text:
In Gin-Rummy, when not to Knock?
In Gin-Rummy, besides several exceptions, in most rounds the best move is to Knock as soon as possible.  What are those exceptions? When it is better to go for Gin / Undercut?
While this question generate some degree of opinion based on expert experience, I presume that some guidelines are a consensus.

From Rummy-Talk: When to Knock, Play for Gin, or Underknock

Knock, under most conditions, should be utilized at the first opportunity, with a few
  exceptions:

When the odds are in favor of your going out in a game, or obtaining
  enough score to catch up to your opponent.
When it is a reasonable
  assumption on your part that you stand to be underknocked by laying
  off his cards.
When you know that your opponent is not in a position
  to knock his hand
When you have your opponent’s hand dead.
When your hand has the characteristics of a gin hand.


text:
How to measure the difficulty of a deal in Klondike-Solitaire?
I want to measure the difficulty of different layouts in Klondike-Solitaire.  My question is, what are the parameters that determine the difficulty of a layout?

By layout I mean the permutation of the cards.  The difficulty is also a matter of the rules (deal-1 or deal-3 cards from the stock to waste (talon), assume the rules are fixed on one of the options)
At Microsoft Solitaire Collection, there are 6 difficulty levels.  From my personal experience, the differences between the first 5 are minor compare to the huge difficulty difference between the second hardest (Master) and the hardest level (Grandmaster).
I thought about the following parameters:

Solvable?
number of moves till solution
number of undo's
time taken (for a given user/solver)

Knowing which percentage of folks who finished a level vs the population who tried that level would seem to give a good approximation of difficulty.

@chicks, that might be the easiest and most accurate solution.  However it needs many players to play that level.  I want to get a difficulty estimation before letting players to play, then adjusting my estimation based on their results.

From a human perspective I suppose one's interpretation of difficulty is quite subjective: what one person might find difficult another may find easy. If, for our purposes, we use a rough definition along the lines of, "how much time the average player spends on a layout", then there are a few interesting things we can say about the relationship between a layout's difficulty and its general features.
As your first point suggests, the solvability of the layout (i.e. does a sequence of legal moves exist that results in a win) is obviously key. Let's consider this case first:
Unsolvable Layouts
If a layout is not solvable then in one sense it's infinitely difficult. However, even for unsolvable layouts, the depth of the search tree can reflect a kind of difficulty.
In the extreme case, if to begin with there are no legal moves available in the tableau (main cards) or the stock, then although the layout isn't solvable, in a sense it is easy because you can immediately give up. Alternatively, a layout may have a huge number of promising moves, only to turn out to be unsolvable in the end.
Using a tool like Solvitaire (full disclosure: I am one of the authors of Solvitaire) can identify unsolvable layouts and record the number of unique states that needed to be searched to prove that there's no solution (often a very large number of states!). Another good solver is Klondike-Solver, although I'm not aware of what metrics it reports.
This isn't a perfect measure by any means. Humans aren't computers, and our brains probably aren't using depth-first search in the way a solver might. Without looking at human gameplay data though, this is probably the best heuristic of difficulty for unsolvable layouts.
Solvable Layouts
As was the case for unsolvable layouts, if one has access to human gameplay data then that's almost certainly going to be the best source of understanding how difficult humans will find a layout.
Let's assume though, that we don't have this data available, but we do have a computerised solver. Just looking at the static cards in the starting layout is unlikely to tell us much, but by running the solver we can deduce more about a deal's difficulty (for a human).
Here are some heuristics one could use instead:
Size of Search Tree
For a computer this is important, but in the solvable case it may not tell us much about human play. For instance, the search tree for a layout could be huge, but have one solution that's very obvious to a human. We would not want to classify such a layout as difficult just because it has a large search tree.
Depth of Shallowest / Best Solution
Based on the above logic, the depth of the shallowest (i.e. "best") solution in the search tree is perhaps a better measure. However, this still may not be ideal. We might label a layout as easy because it has a shallow solution, but if this solution is extremely hard for a human to spot then our heuristic will be misleading.
Number of different solutions
This heuristic could be quite accurate, although it's hard to say. Having a lot of different solutions in the search tree would suggest a human player is likely to come across a solution sooner, but there is no guarantee. This would certainly be something worth exploring further.
Number of states searched until first solution
This is probably the most obvious heuristic: how many states did it take before the solver came across the first solution? However, this measure can be deeply flawed. There is no guarantee that computerised solvers search in anything like the kind of order a human would search in (Solvitaire certainly doesn't).
Say move "A" obviously leads to a solution in a few moves, whereas move "B" leads in a very different direction. There is no guarantee the solver picks "A" first; it might instead pick "B" and try thousands or even millions more moves before it backtracks to the point where it thinks to try "A".
To make this kind of method work one needs a policy for which moves to try. For Solvitaire, which just wants to explore all possible moves, this doesn't matter, but for determining difficulty this is very important! If one could come up with a  policy which reflected closely how an average human would play, then the number of states searched until the first solution would be an excellent metric. But coming up with such a metric is hard, complex work.
What Is A Move: Dominances and K+
In all of these discussions we've talked about moves and states. But how we define a move in Klondike is actually not trivial.
Firstly, consider the stock. The standard rules require we turn over 3 cards in one go. A consequence of this is that at different times there are different groups of cards that are effectively "available" to us.
Bjarnason et al. use a solver which can move any of the available stock cards into the tableau in a single move (they call this the K+ representation). For a human, this is like saying "I remember there's a 5H in there, so I'll just loop back over the stock to get it". From the point of considering difficulty, one may wish to consider this sort of thing to be a single move, rather than several separate moves.
Finally, in our paper on Solvitaire we consider something called dominances. These are cases in which we can prove that one of the available moves is guaranteed to be a good move. For instance, sometimes it can clearly be shown that "putting up" say an ace, is always the right thing to do. This is often obvious to a human player and one might not wish to count this as a move (Solvitaire doesn't count this as an extra move).
In summary:

One must first consider solvability
Then if a layout is solvable, one can begin with easier, but probably less accurate methods like the number of possible solutions, or the depth of the best solution
Finally, and only if one can craft a good (human-like) policy for moves to try at each search step, a better approach may be to measure the number of states the solver searches until the fist solution

