In a multiplayer game and in any Brawl game, the first mulligan a player takes doesn’t count toward the number of cards that player will put on the bottom of their library or the number of mulligans that player may take.")

No. Only Brawl has the free mulligan, and the question makes no mention of Brawl.

Commander has a free mulligan "In a multiplayer game"; the Brawl reference is there because Brawl is two player.

The question makes no mention of multiplayer either.

99% of Commander games are a 4 player pod. Yes, 1v1 Commander exists but it is very much not what people mean when they say "Commander".

37% times the 7 cards you initially draw gives you an average of 2.59 lands per opening hand for 40 lands you get 40% * 7 which gives you 2.8 lands on average. It seems you would need a 43% of lands in your deck to get a 3 lands opener on average. That means 26 lands for a 60 card deck and 37 for a commander deck.


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In what way are cards simultaneously milled by multiple players?
According to Rule 101.4a, "If an effect has each player choose a card in a hidden zone, such as their hand or library, those cards may remain face down as they’re chosen. However, each player must clearly indicate which face-down card they are choosing."
According to Rule 101.4b, "A player knows the choices made by the previous players when making their choice, except as specified in 101.4a."
These rules precisely govern, i.e., how to resolve Rankle's first and third effects.
According to Rule 404.3, "If an effect or rule puts two or more cards into the same graveyard at the same time, the owner of those cards may arrange them in any order."
So when resolving, i.e., the effect of Ruin Crab against multiple opponents, each opponent must decide how their cards being milled will be ordered in their respective graveyards.  Rule 101.4a almost sounds like it governs this, but choosing a card is not exactly the same as ordering a set of cards.  Which of the following resolutions are correct (for a two-opponent game)?
Resolution 1
Both opponents privately view the top three cards of their respective libraries
First opponent in APNAP order privately rearranges their cards into a pile which is later to be placed directly into their graveyard
Second opponent in APNAP order privately rearranges their cards into a pile which is later to be placed directly into their graveyard
Both opponents place their ordered pile into their graveyard
Resolution 2
Both opponents reveal the top three cards of their respective libraries to all players
First opponent in APNAP order privately rearranges their cards into a pile which is later to be placed directly into their graveyard
Second opponent in APNAP order privately rearranges their cards into a pile which is later to be placed directly into their graveyard
Both opponents place their ordered pile into their graveyard
Resolution 3
Both opponents privately view the top three cards of their respective libraries
First opponent in APNAP order rearranges their cards into a pile which is later to be placed directly into their graveyard, and then reveals their pile
Second opponent in APNAP order rearranges their cards into a pile which is later to be placed directly into their graveyard, and then reveals their pile
Both opponents place their ordered pile into their graveyard
Resolution 4
Both opponents reveal the top three cards of their respective libraries to all players
First opponent in APNAP order rearranges their cards into a pile which is later to be placed directly into their graveyard, and then reveals their pile
Second opponent in APNAP order rearranges their cards into a pile which is later to be placed directly into their graveyard, and then reveals their pile
Both opponents place their ordered pile into their graveyard

Related: https://boardgames.stackexchange.com/questions/5367/whats-the-order-of-resolution-for-everybody-discard-in-mtg

This can only matter if playing with certain very old cards, right? Nothing recent or current care about graveyard order?

Well, it matters for compliance.  Tournament policy requires players to maintain the order of their graveyards in formats that permit those older cards, even if no one is actually using them.

On the other hand, outside of tournaments, for example when someone fancies playing a Nether Shadow in Commander, it will be perfectly sufficient for that player to respect graveyard order, while everyone else does as they please.

Also related: List of all cards for which graveyard order matters

It is option 1.
Rule 101.4 is what governs this:

101.4. If multiple players would make choices and/or take actions at the same time, the active player (the player whose turn it is) makes any choices required, then the next player in turn order (usually the player seated to the active player’s left) makes any choices required, followed by the remaining nonactive players in turn order. Then the actions happen simultaneously. This rule is often referred to as the “Active Player, Nonactive Player (APNAP) order” rule.

Note that this doesn't just refer to "choosing a card", but refers to "any choices required". By plain English definition, "arranging cards in an order" is a choice you're required to make.
So all players will make that choice, without revealing anything, because the actual mill hasn't happened yet. Basically, the choice the players are making is how to mill the cards.

I believe you actually mean option 1, not option 3, based on your explanation?

Right, sorry. I missed the "then reveals" in option 3.


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In what category of information are timestamps?
In Magic the Gathering Competitive/Professional REL, there are four categories of information:  status, free, derived, and private.  Under which category of information do timestamps fall?
If timestamp ordering is private information, then if two of Player A's objects with competing static abilities enter the battlefield simultaneously, Player B cannot find out which will take precedence until they actually affect the gamestate.  Player A could also lie about which order has been chosen.
The same applies if timestamp ordering is derived information, except that Player A could not directly lie, but could refuse to communicate the order.  For all practical purposes, these two possibilities imply that Player A does not actually have to make a choice at the time the objects enter the battlefield.  This one also raises the question, "derived from what, exactly?"
If timestamp ordering is free information or status information, then Player B, while resolving Inquisition of Kozilek, can ask Player A for the timestamp ordering of each card in Player A's hand.  It isn't unthinkable that the temporal order of cards in hand could be identified in this way; in Hearthstone, the hand is always ordered with cards to the left being older, and it doesn't break the game.  However, the problem is that Player A is unlikely to remember the order in which each card was drawn in many circumstances.
None of these possibilities are obviously intuitive.  They all feel a bit weird.  In what category of information are timestamps?

What do you mean by timestamp on cards in the hand? The order they were drawn? Timestamps are set by the order permanents entered the battlefield, and as cards in the hand aren't permanents on the battlefield, they have no timestamps.

That's not true. According to rule 613.7d, "An object receives a timestamp at the time it enters a zone." So technically, any object in any zone has a timestamp. It rarely comes up outside of the battlefield, but one example I could find is Torrent of Lava. Its timestamp order with other objects can affect the existence of the ability it grants.

There is a simple answer here that resolves all of the conflicts: timestamps are free information if and only if it makes a difference to the current game state. Otherwise, it is private.
The Magic tournament rules specify that free information includes "Details of current game actions and past game actions that still affect the game state". Any time timestamps are relevant to the game state, they are the result of a previous game action, which makes them details of previous game actions, so they must be public information in that case. Otherwise, timestamps are not included in the quoted category, so they are not free information, and since they do not fit any category of derived information, they must be private.
Rule 613.3 says

Within layers 2–6, apply effects from characteristic-defining abilities first (see rule 604.3), then all other effects in timestamp order (see rule 613.7). Note that dependency may alter the order in which effects are applied within a layer. (See rule 613.8.)

So, the timestamp of an object only matters if the object has a static ability that affects a layer that another active continuous effect also affects. Static abilities never function in the hand, so the timestamp of a card in hand never matters, so that information is private.

The logic here seems to classify timestamps as actions via the rationale that they result from previous actions.  While convenient for the question at hand, surely timestamps are themselves part of the gamestate as opposed to being actions, right?  Otherwise we would have to classify creatures on the battlefield as actions instead of components of the gamestate on the basis that entering the battlefield is an action.