Easiest solution: Let people play those layouts
The best estimator for how human precept difficulty is, well... humans.  If you can let human players play then their win-rate[%] along with time taken and backtracks should produce a very accurate difficulty function.
Reverse engineering spots the hardest difficulty level
I manually solved 3 layouts from each of the 6 difficulty levels and store their initial card layouts. Then used Charlie's Solvitaire to solve those layouts.  I Hoped to be able to distinguish between all 6 difficulties using the output of the solver.  However only the Grandmaster level produce distinct outcomes. The differences between the others 5 levels in terms of Solvitaire output were somewhat ambiguous.
Second attempt: Compare optimal solutions
Solvitaire finds a solution, I thought that a better comparison will be to compare the optimal solution, by optimal I mean least number of moves.   So I adjusted Solvitaire from DFS to search ID-DFS (start with a solution from DFS, then use bounded-DFS that search for a solution with less steps. Repeated with smaller and smaller bounds until there is no solution, which means that the last solution is optimal).
Comparing optimal solutions did not produce better separating results.
However I think that this process could work if it uses more data and better difficulty function.
Solvitaire output has those 10 parameters:

is solvable?
States searched
Unique states searched
Backtracks
Dominance moves
Final stats in cache
final buckets in cache
maximum search depth
final search depth
time taken

I used parameters #1,2,3,5,9,10 from the DFS solution and #1,2,3,5,9 from the ID-DFS
Solvitaire (DFS)

I modified Solitaire to find the optimal solution using iterative-deepening (IDDFS)

Made a difficulty function using the marked parameters:

By any chance is the code for your version of IDDFS Solvitaire public? I'd love to take a look!

Unfortunately it is not public, I did it as a project to a gaming company that made a commercial version of solitaire.  However it is not too hard to transform Solvitaire into IDDFS, I remember there was one pitfall: Solvitaire store the nodes that were found to be a dead-end in order not to open them again, however in a bounded version of it a node could seems a dead end if it has only X moves left but to be part of a solution if it has X+Y moves left.  So If it got to the dead-end-node with less moves it should try to open it.

I will explain it better with an example:   Say you do bounded DFS with a bound of 85 moves.  And say you got to node 'x' after 82 moves, then the DFS found no solution in any of the 3 moves from 'x', so he mark 'x' as a dead-end.  Latter the algorithm gets to 'x' after 70 moves.  But without a update in the code, it will not look if there is a solution from 'x' since 'x' is marked a dead-end.  I added to that check the number of moves it got the node, and if the current visit is lower, is still open up the tree from that node.

source: Stack Exchange

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

sort: votes


text:
Is it possible to go from a Backgammon opening position to a Nackgammon opening position in 6 moves?
Is it possible to go from a Backgammon opening position (left picture)  to a Nackgammon opening position (right picture) in 6 moves? And if not, how many turns would it take?

I don't have a lower bound on number of turns. However, 6 is not enough. (I'm assuming you meant 6 total turns, not 6 turns each.)
You can work backward just from the second player finishing their cleanup. Assuming WLOG that the second player is red.
Red's final move both have to be to 23 or 24 from Bar or 24, and can't take any white pieces.
White's final moves both have to be to 1 or 2 from Bar or 1.
So, in its first two moves, red has to take a minimum of two white pieces in the white home board while moving a piece from 13 to 2 and a piece from 6 into a takeable position. If done as part of the second move, that takeable position has to be 1 or 2.
Every combination I can find that allows red to do this in two moves leaves white unable to fulfill its similar conditions on turn 2.

Yes, it is possible to do it in 6 moves.
This is a problem posted by Stein Kulseth in the year 2000. He also gives the solution and describes all the minimum-length games that produce Nackgammon's opening position.

It is fairly easy to see that 5 moves are not enough, but there are a number of 6-move solutions. In order to have 4 men sent back these must follow this general outline:

Player A leaves a forward blot
Player B hits from the 24-point and also leaves a forward blot
A hits B's forward blot when entering and again hits in his own board while leaving another forward blot on the 2 point
B hits A's forward blot when entering his 2 men and again hits in his own board while leaving yet another forward blot on the 2 point
A enters 2 men, hits and tidies up
B enters and tidies up

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
When is three of a kind a valid set in Indian Rummy?
Is the following a valid set in Indian-Rummy when we're using three decks of cards? What about when we use two decks?
6♠ - 6♠ - joker

Welcome to the Boardgames.SE, be sure to read through our help pages to get an idea of what this site is all about.  https://boardgames.stackexchange.com/tour

@NeilMeyer FYI you can write [tour] to get a magic short link to our [tour] :)

That set is acceptable only in Indian Rummy's three deck variant. (I'll call Indian Rummy by its name Paplu here for some better clarity.)
Two-deck Paplu and regular Rummy only has sets (same card value, different suites) and runs (same suit, sequential card values) and that move isn't legal in this format.
Three-deck Paplu allows the set type you describe, called a tanala:

When three decks are used it is customary also to allow an additional type of combination consisting of three identical cards - such as ♣10-♣10-♣10. This is known as a tanala, and is treated as equivalent to a straight run.
— Pagat's definition of Indian Rummy

The joker is a wild card that can stand in for any other card at all, which means it's legitimate for completing the tanala.

all_licenses: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/3.0/


text:
Online Pathfinder
I'm looking for a website where a GM could create and design a Pathfinder game and host it on that site. The site must include:

Generators of NPC, encounter, dungeon, treasure, etc.
Players play "on that site"! That is to say, the player control the character by clicking the associated interface on that page. Whether it's movement or spell accessing.
Chat box or voice communications to show the interactions between the players.

Is there such a incredible website? Appreciated!!!

This appears to be about the RPG Pathfinder and not the Pathfinder Adventure Card Game.  As this question has escaped notice for so long, I'm closing rather than migrating.  If needed, please re-ask this question over on rpg.stackexchange.com

There doesn't exist anything that fulfills all your requirements, but there are several good options that comes close. One of the reasons are licensing, if anything outside the SRD are to be legally included. What you are looking for is called a Virtual Tabletop (VTT) environment, and here are four that comes close:

Roll20

Website

Free, paid user get some advanced features

Have (free) automated character sheet for Pathfinder (1E & 2E) available, among with tools to run a game, and chat environment where even complex rolls and other things can be outputted

Pathfinder modules with maps, monster stats/tokens, handout available to purchase. (2E: Lots, 1E: limited)

Paid accounts have access to more advanced feature like dynamic map lightning, and can use advanced community-created API Scripts

Scripts exist for treasure generation, and the rest can be created with API Scripts

Astral

Website

Free, paid user get some advanced features

Pathfinder (1E & 2E) character sheets and some modules supported

no methods for advanced generators

Fantasy Grounds

Program: Windows/Mac/Linux

Game Master needs to buy, free for players

Have character sheets for Pathfinder(1E & 2E) available, among with tools to run a game, and chat environment where even complex rolls and other things can be outputted

Extensive catalogue of Pathfinder (1E & 2E) modules & sourcebooks  to purchase; with maps, monster stats/tokens, handouts

Addons exists for various automation, treasure & encounter generation should be possible for PF with some tweaks

D20Pro

Program: Windows/Mac/Linux

30-day trial, one-time purchase

PF SRD and character sheet support for both 1E & 2E, limited modules & sourceboks

Beyond these options with official support from Pazio and content available, you can look at other VTTs that simply have the online rpg framework to play with, but might lack pathfinder-specific help. For some of the smaller VTTs you might need to created yourself, and the available community/documentation will be much smaller. There are VTTs with 3D environments available.

Welcome to Board and Cardgames Q&A. Nice first answer!


text:
RISK The Battle for Medieval England Ambiguous Card
First time playing Risk: The Battle for Medieval England and we encountered a power card that is pretty powerful but also very confusing.  Its a power card that once flipped over must be played immediately and it says:
"Lose 3 Troops From Each Territory In the North."
My opponent flipped it and said we all have to lose 3 troops. I was confused because it was his card so I was thinking just his territories. Does everyone lose 3 troops?
I would have written this card as All Territories in the North Lose 3 Troops.


text:
What are the legacy style games?
What are the "legacy style games"? I've seen them referred to as being a thing, but an actual definition turnd out to be consistently difficult to google due to generic terms involved.
I know a game called "Pandemic legacy" started the trend and "Risk legacy" is one of the followups, but I've never played either of those games (nor their base versions) and while I've read their descriptions, I can't tell which particular aspects of them were introduced to make them "legacy".

The legacy bit is the permanently altering the game during the course of play. Each play session leaves its legacy.

@L.ScottJohnson More accurately it is playing the same game multiple times with each session impacting the next and generally played with the same group of players.

According to BGG listing Risk Legacy was from 2011 and Pandemic Legacy was 2015 afterward.   I don't know if Risk was the first 'legacy' game but it's the first I recall using the title.

"same group of players" is just common practice. It's perhaps better described as an emergent trait of many legacy games. It isn't a required trait or a trait that makes a game a legacy game, however.

@StartPlayer looking at the games BGG labels with the legacy mechanic, it's the oldest.

The short version is that a legacy game is a game that is played as a campaign, where you finish the game multiple times over the life of the campaign. Legacy games are generally played with the same group of players and permanent changes will happen based on the result of each session. As you play this means the game will get a different feeling and at the end it can be drastically different than the start. You could end up altering or destroying various pieces and parts of the game.
As a note there are people who have set up instructions to make a copy of a legacy game replayable from the start.
https://en.wikipedia.org/wiki/Legacy_game

Legacy games are designed to be played over the course of a campaign, usually with the same players, and permanently change over time.[3][10] As such they have been compared to tabletop role-playing games like Dungeons & Dragons.[11] New rules can be introduced as the campaign goes on, allowing for the game to expand both mechanically and thematically.[7] Games can use the expanding campaign as a mode of storytelling; Pandemic Legacy: Season 1 uses a three-act structure to tell its story.[4] Daviau describes legacy games as "experiential" in contrast to traditional games, which are "repeatable".[12] He compared his legacy games to that of a concert where you "buy a ticket for an experience"[13] while Haoran Un of Kotaku describes the idea as "avant-garde performance art".[11]
Legacy games break certain covenants that players expect from traditional board games.[14] Permanent, physical changes can occur to components based on game outcomes and player choices.[2] For instance players might be instructed to write names on cards, place stickers on the game board, or destroy some components.[11] This causes each copy of the game to be unique at the end[3] and has earned the legacy genre criticism in that there is a finite amount of replayability.[7][13] Some games have been designed to be replayable with refill packs or non-permanent stickers while others are still playable with the final permanent changes once the campaign is over.[15]

From BGG:

Legacy Game - A multi-session game in which permanent and irreversible
changes to the game state carry over to future plays.

See BGG:Legacy for a list of 78 (and growing) of games with legacy mechanism.

I can see where it came from, but without recently acquired knowledge it's not a very clear description. Any sufficiently complex game might be played in multiple sessions, so that's not really a signifying feature.

@VienLa, but if you say that, it sounds like you missed the "permanent and irreversible changes" part. Of course it could say that those permanent and irreversible changes also (often) apply to the the game components too...


text:
MTG Mirage Mirror copying Medomai the Ageless
Hey guys just a quick question. Last night I played an EDH game and I have a Medomai the Ageless onto the battlefield. It's my opponent's turn and he uses Mirage Mirror to copy Medomai.
He used Chainer, Dementia Master as the commander. So on his turn he copies Medomai with Mirror, use Chainer to bring back Gary and deals X damage to me and another opponent, he attacks my opponent with Mirage which is a copy of Medomai, the person doesn't have flying blockers so damage gets through triggering Medomai ability to get an extra turn. Then after finishing his turn he sacrifice Gary with a sacrifice outlet and Mirage Mirror becomes the artifact again. Now he starts the extra turn doing the same thing (copying Medomai, bringing back Gary with Chainer, attacking the opponent with Mirage, damage gets through and gets an extra turn, sacrificing Gary, Mirage becomes the artifact again) and the loop goes and goes.
In this scenario Mirage Mirror copying Medomai over and over and getting combat damage through does not get affected by Medomai's second ability (Medomai the Ageless can’t attack during extra turns.)?