No, that's not what I'm saying at all. The category I'm saying they are in is "Details of...past game actions". If an action simultaneously updated the timestamps of multiple objects, the choice of the order of those timestamps is made during that action and is therefore part of that category. For all other timestamps, the relative order of other actions is also part of that category.

Okay, still when you categorize them as details of past game actions, I think this meaningfully differs from my conception.  I've always imagined a timestamp as a sort of "quasi-object" similar to a counter, associated with another object or quasi-object similar to how a counter exists "on" another object or quasi-object.

Two gamestates identical up to the number of counters in play are not the same gamestate.  I assumed two gamestates identical up to different timestamp order were similarly not the same gamestate, but it sounds like you're saying they are (otherwise all timestamp information would be relevant to the gamestate).

Timestamps are, fundamentally, a total ordering on continuous effects, and a separate total ordering on objects and counters. Beyond that, it's not really useful to try to figure out what kind of thing a timestamp is. What matters is that that ordering is always the direct consequence of some details of previous game actions. As for their effect on game state, your "all or nothing" approach just doesn't really work. Timestamps sometimes differentiate game states, if they are timestamps of continuous effects in the same layer, or of objects or counters that create those continuous effects.


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What number is so large that one would never wish to have named a larger number?
When Magic the Gathering players create infinite resource combos, they often name numbers such as 1 million, 1 billion, or 1 trillion.  Usually this results in Player A creating that many permanents, gaining that much life, etc.
Obviously if Player B can also create an infinite combo to produce a counteracting resource (more attackers than Player A's blockers, more damage than Player A's life, etc.), then the original number can always be out-enumerated.  However, it seems plausible to me that Player A could even be in danger if Player B's combo is not infinite.  For example, if Player B could double a creature's power every turn for 40 turns, then it could easily wipe away 1 trillion life with a single attack, nevermind that it could have been attacking each turn along the way.  Player A might be able to survive attacks and doublings over 120 turn by naming 1 duodecillion, which sounds like more than enough turns to cover any game of Magic.  However, if Player B could double their creature's power multiple times per turn, then this still may not be enough.  In fact, if each turn Player B could create a new copy of the source which doubles their creature's power, then greater-than-exponential growth could be achieved.  Here my intuition is that a Googol life may not be enough to guarantee Player A indefinite safety, while a Googolplex may be, but is impractical to work with.
How quickly is it possible for Player B to deal life loss to Player A, using any non-infinite combination of cards?  Of course, I am not only interested in the life case, but also in the cases of creating an arbitrary number of permanents, or other resources, which may have different thresholds for safety.  For each type of resource an infinite combo that does not win a game outright can create, what number is so large that it is virtually impossible to regret not naming a larger number?

Why would you assume that there is a general number large enough that covers every possible game state? You have to factor in that the stack exists and player B can respond to player A if they're holding priority, and if they're not, it's a non-issue anyway. Also, for non-infinite situations, solving it is a matter of applying simple math (not saying it's necessarily possible to do so in a reasonable timeframe), whereas in general, it seems unanswerable to me. Also, you seem to assume that player A cannot simply start their combo again, which isn't true for all infinite loops.

"what number is so large that it is virtually impossible to regret not naming a larger number?" - No such number exists, a finite number can always be exceeded and an infinite number is not allowed by choice (or is a forced draw). This is a non-question in that it will never ever be relevant to the game.

I suggest reading https://www.soniccenter.org/sm/mtg/megacombo.html for an exploration of the limits of non-infinite combos. Note that that article does not provide a practical upper bound, because it applies deckbuilding and gameplay restrictions beyond the ones in the official rules. See also https://boardgames.stackexchange.com/questions/49938/what-is-the-most-mana-that-can-be-produced-by-tapping-a-single-permanent-in-futu for a similar exploration.

@Nij it is not obvious that every finite number can be exceeded by a bounded combo, which is the sort of thing the poster is asking about. That's why it's phrased in terms of regret: if the only way to beat a number X is by a combo which could get arbitrarily large, then you won't regret choosing X because any other number would have been beaten too.

I think James may have a point. There's probably some argument to be made based on computability theory that, since there are a finite number of cards that exist and you could reasonably assume a finite number of them can be involved in any specific game, the number of distinct sequences of actions is finite. Then you can classify them into sequences that allow infinite repetition and sequences that don't, and out of all the ones that don't, there must be some maximum amount of damage dealt, of life gained, of creature tokens created, and so on. (1/2)

But figuring out that amount seems wholly impractical. It'd be a math problem more than a gaming problem, and probably an excessively difficult math problem given how complex Magic is. (We already have a hard enough time doing this for two-state Turing machines which are among the simplest computational models.) (2/2)

I agree that there must be some upper bound, and since Turing Machines can be constructed in an MTG game it must be possible for a non-infinitely-repeatable sequence of actions to produce as much output as the busy beaver machine with however many states you can construct. So, a busy beaver number is a lower bound on that upper bound, which means that the upper bound is necessarily not computable.

There is no number in general
While there are really big numbers, most of those are likely not allowed under tournament rules because they cannot be explained to the average opponent in a reasonable amount of time.
I assert with minimal proof that the biggest number you could expect the average opponent to understand is a googolplex (if you don't accept this assertion, read the sections below).  This is easy to beat with a finite combo.
Here's a 3 card example:

The emblem from Liliana, the Last Hope's <-7> ability
Doubling Season
Extravagant Replication

Each upkeep, have Extravagant Replication copy Doubling Season.  Each doubling season doubles the number of doubling season tokens you get from Extravagant Replication, and each one in turn doubles the number of zombies you get from Liliana's emblem.  All of this compounds every turn.  This is superexponential growth.  After 3 upkeeps, you have 2059 doubling seasons and 2 * 10625 zombies.  After 4 upkeeps, you're already over a googolplex zombies.  Also of note, at this point, you have crashed any digital implementation of MTG because even the scientific notation for how many zombies you have has more symbols than there are atoms in the observable universe.

What about 1010googolplex?
Still not high enough.  Even if you keep stacking tens, it won't be high enough.
Let's take the combo from the previous section and run it for 30 turns.  That's crazy big.  Add Paradox Haze to increase the rate that the combo iterates and another Extravagant Replication to copy Paradox Haze each upkeep.  Now, your unfathomably large numbers of Doubling Seasons will each double the number of Paradox Hazes you get, and each Paradox Haze will cause the Extravagant Replications and Liliana's emblem to each trigger an additional time on each subsequent turn, causing you to have a super-exponentially increasing unfathomably large numbers of upkeeps every turn, each of which causes super-exponential growth of your mass of Doubling Seasons and zombie tokens.  You rapidly get into numbers that need advanced math notation to even represent.  By the way, we're only at 5 cards so far.  We haven't even started copying Extravagant Replication, or added Devilish Valet, Fungal Sprouting, and Panoptic Mirror.

What if both players are math PHDs?
This way, you can name incomprehensibly large numbers.  Then we get into murky territory, but I still think that a reasonable interpretation of game rules says there isn't a valid number that will be high enough.
Some of the biggest mechanisms formally defined numbers result in numbers whose precise values aren't currently known (recommended supplemental reading).  So, just being to name a really big number isn't enough; I would claim that your number needs to be computable in order to use it in a game of Magic.  And once we have that stipulation, then no number you can name is high enough because you can implement a Turing Machine in Magic.  Thus, any computable number you name can be exceeded by my finite combo of a truing machine that calculates your number plus 20 and deals you that much damage.