Somewhat surprised no one has pointed out that Medomai is a legenedary creature, so it (or the mirror) goes to the graveyard upon copying.  Not a big deal if you have enough mana and Chainer on the battle field.

@John, that's because Mirage Mirror is copying an opponent's Medomai, so the Legendary rule isn't relevant.

@John with M14 the Legend rule changed from one in the game to one per player - clone effects no longer kill opponent legendaries.

"[Cardname] can’t attack during extra turns" places an attacking restriction on the card.

508.1c The active player checks each creature he or she controls to see whether it’s affected by any restrictions (effects that say a
  creature can’t attack, or that it can’t attack unless some condition
  is met). If any restrictions are being disobeyed, the declaration of
  attackers is illegal.

Therefore, any creature with that restriction can not be declared as an attacker during extra turns. It does not matter how the extra turn was created.
Note that Medomai has two separate abilities. They happen to work together by design (or rather against each other), but they are not connected ruleswise.
Now the restriction only applies while declaring attackers. If Mirage Mirror was somehow attacking before it turned into Medomai (e.g. March of the Machines), that player would continue to get extra turns from Medomai's first ability. Similarly, it will also work if Medomai or a copy thereof is put onto the battlefield tapped and attacking (e.g. Ilharg, the Raze-Boar).

Thanks that explanation was clear and complete. Thanks :)

It totally does. Medomai doesn't care how those extra turns were created, or even whose extra turns they are. That loop doesn't work.
Source: The ability says "~ can't attack during extra turns." They were attacking with it on extra turns.

But since Mirage Mirror becomes a new instance of Medomai every turn, it will ignore that second line of the creature? I think it will apply if Medomai is still on the battlefield after being cast...

@JorgeArturoAdolioCascante no it wouldn't.  All that matters is that you are currently in an extra turn and the Mirage Mirror is currently a copy of Medomai.  You are not allowed to attack with Medomai on extra turns, even if it just came onto the battlefield or didn't start the turn as a Medomai.

@JorgeArturoAdolioCascante Why wouldn't it apply? It doesn't say "~ can't attack on extra turns granted by ~" it just says "~can't attack on extra turns" They are still extra turns even if they were created with a different Medomai, or by say a Lighthouse Chronologist, Medomai still can't attack.

"Medomai the Ageless can’t attack during extra turns." All that Medomai cares about is "Is this an extra turn" it doesn't care what created that extra turn, so it can't attack on an extra turn it created, a different Medomai created, or one that was created by say Nexus of Fate. If it had been written "Medomai the Ageless can’t attack during extra turns created by Medomai the Ageless." it would be self referential and only care about the extra turns that copy of Medomai gave, and this combo would work.
Saying a creature can't attack means you can't declare it as an attacker as normal, however it can be put in tapped and attacking legally. In this way Medomai combos well with Ilharg, the Raze-Boar or Yore-Tiller Nephilim (with a way to get it back in the graveyard). It also can create more than one extra turn, if it were given double strike or extra combat steps (I have one in my Rafiq of the Many EDH deck along with Finest Hour to give me 4 extra turns) since it can attack on other combat steps and gives turns every time it hits an opponent.


text:
MTG: Arena. Is there any way to use packs other than open them?
I was expecting there to be someway to Draft with the packs I have, but I can't find anyway to do that.
Is there ANYTHING that I can do with packs other than open them and claim the cards.
If not, is there ever any advantage to delaying opening packs? (Any times when packs are more 'efficient' to open? Any advantage to having a surplus of packs etc.)

One reason (said with light quotation marks) you can't use Arena's packs for drafts is that they contain different amount of cards than the packs used for drafting. I'm sure they could work around this somehow, but they have decided not to.

I find it more economical to save my ingame currency for a draft instead of buying boosters. I get more cards, I tend to get more rares (since the KI sometimes doesn't pick rares in 2nd and 3rd boosters), and I get at least one additional booster.

No, once you have a pack in your pack inventory on Arena there is nothing to do with it other than opening it.
Arena's packs aren't the same as paper magic draft packs, where draft packs have 15 cards (1 mythic/rare, 3 uncommon, 10 common, 1 basic land (yes you ARE officially supposed to draft the basic lands too)) arena packs contain 8 cards (1 rare, 2 uncommon, 5 common) which are not useful for drafting. Arena does use regular size (excluding basic land slot) draft packs for their limited format games, but you only get these packs when entering one of those sealed or draft events in game.
As for your second question, if there is any advantage to delaying opening packs, yes there is. Arena's system isn't totally random when opening packs, it intentionally reduces giving you more than a playset, this doesn't apply to draft packs which is properly random. This means drafting more first then opening packs later gives you a lower chance of getting unusable duplicates than opening packs and drafting later. The trade off is you have less card inventory for constructed formats while packs are sitting in inventory unopened, so which method is better depends heavily on what formats you prefer.
Arena has two mechanics for dealing with 5th+ copies of cards, something that is very common for heavy drafters who will pick the same good cards often. First, the vault, handles low rarity duplicates, commons give one point, uncommons give 3 points, and for each 1000 points you get to open the vault, awarding 6 wild cards (1mr, 2r, 3uc). The second mechanic handles high rarity cards, and awards gems, 20 for a rare, 40 for a mythic rare.


text:
Is the scry mulligan rule implemented in the rules at large yet?
On Friday, my local store said the mulligan rule being tested at PTORI (Before starting the game, any player who mulligans to less than 7 gets to scry 1) had been implemented into the game proper. But there's been no announcements on wizards.com to that effect, and no source hosting the comprehensive rules reflects this in 103.4. The TO just said that "mtgsalvation confirmed it", but their wiki doesn't, their front page doesn't, and the last relevant forum post I found was from mid-July, before the pro tour actually happened. I know it's looking likely that it'll stick around, but they were pretty clear that only PTORI would be affected for the time being. Am I missing an announcement somewhere?

The new mulligan rule was officially added starting with Battle for Zendikar prereleases on September 26-27, 2015 (or October 9-12, 2015 for Magic Online).

As of this question being asked, it was definitely not a rule yet. In fact the Latest Developments column on August 7 said:

We will be looking at both how the Pro Tour played out, as well as the reactions from the players regarding the rule. I don't know when we will have a decision based on this, but we hope to have new information fairly soon.

Anything official will surely be on the official site, so I wouldn't bother digging on forums or anything - things like this happen in official ways, not just via rumors and forums.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/3.0/


text:
Why would you want to play with a deck that had 3 or more colors in it in Magic? (limited and constructed)
Background: I'm currently writing a strategy guide for magic as my pet project, and I'm trying to explain decks with multiple colors. Here's what I have so far:

Dual color decks are useful because they allow you to take advantage
of the best of 2 different colors instead of just 1. This allows you
to minimize the weaknesses a mono-colored deck might normally have.
For example, red decks usually have trouble dealing with
enchantments, blue decks tend to have pretty bad creatures, green
decks tend to be fairly bad with removal spells, etc. When
combined with another color, most of these disadvantages are
drastically reduced as you can use cards from each color to compensate for the
other's weaknesses.
Some decks also have more than 2 colors, although they are much less
common. With 3 colors your deck will have plenty of options to reduce
the shortcomings of any individual color as odds are there's a card in
at least one of the other colors you're playing that will fill that
gap nicely. The downside of a tri-color deck is that it's substantially
more difficult to get the mana you need to play the cards in your
hand. This means you'll need to devote more of your deck to cards that
can generate mana of multiple colors, which detracts from your overall
focus of getting cards out onto the field that actually win the game
for you. Tricolor decks often include green because it is the best color for generating
mana in its own and other colors, making it great support for the rest
of the deck. These decks often rely on tricolor spells for finishers as they are
often extremely powerful to make up for the difficulty of casting them,
to the point where some can almost win the game on their own.
Decks with more than 3 colors are extremely rare as there is usually
relatively little advantage a 4th color will add that couldn't be
covered by the others, and it is VERY difficult to create a
successful 4 color mana-base that will let you play all of the cards
you need to in a timely fashion.

When I read this, it sounds like I'm saying "Tri-color decks are bad because they're so hard to get to work correctly and people only play them because of powerful gold cards." However, I'm pretty sure this isn't true and is stemming from my own lack of experience with tri-color decks.
Question: Why would you want to play a tri-color deck? What advantages does adding the third color often add that makes the deck better than if it only had 2? What should I add to this section of my strategy guide to make it a little more complete/useful?

I cant speak for everyone else, but most of my decks tend to be at LEAST 3 colors. And a good amount of them are all five. It can totally work well, if you build right for it.

Your mana base is the chief constraint, but it's not always as restrictive as you think.
This really depends on the format and your budget, but, sometimes, it's really not "substantially more difficult to get the mana you need to play the cards in your hand" with a three-color deck.
In something like 20 playtest games with Domain Zoo, which is (albeit nominally) a five-color deck, I never once got color-screwed. The fetches+shocks mana base was just that good. Why did the deck play a fourth color? Because Dark Confidant, by itself, was worth it. Why did the deck play "Islands"? To get 1 point of extra damage out of Tribal Flames.
How far you can push the color distribution in your deck depends on the particular mana base available to you, and the overall "speed" of the metagame (which determines whether you can afford to devote your early turns to playing tapped lands like Zagoth Triome or color-fixing spells like Farseek).
In Scars + Innistrad Standard, the distribution of multi-colored lands (one cycle of allied-color "fast" duals in Scars, one cycle of allied-color "buddy" duals in M12, one cycle of enemy-color "buddy" duals in Innistrad; Evolving Wilds as the only fetchland) makes a friendly-color three-color mana base comparable to an enemy-color two-color deck's. The reason most tournament decks are playing two allied colors instead is that this lets them cut colored-mana lands to run more utility lands like Nephalia Drownyard, Moorland Haunt, and Kessig Wolf Run + Inkmoth Nexus, which are some of the best ways to add depth and resilience to your deck.
Indeed, it's when you need your lands to do something special that you're most constrained on colors. Some examples:

Valakut, the Molten Pinnacle requires lots of Mountains to operate. Valakut, the deck, consequently, had a pretty awful mana base for a two-color deck, because most of the cards were green but the deck needed to max out on plain old basic Mountains to do get its combo.
A deck with Dungrove Elder will want copious Forests regardless of its color distribution. If you pick him in M12 draft but can't force mono-green, it still behooves you to play as many Forests as reasonable if you want to make use of the Elder.
Vedalken Shackles is, sometimes, considered reason enough to go mono-blue all by itself.
Tron devotes half its lands to the (colorless) Urza's lands, leaving it with very starved for colored mana. Balancing two colors and a ton of colorless lands is actually much harder than getting a working three-color mana base, because there's not a lot of flexibility in how you can pay for your spells.
Metagame-hater decks playing lots of non-basic-land hate (e.g. Back to Basics, Ruination) can find it limiting their own mana base.

The main benefit is, as you said, access to stronger and more varied cards.
Some prominent historical examples of very different applications:

Weissman's original archetypal control deck was a five-color deck. The majority of the cards were blue, but it used the other colors to cast "silver bullet" cards like Moat, meant to nullify entire strategies.
Many of the Alara-era Cruel Control decks you're probably thinking of when you talk about tri-color finisher cards were actually four- or five- color decks, in order to run cards like Wall of Denial, Oblivion Ring, Celestial Purge, and Cloudthresher -- not quite silver bullets, just efficient answers to the metagame.
In the Mirage era, World Championship winner Jakub Slemr played a black weenie aggro deck that used City of Brass and Gemstone Mine to play all the different-colored "ETB" creatures, which were some of the best weenies in the format. Of course, this also enabled him to stuff his sideboard with lethal hosers from all colors.

For a more recent (2012) example, consider control in Scars + Innistrad Standard. These days, blue-based control decks usually use other colors to supplement their counterspells with strong finishers and removal. Assume you're choosing between U/W, U/B, or W/U/B ("Esper").