Beyond computability
In a lot of formats, there is no maximum deck size.  If we ignore the "you must be able to shuffle your deck unaided" bit in the rules and have no time limit on the game, then for any number you pick as your life total, I can design a deck that will beat it with a finite combo.  As a trivial example, I can make an arbitrarily large deck of Swamps and Relentless Rats which will eventually get there.

Joke Suggestion
Define "Garfield's Number" to be equal to one greater than the largest amount of damage you can deal with a finite combo in any current Magic format, with the sub-definition of the maximum deck size for formats that don't define one is the largest deck than can be shuffled without assistance by any human that has or will ever live.  Set your life total to be that.  If you can get a tournament judge to accept this as a valid number, congrats.

A few MtG formats have limited deck sizes, where at least your highlighted phrase would not apply. In addition, the conclusion that there is no general number just because there is no maximum deck size (meaning that if there were a maximum deck size, the question could more or less easily be answered) doesn't seem to be feasible, going by the other answers.

@TheThirdMan fair point; I've de-emphasized the maximum deck size

Please don't advance the idea of using "advanced math notation" to pick a number in a game of Magic. It's most likely obfuscation of free information and thus against the tournament rules.

@Hackworth would you force someone to write out 10^100 longhand? 10^10000? 10^1000000? At some point writing it longhand is slow play, and were only at high school notation. You pretty quickly get into numbers whose decimal representation doesn't fit on the largest storage device imaginable

@Caleth If you can't write out a number without a defintion/notation that the average player wouldn't know, then the MTR forbids it on communication grounds.

@Hackworth only if the player using that notation can't explain it to their opponent and/or the judge

@hackworth let's say you run the combo where you make a copy of doubling season and paradox haze every upkeep in a tournament.  How would you represent the number of permanents generated?  You rapidly exceed what can even fit on a standard sheet of paper using exponential notation.

@Zags That's another situation entirely. We were talking about players choosing arbitrarily large numbers, not numbers the game mechanics force you to calculate.

@Caleth As per the MTR, yes, if you can answer all possible questions about your number, then it's not obfuscation. But that's obviously not what I'm objecting to.

@Hackworth I'd be reasonably confident in being able to explain Knuth's arrow notation to someone who understood exponentiation

I suspect a judge would have to rule the same for all opponents, regardless of their math proficiency.  The Math PhD could effectively "fail to find" the definition of hyperexponentiaion while "searching their memories."

There isn't one.
The question you're asking is a bit like asking this:

My friend and I can both eat infinite numbers of hotdogs. What number of hotdogs do I have to eat such that my friend can't then eat a bigger number of hotdogs?

There is no such number. Your friend can always choose a bigger number: they can choose the number you chose plus one, or squared, or times/plus any arbitrary number. I could pick an immense number like Graham's number, and they can just name Graham's number plus one. That's by the nature of how infinity works—there's always a bigger number.
You're proposing a scenario in which both my opponent and I might have access to infinite combos. (Using a finite amount of cards, but that's true of every deck.) If I have an infinite combo to gain an arbitrary amount of life during my turn, but my opponent has an infinite combo to attack with arbitrarily many creatures or deal arbitrary much damage to me, they can always pick a larger number than I picked. That's just how it works.
Once your opponent has an infinite combo they've probably won. The exception is if you can completely negate whatever they can do infinitely: e.g. they can do infinite Grapeshot, but you have Protection of the Hekma. There's no out-infinity-ing them on the defensive side.

The question did specify "using any non-infinite combination of cards", so I thought that infinite combos (which I interpret to mean combinations of cards that can produce an arbitrarily large amount of damage/life/permanents/etc.) were ruled out.

@DavidZ To me the question seems to be asking pretty clearly about infinite combos. They introduce it talking about naming numbers for infinite combos, talk about how being able to do something an arbitrary number of times doesn't guarantee survival against an opposing infinite, and then ask “for each type of resource an infinite combo that does not win a game outright can create, what number is so large that it is virtually impossible to regret not naming a larger number?” and that's a question about infinite combos. All combinations are finite; my infinite loop with 5 cards is 5 cards.

Or to summarise: that single sentence in the last paragraph is either about infinite combos and phrased weirdly, or is out of place among every other part of the question including the question that comes immediately afterwards. The question's ultimately about naming a big enough number when you can do a thing an arbitrary amount of times, which is an inifinite combo, and there is no such number you can name.

My interpretation of the question is that player A has an arbitrarily repeatable combo (for which any number can be named) and player B has a combo that is not arbitrarily repeatable but may be fast-growing. It then asks whether there is a number that player A can name that exceeds whatever player B can accomplish. It's obvious that you can't just choose a number that cannot be exceeded by any number the opponent could choose, and by my reading the first sentence of the second paragraph of the question directly acknowledges that.

Essentially if the question isn't asking "what number can I name that won't be out-infinity-ied", I just have no idea what the question's asking and would consider it needs to be reformulated to clarify what's going on. As-is, this is my answer to my reading. Whether or not the combos are truly infinite in practice, they may be arbitrarily large for any theoretical number up to infinity, and at that point they're the same thing.

Player A has an infinite combo and player B doesn't have an infinite combo. Is there a number player A could choose that player B can't exceed without an infinite combo? That's the question, as I read it.

I think this answer is sufficient since it covers the essence of the issue with the question very well, but I'll happily try myself at an alternative phrasing (with more or less the identical conclusion).

There isn't a specific number that generally applies, or a way to phrase it, other than "high enough".
Assuming player B has a non-infinite combo which they will fire after/in response to player A's infinite, but non-repeatable-after-finished combo, and the two combos are able to interact with each other (in all other scenarios, this is a non-issue, and @doppelgreener's answer applies), then it will always depend on player B's specific combo, and the answer always will be "you have to choose a number high enough so that player B's combo won't be able to top it"... which is rather obvious, and a predicament that's more or less already contained in the question.
There are two situations this can be broken down to:
Player B's combo doesn't involve hidden information
In the event that player A sees all combo pieces, player A can determine the amount of times they need to execute their combo by calculating the exact results of player B's combo. At this point, the limitations to figuring out a number are player A's math skills, time contains (especially in tournament scenarios when you're not only encouraged but required to play at a reasonable pace), or a combination thereof, but there's always a number that can be determined that's high enough to beat player B's combo.
Player B's combo does involve hidden information
In the event that player A can't know how many times player B can repeat their combo, there is no way for player A to know what number to choose - they will have to guess.


text:
Does collecting rent later than the turn which triggered it, but before the second player following throws the dice, recompute dependent game events?
According to the official rules of Monopoly,

The owner may not collect the rent if he/she fails to ask for it before the second player following throws the dice.

Suppose Player A's property is landed on by Player B, and neither player acknowledges that rent is owed.  Then Player B rolls and lands on Income Tax, and pays 10% of his/her total worth.  Between Player B's turn and Player C's turn, Player A asks for the rent.
Should the game be rewound to just before Player B paid Income Tax, and the 10% figure be recalculated?  Is the rent in this situation considered to be paid at the time Player A asked for it, or at the time Player B triggered the debt by landing on Player A's property?
If Player A acknowledges the debt intentionally at this time as a strategy, is there any tournament or other official precedent for disallowing or penalizing the play as bad sportsmanship?  Similarly, is it legal for Player A to wait until the last possible instant to ask for rent in hopes that Player B will, i.e., purchase a house on his/her turn and then remove it to pay the rent?
If these are not disallowed, is it legal for Player B to force Player A to collect payment before Player A asks for it?  Can Player B pay Player A without Player A's consent as soon as Player B lands on Player A's property, or between their turns?  Can Player B perhaps even pay Player A after Player B rolls and lands on Income Tax, but before the 10% is numerically locked in?

I could be wrong but the highest rent value is $2000 (Boardwalk with a hotel) which would make the difference in property tax $200 and while that is the most I am not sure how much of a difference it would make outside of the player no longer being able to afford to pay the rent.

I would not consider $200 trivial, perhaps situationally relevant.  And this situation could occur multiple times throughout the game.

$200 is only if they land on Boardwalk with max hotels and most likely it will be much lower then that. As for it happening multiple times in a single game I would think that is pretty rare as a player would have to land on the property, have another roll and then land on income tax. Personally in all the years that I played the game I didn't land on income tax all that often.

This brings up an interesting question. If a player lands on two different properties and can only pay the rent on one (or neither) which player would they go bankrupt to? Would it be the order that they landed on the property or the order that they got asked to pay? It would seem like bad practice to let them go bankrupt to the first player if that rent was missed and the player chose to not ask for it until it could not be paid.

The game does not rewind.
Here is the most relevant bit from the 2009 Monopoly World Championship Rules:

10. What happens if someone calls for rent while the next player has just thrown the dice?
If the dice are already on the table when someone calls for rent, the roller moves their token and takes action for the space they land on. Once their turn is over, the other player can collect the rent owed from earlier in the game.
If the dice are still in the air when someone calls for rent, place the roll to one side and settle the rent immediately. The roller then moves according to their roll (unless they were the rent-payer and were bankrupted on that turn).

It's clear from this that actions that occurred before the collection of rent are left as-is.
Whether or not a player can tactically delay asking for rent in order to force another player into bankruptcy is not directly addressed in the rules.  The tournament rules specify:

7. How long do I have to ask for rent?
In Championship Monopoly, you can ask for rent until the second player to your left throws the dice. This is to encourage good sportsmanship.  E.G. Player A lands on your property, Player B takes a turn. You can still ask for rent. As soon as the dice leave Player C’s hand, you have missed your chance.

The rule explicitly mentions that it exists "to encourage good sportsmanship".  The spirit of the rule is so that the next player can't grab the dice and roll to prevent a player in the lead from being able to collect on rent.  If the payer pointed out that they owed rent and the payee refused to acknowledge that until one turn later to gain material advantage, this would be a pretty cut and dry case of bad sportsmanship.
A payee waiting to collect on rent when the payer hasn't mentioned it is a murkier area.  I think this is unlikely to be a sound strategy due to the risk of losing their opportunity to collect if one of the following players is particularly fast at beginning their turn, but it may be legal.

This answer is helpful and addresses the question.  I would just note that "using an intentional delay to tactical advantage is clearly not in line with good sportsmanship" seems to me an overdrawn inference, at least in light of the linked document by itself.  It is perfectly plausible that tactically utilizing a rule designed to prevent would-be bad sportsmanship is not itself considered bad sportsmanship (I recently asked an MTG question which I believe falls into this category).  In fact, if I was in charge of Monopoly, I would value incentive compatibility over intuition adherence here

and consider the tactic to be fair game, so as not to give players an incentive to lie about their intentions.  On the other hand, I would not permit players to tactically utilize the rule that if the wrong type of card is picked up, it is placed on the bottom of the deck, so ultimately it's a subjective judgment of how to manage the tradeoff until official clarity is provided I guess.

@user10478 I've toned down the sportsmanship claims a bit.  I think that delaying the collection of rent after the payer has pointed it out is bad sportsmanship, but simply delaying for lack of anyone saying anything is much murkier.  It's unusual for a rule on game procedure to mention sportsmanship in the first place (other than delay of game), so there's not a lot of general game rules precedent to go off of here.

Note that the Championship rules explicitly differ from the standard Monopoly rules on the time window for collecting rent.

@L.ScottJohnson how so?  It's still until the second player following the player who lands on the property rolls the dice.

Because in the regular rules, it's just until the next player. See my answer.

@L.ScottJohnson The current monopoly rules say "The owner may not collect the rent if he/she fails to ask for it before the second player following throws the dice".  If this has been different in different printings, then it makes sense that it was specified in the tournament rules.

Current rules on Hasbro.com say "If they don’t ask before the next player rolls the dice, you don’t have to pay!"

@L.ScottJohnson Both  https://www.hasbro.com/common/instruct/00009.pdf and https://www.hasbro.com/common/instruct/monins.pdf say "second player following".  The only ones that say "next throw of the dice" are the "vintage" rules: https://www.hasbro.com/common/instruct/Monopoly_Vintage.pdf

You misunderstand the game rules:

When you land on a property that is owned by another player, the owner collects rent from you in accordance with the list printed on its Title Deed card.

Thus the rent due by Player B to Player A is due when Player B lands on a property owned by Player A; not when Player A demands or acknowledges it. This puts a dual obligation on both players to ensure the correct rent is paid.
As a game expecting ethical behaviour by al participants, it is incumbent on Player B to make prompt payment of rent due regardless of how quickly, or not the owning player demands such. Any player intentionally avoiding payment of rent due is playing unethically (by the official rules). It's of a kind with stealing from the Bank when other players have their attention elsewhere.
The guidance for rectifying the irregularity of payment not being made when due is simply that: How best to restore the proper game situation, and setting a limit on how far back such rectification should be attempted.
This of course assumes play by ethical, attentive, and sober participants. Frat house games played on a temporary table of 6 or more empty 2-4's should simply agree on what rules all participants can be expected to recall and abide by in their progressive state of insobriety or inattentiveness.

This does not appear to answer the question of what to do with the property tax payment that was calculated and paid before rent was paid. While what you say is true about paying rent the issue is it can be missed at times and the rules even have a part for how long it can be missed and still paid.

Something that I just realized is there is another semi related question about players being able to bid more then they can afford on property up for auction that might be relevant to the ethical behavior argument. https://boardgames.stackexchange.com/questions/57400/can-you-overbid-in-monopoly-as-a-tactic

No rewinding. The rent is paid when it is demanded (within the allowable window of time). That's it.
And the rent is not paid until/unless the player asks for it, by virtue of the phrasing "The owner may not collect the rent if he/she fails to ask for it before the second player following throws the dice.."
Also, a tangental note: "second player following" means "next player" in this case. Other editions make this clearer.
Current rules (as of this answer) at Hasbro:

If they don’t ask before the next player rolls the dice, you don’t have to pay!

Wiki:

When the non-owner lands on a property that somebody else owns, the owner has to do something. A property owner must ask that non-owner to pay the rent before the next player rolls the dice.

As well as others editions:
1972:

Note: If the owner fails to ask for his rent before the next throw of the dice no rent may be collected.