Black offers a variety of instant-speed spot removal for creatures (Doom Blade, Go for the Throat, Dismember, Tragic Slip, Virulent Wound) but only rather slow sweepers (Black Sun's Zenith, Life's Finale).
White has superior sweepers (especially Day of Judgment, but also Elesh Norn, Grand Cenobite) and can answer any type of permanent (Oblivion Ring, Revoke Existence, Celestial Purge), but really can't match the efficiency of black's creature removal.
Three-color "Esper" control gives you the best of both worlds, and the ability to play a couple of good white-black effects (Sorin, Lord of Innistrad, Unburial Rites, Lingering Souls, Vault of the Archangel), but greatly curtails your ability to play utility lands -- in particular, Ghost Quarter, which is a critical answer to a class of cards you otherwise can't deal with directly.

In the Modern format, where players have access to Path to Exile, the strategic calculus changes: white is just as good, if not better, for powerful spot removal, while black's advantage is powerful targeted discard (Thoughtseize, Inquisition of Kozilek). At the tournament level, it only takes a few strong cards to effectively shift the color pie.

Great mini-essay there, easy +1.  I even like the (presumably unintentional?) "plane old basic Mountains" pun...

I originally wrote that you "can't deal with" utility lands without Ghost Quarter, but that's not really accurate: you can't remove them directly, but you can nullify their abilities, e.g. using Spellskite to stop Wolf Run or graveyard removal to shut down Moorland Haunt.

I'd love to fix the plain/plane error, but that is not enough characters, could someone do it when they have another edit to do.

@LovesTha Well, someone who's actively playing Standard right now can rewrite the "modern-day example" to be something fresher than five-year-old Esper. (I'm not the person to do it, at the moment, but anyone who wants to 100% has my blessing!)

"shocks" link is broken.

The simple answer to this question is to cover the weakness of the colors and when there is enough fixing you can just play the best cards.  Honestly if you look back at the past few year most of the broken standard decks have been at least 3 colors.

I'd suggest avoiding using the word modern in some places here. Distinguishing between Modern (the format) and modern (recent examples) can cause issues. I was wondering why a Sanctum of All Modern deck would only run the newest 5 shrines and not some Hondens, and how it'd be competitive in Modern, when I realized you were using a different meaning of the word.

That last comment should go to @Wildcard, and you shouldn't directly edit other answers to add content to them, write your own to give added examples.

@Andrew, ah, I didn't realize "Modern" was a format.  I meant "current (late 2020) standard" I suppose.

@Wildcard When Alex said Modern the paragraph before, he is referring to the constructed format for all cards from 8th edition forward, i tgink its now the most palyed 60 card constructed format.

@Wildcard I removed the mention of Sanctum of All decks for now — I'm not up-to-date with the current metagame, so I'll gladly put it back, but I'd like some confirmation in the comments that it's actually a top-tier format-defining competitive deck first (as opposed to fun but fringy stuff like the Bolas Door-to-Nothingness Fog decks ca. 2013). Spliced some of the stuff about triomes and color-fixing into the earlier section about the breadth of your mana base.

@AlexP I'm not a pro player, but the deck shows up with great regularity in ranked MTGA constructed games at the diamond tier level.  Does that count as confirmation?

Alex P's answer is fantastic, but I think I can add to it somewhat.
Limited:
You mentioned limited in your answer, and in limited it is very frequently necessary or nearly necessary to play 3 colors to get decent cards.  Of course even then your third color should often be a "splash" with just one or two cards and keep a heavy emphasis on the two primary colors.  Even when I could make a decent 2 color deck in limited, I am willing to splash a third color if it gets me a good removal card..
Constructed:
Alex P covered this one far better than I ever could, but I'll add that if green is one of your primary colors it becomes much easier to support a third color.
Casual:
This probably does not matter much if you are writing a strategy guide, but in casual play a lot of decks have themes or self imposed constraints that nudge them towards more colors.  I used to have a 5-color dragon deck.  Most of the cards in the deck were mana fixers/accelerators and the rest were dragons.  It hardly ever won, but it was fun to play and when it did win it was because I hard cast a massive dragon that proceeded to eat my opponent.  I also had a 4-color (no blue) angel deck that did a bit better than the dragon deck, but was also fun in a casual sort of way.
Format: Ok, this is obvious, but I'll say it for completeness.  Some formats (prismatic, etc) require multiple colors.  Also, some blocks encourage it.  Many Ravnica block decks benefitted from a third color.

During Dark Ascension prerelease Sealed, I pulled two Evolving Wilds, which allowed me to run a solid two-color deck plus 1x of any land I needed to pay flashbacks. It was wonderful.

Ravnica isn't the best example for blocks to encourage using a third color - Tarkir, with it's three color Khanates is a good one, but the best one of all is Alara, the block containing the only set in Magic where every single card was at least two colors, Alara Reborn.

I'll add just a little bit to what Alex and Timothy already said: multiple colours decks are playable with no huge problems, especially if you limit the cards with high casting cost in one specific colour.
Several legendary cards have multiple colour casting cost and... they are there to be seen in play!
I think multi-coloured decks are special fun when you construct a combo deck (that, let's say, grants you automatic victory when you manage to play the combo-cards together) around a three cards combo, where the cards are of three different colours.
If black is not one of that three colours, you may want to add also that four colour just for adding Demonic Tutor, etc.
In general, being this a card game that you play for fun and to confront different playing styles, having more colours may be fun and add to the variety of the game, otherwise everybody would play the very same deck that won last world championship or so ;-)

Another example that Alex P didn't cover is that adding another color can, in the right application, make your deck a lot more powerful.  Consider all of the flackback costs in Innistrad and Dark Ascension.  A decent amount of them are off-color of the main casting cost of the card.   If you don't play the right colors, the card is a one-off.  However, if you play both colors then the card is a two-fer, giving you a decent amount of card advantage (especially considering the effect of the flashbacked spell).

As the size of your card pool increases, your ability to support more colors in your deck similarly increases.
You could say the same thing for your ability to consistently kill the opponent on turn 1 before they have even played a card.
The more I play, the more I really appreciate basic land, particularly when it is untapped the turn you bring it into play.  It isn't always!
To help understand why you might want to do this, I would suggest you make an enemy color deck, something like white/black or red/blue, and add 1 basic land of the type in the middle of those two.
Add some kind of way to search the whole deck and pull the middle land out and put it into play.  Something like Terramorphic Expanse or Rampant Growth.
With the rest of the card slots, try to find some cards that are good without that extra 1 off land being in play, but they would be really good if it was in play.
Something like Ribbons of Night where you get to draw a card for free if you have that land included in the casting cost, but it's castable for the same cost even if you don't have it.
If you build a mana base that's something similar to 10x plains, 10x swamp, 4x terramorphic expanse, 1x island you would be surprised about how consistent it can be while still offering you potentially more raw power.
In a general sense, every time you add a color you are also reducing your deck's consistency.  There is no reason to do this in a general sense, unless you get compensation for that.
If you had a deck with 24 mountains and 36 red cards, it wouldn't make sense to change one of those mountains out for an island just for the heck of it.  The only reason to do this at all would be that you increased your raw power enough that it was worth sacrificing a little consistency.
It's actually quite easy to do that.  That's part of the reason that it would be hard to ever convince me to run a one color deck (especially non-white).
Having a second color adds a lot in the way of raw power without requiring the player to give up much of anything.  The card pool is such that one can make ultra consistent 2 color decks.  There's essentially no downside to the second color and plenty of upside.
If one is willing to spend a substantial amount of $RL on their mana base, the same thing more or less applies to having three colors as well.  It's not possible to make 3 colors as consistent as 2, but you can get in the ballpark at least and it's often worth doing.  Many formats are filled with 3 color decks for this reason.

"As the size of your card pool increases, your ability to support more colors in your deck similarly increases." The reasons to support more colors in a larger card pool also decreases in this case though, you have more options in each color so less need to go into more colors. - People go into a third (or in some sets fourth) color in limited out of need due to having not enough good, playable cards in two colors in their card pool.

I use a three color deck (white,blue,green). It works well with me because I have worked out a good manna base for every type of deck. on my main I use 45 lands 15 from each color I have in my three color deck .it works really well for me and my strictly 100 card deck


text:
Are the A-sides and B-sides of the wonder boards really balanced?
I know the rules claim the sides are balanced and can be mixed in the same game, but I have my doubts. In general, the B-sides feel a little stronger to me, but it's sometimes hard to compare because they are so different. In most cases, it probably comes down to style of play and strategy to be used in a particular game. However at least in the case of Rhodes, the B-side seems to have an advantage. Here's a comparison:
Completing all wonders of A-side yields:

10 Victory points
2 Shields

Completing all wonders of B-side yields:

7 Victory points
2 Shields
7 Coins

Just looking at this alone, there is little difference between the benefits of the two sides. Since the 7 coins can be converted to 2.33 victory points, the A-side has only a very slight edge. However, there are some other things to consider which tip the scales in favor of the B-side:

3 coins are worth more than a victory point. At a minimum 3 coins equal a victory point, but coins are the tie-breaker, so 3 coins is greater than one victory point. Also, coins have more utility as they can be used to purchase other things, if desired.
The B-side gets a Shield bonus and 3 coins after building the first wonder, making it much easier to get a military victory at the end of the first Age, and the coins are very helpful early on. The A-side doesn't give any benefit for the first wonder (aside from the victory points, which don't help until the end of the game).
The A-side requires 2 Wood, 3 Clay and 4 Ore to build its three wonders, respectively. The B-side only requires 3 Stone and 4 Ore. The 3 Clay and 3 Stone are basically equivalent, so the B-side avoids needing two of a third resource (Wood).
The B-side only requires two turns to build all wonders, while the A-side requires three. With the extra action a B-side player could simply discard a card for three coins, giving it a total of 10 coins, which then puts its rewards ahead of the A-side as the 10 coins would give 3.33 victory points. But the B-side player also has the option to put an additional card into play, which could be worth much more than 3 coins.

All of this makes a strong case that the B-side of Rhodes is not balanced with the A-side. About the only advantage I can think of that the A-side has is for cards that count number of wonders built: Arena and Builder's Guild. Depending on the setup, Builder's Guild may not even be in the game. And, the B-side has the advantage of weakening that card for your neighbors. Arena is a nice card, but it's not a big loss, IMO. Does the A-side have some other advantage I'm missing?
I'm not sure if the A-side of Rhodes is underpowered or the B-side is overpowered, compared to other boards. When using the wonder cards to randomly determine which wonder board each player will play, we've taken to letting players choose the side they want to play because the consensus is some sides are just weaker than others.
Do any of the other boards exhibit similar imbalance? I haven't examined the others that closely.

I haven't played enough to write a real answer, but I did notice that as well. If nothing else, the A sides were terribly boring in comparison to the B sides.

I wouldn't expect the two sides to be balanced. I've never played the B side but from the rules I got the impression that they are simply more complex than the A sides... I always expected that you'd either all play the A side or all play the B side.

I would not even say that all A or all B are balanced to each other,  Makes it hard to claim A and B being balanced

@Kempeth The rule book claims they are balanced: "The B sides of the Wonders are slightly more complex. Nevertheless, both sides remain balanced and can be used in the same game." (pg. 9) Sounds like no one's buying that claim, however.

Related threads on boardgamegeek.com: 1, 2

From what my friends and I can ascertain, the board sides are not designed to be strictly balanced per se, but are instead designed to work best with different play strategies, or at least different skill levels.
In my experience, the "A" sides provide fewer opportunities for specialization than their "B" side couterparts. For example, if a player is playing with Rhodes, but fails to take advantage of the early military points and never plays a military card, that player would have probably been better off with side "A", which provides a few extra victory points and more closely matches the costs and benefits of the other wonders.
That said, it doesn't take much more than a basic working knowledge of the game to make the "B" sides more than pay for themselves, especially with a wonder as unbalanced as Rhodes (other wonders' "B" sides aren't quite as obviously better). In our games, even novice gamers rarely select side "A" after their first couple run-throughs, and when they do, it is rarely a winning strategy.

No, they are not strictly balanced.  It's impossible for such a thing to be true.  Still, the degree of imbalance in a global objective sense is not as great as a the perceived imbalance of the boards given any particular set of players.
7 Wonders is highly interactive, so your choices affect your opponents and vice-versa.  Due to a particular set of player's strategies, some boards will certainly provide an advantage over other boards.  If you are finding one side wins more than another try adjusting your strategy (and getting other players to do so as well) to compensate, and you'll see a significant change in winning stats.
Even if a large number of games, with a wide and varying number of players with different skills in different groupings is played and collected (e.g. on BSW) differences in wins for different sides of the board could be due to the skill-level/complexity of playing that side of the board, and not an intrinsic imbalance.