2005:

IF THE OWNER FAILS TO ASK FOR RENT BEFORE THE NEXT THROW OF THE DICE, NO RENT IS COLLECTED.

Apparently, one World Championship (in 2009) used yet another rule:

In Championship Monopoly, you can ask for rent until the second player to your left throws the dice.

(emphasis added).
Which is a bit odd, because as written, it keys off of players' position relative to you the property owner rather than relative to the player landing on your property.
Those rules go on to "e.g" their way into the more sensible framework:

E.G. Player A lands on your property, Player B takes a turn. You can still ask for rent. As soon as the dice leave Player C’s hand, you have missed your chance.

But note that this is an explicit change for "Championship Monopoly" to the standard rules. The reason given for the change is "to encourage good sportsmanship".

To me, "second player following" is unambiguous (for the purposes of this question, I only care about regular Monopoly, not spinoffs).  I assume the rule is in place to prevent the next player from racing to roll the dice before rent is acknowledged, which may be indirectly beneficial to them in some situations.  The Wiki page does not have citations for some reason, so I don't know why it says that.

Well, again, that's not important to the answer, but "second player following" is ambiguous. If you take "first player following" as the next player, then "second player following" would be the player after that. So B lands on A's property. Finishes turn. Then the first player following, C, takes their turn. Then the second player following is D. See also "second to last" in comparison to "first to last" (next to last) as in https://english.stackexchange.com/questions/56344/usage-of-second-third-fourth-last#answer-186827

In my mind, your explanation that D is the second player following B is unambiguously correct; the incorrect hypothesis would be that C is the second player following B.  I acknowledge that other similar phrases may be ambiguous.  For example, it is often said that Christ was resurrected "on the third day," referring to the Sunday following Good Friday.  If you interpret this as something like "on the third day of being dead," then the timeline makes sense, whereas if you interpret it as "on the third day following the crucifixion," then it should be Monday.

I do think this matters because the question doesn't arise if C is the second player following B.

Ah, I didn't realize you were reading it the other way (that's ambiguity for you). Yes, C is the second following player. See also this rulebook:https://www.hasbro.com/common/instruct/Monopoly_Vintage.pdf

I read those rules as you can ask for rent later but not what it means in terms of sequence of actions. Such as asking for rent after they pay rent to another player causing them to go bankrupt to you instead of that other player.


text:
Additional Official Rule Sources For Monopoly
Some questions about how to play a game of Monopoly are not answered in the official rulebook that comes with the board game (not surprisingly).  For example, under the header of Building Shortages...

When the Bank has no houses to sell, players wishing to build must wait for some player to return or sell his/her houses to the Bank before building. If there are a limited number of houses and hotels available and two or more players wish to buy more than the Bank has, the houses or hotels must be sold at auction to the highest bidder.

Several questions come to mind...

Suppose multiple houses are available that two players wish to buy.  Is there a single auction for all houses in contention (i.e., with bids like "$100 per house"), or is each house auctioned one at a time?

If houses and hotels are in contention, the result of one auction may influence player decisions in another, or if hotels are auctioned first, even make auctioning houses unnecessary as more return to the Bank.  Who decides the order of auctioning between houses and hotels?

Is there a specific order that actions between turns are executed?  For example, suppose Player A wishes to buy the last house in the Bank and has enough money on hand to purchase it for at least one of their properties they could put it on, and Player B wishes to buy the house but would have to mortgage or trade with another player to acquire the cash to purchase it for any of their properties they could put it on.  Presumably, Player B cannot immediately force the house to go to auction, since they could not legally buy the house normally, but can Player B perform other between-turn actions, such as mortgaging or trading with another player, before Player A has a chance to purchase the house?  How is priority to act between turns managed?

Broadly, I am interested in how these types of questions should be answered as objectively as possible.  I know there are official Monopoly tournaments, so presumably answers to these types of questions have been worked out.  Is there a more advanced rulebook or other official rule source that teaches how to play Monopoly in more detail?

I have found this Monopoly World Championship rulebook and accompanying FAQ. I am unclear on whether or not this was written or endorsed by Hasbro, but these rules seem as good as any until a more definitive source can be found (I'm pretty sure such sources do exist).
To answer the questions according to these resources (replace "houses" with "hotels" as appropriate):

The FAQ states that each player declares in advance how many houses they wish to buy. The houses are auctioned one by one, and once a player has acquired as many buildings as they declared, they are not entitled to bid on other houses. Curiously, the FAQ states that trading with other players is not allowed during this type of auction.

The rulebook and FAQ are silent on this. But I'll note that buying hotels adds houses back to the bank, so the players might agree to resolve the hotel shortage first. But if a player wants to buy hotels but needs to buy more houses first, then they might prefer the shortage of houses be resolved first so they can have a chance at a hotel. So if the players cannot agree on which to resolve first, I think—this is just my opinion—the most sensible course of action is to resolve the situation that came up first. If it started with "I want to build a hotel" (and the player didn't need more houses first), then resolve the hotels first; if they said "I want to buy a house" (or "I want to buy a hotel", but they needed houses), then resolve houses first. For whatever it's worth, the old NES and SNES versions had buying houses and buying hotels in completely separate rounds; you could not try to buy a hotel while other players were declaring their intent to buy houses or vice versa, and you couldn't try to buy a hotel until you already had 4 houses on each property of the color group.

The rulebook and FAQ do not discuss this exact situation, but the aforementioned rule that players may not trade during housing shortage auctions suggests that the intent is that players may not conduct any kind of trading operations during a housing shortage, including mortgaging property. Still, this is only a guess.

I'll go ahead and accept this answer to bring finality to the question.  I actually considered deleting this question, as I have since discovered that there are many additional rule sources for Monopoly varying in how official they are, which include some mutually contradictory policies.  I think the most helpful project in this vein would be to write an expanded version of the official rulebook that incorporates all information from additional sources with citations, and simply acknowledges the contradictions where they exist.  Maybe I'll try to compile such a resources in the future.


text:
How to classify specific instructions as double one-shot effects versus continuous effects?
According to CR 610.3...

Some one-shot effects cause an object to change zones “until” a
specified event occurs. A second one-shot effect is created
immediately after the specified event. This second one-shot effect
returns the object to its previous zone.  If a resolving spell or
activated ability creates the initial one-shot effect that causes the
object to change zones, and the specified event has already occurred
before that one-shot effect would occur but after that spell or
ability was put onto the stack, the object doesn’t move.  If a
resolving triggered ability creates the initial one-shot effect that
causes the object to change zones, and the specified event has already
occurred before that one-shot effect would occur but after that
ability triggered, the object doesn’t move.

However, this rule does not seem to express itself in maximal generality (or perhaps there is another similar rule which broadens the rules' joint scope).  We can infer that the object changing zones is not a necessary condition for applying the prescribed behavior, as Oubliette has a ruling which states...

If Oubliette leaves the battlefield before its triggered ability
resolves, the target creature won't be phased out or tapped.

On the other hand, if my understanding is correct, then if Act of Aggression is legally cast during the first Cleanup Step, its controller will gain control of the target until the relevant turn-based actions during the second Cleanup Step; the game will not look back to the relevant turn-based actions from the first Cleanup Step and prevent control from changing on the grounds that the effect of Act of Aggression should have already expired.  We've seen that the key difference is not the zone change; it is also presumably not the bit about being a specified event, as CR 700.1 states, in part, "Anything that happens in a game is an event," and uses another turn-based action (blocking) as the example, so "until end of turn" should also qualify as a specified event.  The key difference must be that Oubliette is treated as a double one-shot effect as detailed in CR 610.3, while Act of Aggression is treated as a continuous effect.
Is there either some set of necessary and sufficient conditions within an instruction, or some online resource, which classifies it as either a double one-shot effect, or a continuous effect?  Something (kind of) analogous exists in Yu-Gi-Oh! on its wikis, but I have not been able to find this information on Gatherer or Scryfall, or even in the Scryfall API logs.

Act of Aggression is not governed by 610.3 because it doesn't cause the targeted object to change zones. Even if it was, the "the effect doesn't happen" clause would still not be applied.
First, for Rule 610.3 to apply it requires that the object changes zones. Since AoA targets creatures, that object has to be on the battlefield, and the battlefield is a zone that is shared by all players. Changing the controller of that creature does not cause it to change zones. That is why 610.3 does not apply to Act of Aggression.

400.1. A zone is a place where objects can be during a game. There are normally seven zones: library, hand, battlefield, graveyard, stack, exile, and command. Some older cards also use the ante zone. Each player has their own library, hand, and graveyard. The other zones are shared by all players.

Second, even if AoA was governed by 610.3, it would still ignore the first instance of the "until end of turn" event and have its effect of changing control. I believe you misread the relevant clause of 610.3:

If a resolving spell or activated ability creates the initial one-shot effect that causes the object to change zones, and the specified event has already occurred before that one-shot effect would occur but after that spell or ability was put onto the stack, the object doesn’t move.

So AoA would fail to change controller only if the "until end of turn" event happened between casting and resolving AoA. However, that can never happen because that event only happens when the stack is empty and all players pass priority.

Cleanup step

514.2. Second, the following actions happen simultaneously: all damage marked on permanents (including phased-out permanents) is removed and all “until end of turn” and “this turn” effects end. This turn-based action doesn’t use the stack.
514.3a At this point, the game checks to see if any state-based actions would be performed and/or any triggered abilities are waiting to be put onto the stack (including those that trigger “at the beginning of the next cleanup step”). If so, those state-based actions are performed, then those triggered abilities are put on the stack, then the active player gets priority. Players may cast spells and activate abilities. Once the stack is empty and all players pass in succession, another cleanup step begins.

So you cast and resolve AoA in 514.3a, and it will end in the second instance of 514.2. This is the completely normal order of events and not what 610.3 regulates.

Finally, Oubliette is irrelevant to the 610.3 discussion because, contrary to your assumption, Oubliette is not governed by 610.3 but by 610.4, an analogue to 610.3:

610.4. Some one-shot effects cause a permanent to phase out “until” a specified event occurs. A second one-shot effect is created immediately after the specified event. This second one-shot effect causes the permanent to phase in.

In addition to this, the relevant part of 610.3 talks about whether the event "has already occurred", but the end of turn happens repeatedly forever, so there's no meaningful point at which you can say that it "has already occurred" vs "has yet to occur".

@murgatroid99 not quite, I believe. There is no perpetual "end of turn" state akin to game-state triggers, if that's what you meant. The "end of turn" event that ends AoA happens at a distinct point during the cleanup step, in 514.2. If you cast AoA during 514.3a, it will have its control-changing effect until the next cleanup step and 514.2 comes around.

That's not what I meant. I meant what you said in your expanded explanation. My point was that after the end of the first turn of the game, there has always been an "end of turn" event in the past, and there will always be another "end of turn" event in the future, so there's no reasonable point at which you can say that "the end of turn" has already occurred, which is part of the premise of the question.

Yes I see what you mean. I guess that's why, even though 610.3 is defined very broadly in that it any event can qualify, in practice, applicable cards such as Banishing Light all have events that can only ever happen once, such as when Banishing Light leaves the battlefield.

Technically I guess it would be possible for Wizards to have created a rule here on the same principle as CR 503.2. "If a spell states that it may be cast only 'after [a player’s] upkeep step,' and the turn has multiple upkeep steps, that spell may be cast any time after the first upkeep step ends."  It could have been that "has already occurred" looked for the first occurrence that turn.


text:
Does a maximally clumped deck or a mana woven deck require require more shuffling to sufficiently randomize?
I've been under the impression that a mana woven deck (i.e., nonland nonland land nonland nonland land...) reaches a state of randomization after fewer riffle shuffles than the same deck with all nonlands on the top and all lands on the bottom.  More generally, the idea would be that more clumping requires more shuffling to achieve sufficient randomness.
Recently, however, I've heard it claimed that a certain style of weaving (or double pile shuffling) requires twenty riffle shuffles to make random.  This struck me as odd, since 7-9 riffle shuffles are generally considered sufficient.
Do clumped or declumped decks require more shuffling to randomize, or equivalently, given a fixed number of riffle shuffles, is one more likely to end up with a sufficiently randomized deck starting with a clumped or declumped initial distribution of cards?

I think there's some tricky subtext here: you're describing a technique for trying to un-randomize a deck in a very purposeful fashion, and then asking how much randomization will defeat it afterward. Having some links to the specific claims might help because I think their context (e.g. a math article, Judge guidance for tournaments, &c.) changes the character of the question a bit.

You can't un-randomize a deck except by looking at it and stacking it according to card faces. Obviously, doing that resets the effect of any previous shuffles, but any other procedure that does not involve looking at cards cannot reduce the randomization of a deck.

@AlexP The source is pretty low quality, just a random Youtube video:  https://youtu.be/2nXAXMK3V40?t=441 The opposite conclusion (that declumping reduces the amount of shuffling needed to randomize) came from a simulation someone did, but I can't find it now.

Most people underestimate the number of shuffles required to get a deck to be truly random.  Persi Diaconis has spent his career studying shuffling, a good summary article on his results, and the tumult his findings caused is given here:https://www.nytimes.com/1990/01/09/science/in-shuffling-cards-7-is-winning-number.html

I find it amusing that you know 7-9 is sufficient.  20 years ago, that was NOT known, and most people did 3-4.  7-9 is truly sufficient.  What caused this change?  Persi.

The claim in that video is absurd. 7-9 shuffles is sufficient to randomize a perfectly ordered deck. There is no way to get "such a highly ordered space" that you need 20 shuffles.

@user10478 Thanks for the link. If I had to guess, I think the video creator is just saying "20 shuffles" under the assumption that many people's riffle shuffles aren't quite as good as the canonical riffle shuffle technique. (In contrast: if an opponent actually shuffles a deck and then hands it to me to shuffle, I don't really think twice about just riffling it 2-3 times and then handing it back in a friendly / low-stakes game.)

It doesn't make a difference.
No deck in a known order is more or less random than any other deck in a known order. A deck with all of the lands together requires just as many shuffles for randomization as a deck with the lands perfectly distributed. A mana weave starts with looking at the cards to see which are lands and which are not, and doing so resets the deck to a "known order" state and undoes any previous shuffles.
To see why this is true, we can characterize the degree of randomization of a deck by the number of possible permutations it can possibly be in. Any deck with a known order has one possible permutation, and a fully shuffled deck can have every possible permutation. A partially shuffled deck is somewhere in between.
A pile shuffle deterministically maps any given permutation to a specific other permutation, so it changes the set of possible permutations, but always preserves the size, and therefore preserves the degree of randomization.
If you experience that reordering your cards before shuffling (such as by mana weaving) results in a noticeable change in your draws, then you are not shuffling enough.

I would probably note that while a pile shuffle doesn't change the randomisation, mana weaving (or otherwise arranging the cards based on their front face) decreases randomisation. A side note: randomisation should also factor in uniformity in the chance of each permutation.

(I see you did mention that in a comment on the question)

The way I see it, mana weaving and just looking at the front faces of the cards have the same effect of returning the deck to the "known order" state. A mana woven deck isn't less random than any other known order. To the side note, I did say the proof was rough; I'm just assuming that all possible permutations are equally likely, which is probably true of most common shuffling procedures.

Not sure why my previous comment was removed. Do you have any actual proof (mathematical proofs, links to sources, or things of that nature), or is this just your opinion?

I added a paragraph with a proof. If you are unsure about something specific in it, I can try to address it.

"it changes the set of possible permutations, but always preserves the size," is that actually true? Any given permutation after the mana weave can have multiple possibilities for pre-woven states, even if each pre-mana weave state will only produce one state after the mana weave.

A mana weave is a different thing from a pile shuffle.

This question is about mana weaving, though.

Yes, I know. I realized after I made that comment that my answer focused mainly on the pile shuffling point that was a relatively small part of the question, so I edited it again to address the mana weaving more directly.

murgatroid99's answer sufficiently answers the question, this is more of a philosophical take on deck randomisation.
A perfectly random ordering is such that every possible order is equally likely. This is exactly equivalent to each position containing a uniformly random card, or each card being in a uniformly random position.
Another important factor to shuffling a deck, is the amount of knowledge you have about the order of the deck. Eg. If you somehow perfectly shuffle a deck then look at it, from your perspective the number of possible permutations went from N! to 1.
Does this mean that just leaving your deck as is, would be random? Kind of... It's a bit like looking the face of a dice from the last game you played instead of rolling it, but a deck of cards has more possibilities, therefore more room for unintentional ordering. From last time you played, your lands and creatures are probably in clumps. Not to mention, you may even remember what you had last game, and where they are in your deck. Shuffling is a way to mitigate this.
One way to perfectly shuffle a deck, (although time consuming) would be to have another person take your deck face-down, select a card at random (using a dice or other RNG), and put it in a new pile. They repeat this until every card is in the new pile. This satisfies every aspect of being perfectly random; each card could be anywhere, and you know nothing.
Obviously, this is impractical to do, so you want something as close as possible. Multiple (7-9) riffle shuffles are usually sufficient to come close enough to uniform randomness, and to obfuscate your memory of card positions. Since this is the physical world, there are some caveats. It depends on a certain amount of dexterity. If your riffles consistently leave the top or bottom cards in the same position, a few overhand shuffles between riffles should suffice. Or, if your cards are prone to sticking together while riffling, one pile shuffle (without looking!) can separate them again. Note that overhand or pile shuffles alone are generally considered insufficient forms of shuffling.
Pile shuffling alone, deterministically rearranges the deck, thus not increasing randomisation. Equivalently, it's flipping a dice onto its opposite face instead of rolling it.
Mana weaving is more detrimental. It's actively arranging the deck according to a rule. It can never increase the randomisation. Moreover, it introduces information to you about the order. If not specific card places, you now know that after drawing 2 nonlands, the top card is likely a land. It's the equivalent of turning your dice to a face you want instead of rolling.
A perfectly shuffled deck is very unlikely to have a neat pattern of land/nonland cards. Just like how a standard deck of playing cards is unlikely to be shuffled in order of black, red, black, red, etc. You should expect some clumping even in a perfectly shuffled deck.

I like that this answer points out some of the practical reasons why different kinds of shuffling are used for physical cards.


text:
Replacement Effects Modifying Non-Effects
In this video, it is explained that replacement effects do not transform non-effects into effects; an event such as a turn draw modified by a replacement effect is still a turn based action, not an effect.  This is in concord with the definition of replacement effects.
However, this seems to contradict CR 614.16, which states that...

Some replacement effects apply “if an effect would create one or more tokens” or “if an effect would put one or more counters on a permanent.” These replacement effects apply if the effect of a resolving spell or ability creates a token or puts a counter on a permanent, and they also apply if another replacement or prevention effect does so, even if the original event being modified wasn’t itself an effect.

I would have expected such replacement effects to apply if and only if the original event being modified was itself an effect.  What is the reason for this apparent discrepancy?  Is the information from the video outdated, or is the CR quote a thing which literally only applies to token and counter creation because Wizards says so?

Possible duplicate of https://boardgames.stackexchange.com/q/38582/9999; certainly related

There is no contradiction, and there is only apparently a contradiction because you paraphrased the statements that were made in the video you linked in ways that substantially changed their meaning. In particular, the video makes no reference to "effects", and that word is critical to the application of rule 614.16.
The statements actually made in the video are accurate. To be specific, starting at the linked timestamp, the person in the video says

Amy activates Words of Worship during her upkeep, and then goes to draw a card for her turn, while Nick controls Rain of Gore. What happens? So, in this case, it's just like with the lifelink example. Even though the Words of Worship is the thing that's causing Amy to have the instruction modified to become something that involves life gain, the Words of Worship isn't in fact causing Amy to gain life. Rather, what's happening is: there's a turn-based action in the game that makes Amy draw a card, and Words of Worship is modifying what happens as a result of this turn-based action in order to include some life gain. Rain of Gore is not going to kick in here, and that's because it only cares if a spell or ability Amy controls is going to cause her to gain life. In this case, there's a turn-based action that's causing Amy to gain life. And the fact that an ability Amy controls resulted in that turn-based action being modified to include some life gain, well, that's immaterial. The game doesn't care about that, because that's not the actual thing that's causing her to gain life. The thing that's causing her to gain life is the turn-based action, and therefore Rain of Gore cannot affect this event.

For reference, Rain of Gore has the following text:

If a spell or ability would cause its controller to gain life, that player loses that much life instead.

Notably, the text of this replacement effect specifically checks whether the event is caused by a spell or ability. As the video describes, in a situation where an event that is not caused by a spell or ability is modified to include life gain, Rain of Gore cannot effect it.
Rule 614.16 has no relevance to that example. As the first sentence of that rule indicates, it is only relevant to replacement effects that apply "if an effect would create one or more tokens" or "if an effect would put one or more counters on a permanent". There are only three cards that that rule applies to directly: Anointed Procession, Doubling Season, and Parallel Lives.

Good catch, I totally missed that!  Having played much Yugioh before learning Magic sometimes glitches my brain on the ability/effect distinction ("effect" there refers to both).


text:
Do sentence breaks, conjunctions, or "also" have any rules meaning?
Oracle rules texts have several suspicious variations which do not appear to correspond to any differences in rules meaning.
For example, they may have any of the following forms...
[Do Action A] and [Do Action B].
[Do Action A], then [Do Action B].
[Do Action A].  Then [Do Action B].
[Do Action A].  [Do Action B].
Additionally, the word "also" is sometimes, but not always, embedded into an Action B.
Does any of this affect card functionality?
I know that for an instruction to draw or manifest multiple cards, each card draw or manifestation constitutes its own action, i.e., if you manifest two cards from the top of your library you do not get to choose which has an earlier timestamp, whereas if you Collected Company them onto the battlefield you do.  This distinction can also affect how triggered abilities or replacement effects apply.  However, this is determined by the nature of instructions in question; I'm not sure whether their wording in the context of the rules text also matters.
Secondly, there are differences in information revelation determined by the nature of instructions, i.e., players later in APNAP order applying an instruction for all players to make choices simultaneously learn earlier players' choices if cards are being chosen in public zones, but not if cards are being chosen in hidden zones (Rankle, Master of Pranks exemplifies both), but again, I'm not sure whether wording can also matter, such as when players are making choices across multiple actions within a single instruction, or across multiple instructions.
Do these wording differences in rules text matter at all?

Do you have a concrete example of a card where the answer to this question would make a difference in resolving that card's effects?

@murgatroid99 I think you're just asking OP their own question back. If OP knew of cards where these variations in wording would make a mechanical difference, we wouldn't have this question to begin with.

No, I think you have that backwards. If we knew of a card for which we are unsure how to resolve it because we don't know the answer to this question, that provides motivation for resolving this question. On the other hand, if there is no such card (as I suspect), then the answer to this question is essentially independent of the Magic rules: the answer could just as easily be yes or no, and it would have no bearing on any possible game of Magic.

I think Hackworth did get the gist of the question, essentially, "is there ANY rule which treats the given rules text templates differently?"  It must be the case that there either is or isn't such a rule.  The danger I faced in asking less abstractly was failing to instantiate the appropriate rule, i.e., if it turned out that "discard 1 card and your opponent discards 1 card" did not allow the NAP to see the AP's choice before choosing, but "discard 1 card, then your opponent discards 1 card" did, I would run a substantial risk of listing "and" VS "then" examples without the relevant effect.

Consider this: rule 608.2e effectively answers this question for instructions that call for multiple players to make choices. If the only actual cards for which this question makes a difference are cards with instructions like that, then that rule answers the question. Otherwise, it doesn't. So, it does matter what existing cards this question makes a difference for.

No, you simply apply the rules of the English language unless the wording matches a specific pattern in the rules that would signify some property, such as an ability being triggered, or it creating a replacement effect. The differences in wording you list are for better readability/clarity, to create a more compact and comprehensible card text.
Grammatically, the things you listed are conjunctions. The are used to join together separate sentences that have a relation to each other, such as when they happen in a certain logical order, or they refer to the same thing in the same or a previous sentence.
Lobotomy has all 4 of these conjunctions you listed combined into one card. It serves as a convenient example of why the different wordings are applied even in the same card.
Let's break the card text down into clauses first:

Target player reveals their hand, then you choose a card other than a basic land card from it.

Search that player's graveyard, hand, and library for all cards with the same name as the chosen card and exile them.

Then that player shuffles.

In clause 1, we could get rid of ", then" by rephrasing it as the equivalent:

1a Target player reveals their hand.
1b Choose a card other than a basic land card from that player's hand.

In clause 2 we could get rid of the first instance of " and " by rephrasing it as the equivalent:

2a Search that player's graveyard for all cards with the same name as the chosen card and exile them.
2b Search that player's hand for all cards with the same name as the chosen card and exile them.
2c Search that player's library for all cards with the same name as the chosen card and exile them.

We could further rephrase clauses 2a-c  to get rid of the remaining instance of " and " by separating the "exile them" instruction into its own clause, but you can already see where this would be going.
Clause 3 could simply be rephrased as:

3 That player shuffles.

You can imagine the huge chunk of text that would result from the omission of these conjunctions. It would still be perfectly functional card text, but, apart from the fact that you can't fit so much text on a physical card in standard font size, it would be much more difficult to comprehend what the card is even saying mechanically.


text:
Has there been any attempt to learn the impact of replacement on win rate?
Deck A is a 60-card deck, with each card present in only one or two copies.  Deck B is a 120-card deck, consisting of two additional copies of Deck A merged together.  As a simplifying assumption, let's say there are no cards in the decks that allow players to search; every instance of accessing the library is random.
Deck A and Deck B should function very similarly.  In terms of win rate against other arbitrary decks, my hypothesis would be that Deck A performs somewhat better, as it should tend to be bad, all else equal, to draw multiple consecutive copies of the same card or even card type (having lands and non-lands alternate seems optimal), and drawing a card in Deck B effectively only removes half of that card from the library, relative to drawing a card in Deck A.
We could extend this setup by merging any number of copies of Deck A, ignoring the actual game rules, and in the limit, the result should be identical to running Deck A, but drawing with replacement, instead of without replacement as trading card games are always played in reality.
Has there been any attempt to measure the impact of anything in this ballpark, either drawing with replacement versus without replacement or using the notion of partial replacement captured by merging a finite number of copies of a deck, on win rate?  I have seen a few A.I. algorithms trying to emulate games of Magic to compare the impact of specific decisions on win rate, and I was wondering if this has been tested, or if it is even possible to use mathematics to put some bound on the answer.
I find this question interesting in the analysis of Yorion decks, for example.  People often discuss the drawback of not drawing your best cards as often in larger decks, since the ratio of those best cards to total cards is lesser, but I am interested in how much of an additional detriment the replacement factor is.

To my understanding of mathematical terms, none of what you describe is "replacement."  (Cards which shuffle back into the library after being used would be replacement, and that's actually a mechanic...)

There is discussion about "deck thinning", which is reducing the likelihood that you will draw additional cards which are less desirable later in the game (like using a fetch-land to remove an additional land from your deck and improve your chances of drawing "business").

@Jedediah My claim is that if you take an arbitrarily large number of identical decks and shove them together into one huge deck (ignoring rule violations), drawing multiple cards from the huge deck without replacement is functionally equivalent to drawing multiple cards from one of the initial decks with replacement.

@Jedediah I'm guessing this is referring to a technical term from probability theory: to "draw with replacement" means to select a random element (a card) from a set (deck) without changing the probability of getting each element in subsequent draws. It doesn't necessarily mean that cards drawn would actually be placed back in the library.

I'm not sure of the math exactly, but I'm sure you can demonstrate that as the number of copies increases and you approach "drawing with replacement", you will see an increase in the variance of the distribution of the first N draws, for a fixed N. And I would say it is a generally accepted fact that consistency is valuable and variance is detrimental to winning games.

As a very simple example, if N=2, then if you draw from a 60-card deck, there are 60 possibilities for the first card, and 59 for the second. On the other hand if you consider drawing with replacement, there are 60 possibilities for the first card, and 60 for the second. So the first case is more constrained, and the difference increases as N increases.

I probably would have voted for ^ as an answer :P  Is the OP looking for specific MtG vs generic probability?  Not convinced this is a MtG specific Q now, just maybe bounded by Magic rules.

@joedragons murgatroid99 is indeed describing the distinction between replacement and non-replacement I had in mind.  What I'm hoping for in an answer is some research into how much of an impact this has on win rate, at least for a few example decks.  Similar questions have been approached via simulating actual games of MTG.