It's a good point that many games like this are self-balancing, but I don't think that Seven Wonders is particularly one of them; first of all, the boards look sufficiently close to balanced that I don't think anyone ever cries "Uh oh, Bob drew Egypt, everyone gang up on him!"; and, more importantly, what can you do if you're sitting on the other side of the table from someone with an "unbalanced" civilisation?  Very little that's relevant, apart from maybe complain loudly about their good fortune.

I didn't mean to say the game is self-balancing.  In fact, 7 wonders (like many games) is one that can turn early small advantages into big victory point swings without the ability for opponents to do much about it.  From a game theoretic sense, it's quite interesting as, even if a neighboring player could do something about an advantage, its often still rational not to, since it would hurt you (relative to all other players) more than it would help him.

[EDIT]
As promised I'm getting back with more stats.
TL;DR: You can view an interactive version of the stats in this js fiddle
There are 3 tables. One for sides only, one for wonders only and one for wonders and sides.
Long version200 games played.
Taking into account sides only (ignoring wonders) results into a pretty good balance in points average. Side A wins by 0.1 points, but side B has a better winning percentage.
Looking at wonders only, ignoring sides, there are differences.
The stats contain also some extension packs and the funny thing is that the wonders from the extension packs are at the top or bottom of the list.
For the standard wonders the difference in points average between the "best" (Ephesos) and the "worst" (Rhodos) is a bit below 4 points.
Combining both wonders and sides the results are all over the place.
Some wonders look balanced, some don't.
But, again, one funny thing. Contrary to the OP's example, Rhodos looks like the most balanced one. The points average are the same up to the second decimal point 62.94, though nobody in our group has ever won with Rhodos A. (this has been debunked 100 games later.)
most balanced wonder seems to be Alexandria with 0.34 points in favor of B.
Most unbalanced one is: Halikarnassus with a difference in average of 4.5 points in favor of B.
Disclaimer: My answer should not be taken as proof of anything. The values are based on the games we played in the same group of 12 people. Almost all games were played with leaders and cities extension (3 were played with standard game only) and the wonders were assigned randomly and in most of the games the sides were random as well
[/EDIT]
Original answer:
I know this is an old question, but I wanted to add my input on this.
We play 7w occasionally at the office and I've built an app to keep scores (because we ran out of score papers).
We don't have a lot of games recorded, so this may not be conclusive, but the equilibrium has been there since the beginning.
The numbers are not exactly the same, but they are really close.
All of the games have been played with the standard game plus extensions for leaders and cities and wonders pack.
The wonders were assigned randomly, but in some of the games the sides were chosen and in others were assigned randomly.

Here is the text version:
Side    Games Played    Games Won   Win %      Total Points     Average     Max Points  Min Points
A       127             21          16.54%     8121             63.94       91          36
B       131             24          18.32%     8507             64.94       90          34

What "scares" me the most is the points average.
It differs by 1 point.
If I choose only the games played in 3 to 7 people (exclude the ones in 8) so they all have 7 rounds per age the average is this:
A: 66.25
B: 66.72

This means less than 0.5 points in difference.

If you are interested in the scoring app you can get it here: https://github.com/tzyganu/7-wonders-score. The app is free and I get no financial benefits out of it so this should not be considered self promotion (spam). Some dev skills may be required to install the web app. (some bugs may occur as this was build in a rush).

Finally, some evidence, and not just vague claims of "this could be used for this" or "these are pretty similar so their effects are similar" or even "this isn't as useful as that" or "they don't balance at all".

@Nij. Thanks. Will update the post when we reach 100 games. I think it would be more relevant than with 45 games. Will post more stats then if needed. For example on wonder or on wonder and side combined. The only think that worries me so far is that nobody won with Manneken Pis (either side) out of 16 games played with it.

The two sides are not designed to be balanced. On 6 out of 7 wonders in the base game, side A has been in fact created to ease new players in the game, by providing a common pattern among them:

3 Victory Points
Variable power
7 Victory Points

Side B is almost always strictly better than side A. Here's a detailed approach, from my point of view.
Alexandria
Probably the most versatile wonder in the base game, you can seldom get locked out of resources with this one. Side B gives you:

The wild raw material in the first stage, which is extremely important to have in the early game. When constructing stage #2 of side A it's usually too late to take full advantage of it.
Gives you a wild advanced material, which is usually welcome, especially in 4-player games, where you can easily get locked out of one.
3 Victory Points less, in exchange for aforementioned infrastructure. Is it worth it? I'd say yes.

Furthermore, side A is really weak, as you need double Glass for the final stage. There is no card or other Wonder stage in the base game that needs a double glass and your neighbors, if playing smart, will refuse to take one for you. So, unless you come across a Forum in Age II, you're forced to lose an action to take a second Glass, so that you can take 7 Points, making it 3.5 points per move in the late game, which is very suboptimal. Ephesos and Halikarnassós suffer from a similar problem.
Babylon
Overall, the weakest Wonder in the base game, from my point of view, and side A makes things worse as it forces you to get science cards, by placing the wildcard science symbol in the middle. Side B gives you:

The ability to play the last card of each Age, so up to 3 more cards than other players.
The wildcard symbol is placed at the last stage, so you may opt out from science completely.
7 Victory points less, in exchange.

Is it worth it? Well, both are weak.
Ephesos
Nothing to discuss here. Side B also gives you 10 points, more money, 12 coins in total instead of 9, and the money is distributed over the 3 stages, making Ephesos more versatile. Furthermore, side A suffers from Alexandria's side A problem, by requiring a double Paper for the final stage. Therefore, side B is strictly better than side A, no arguments about that. This is probably my second favorite Wonder in the base game, following Alexandria B.
Gizah
That's probably the only wonder with almost equally balanced sides. Side B gives you 5 more points in exchange for one more wonder stage. However, I would still claim that side B is more useful as:

It tends to increase the value of the Arena from 6 Victory Points to 8 Victory Points, including money, and the value of Builders' Guild, if you get it.
Makes it a perfect anti-science Wonder, as you can tuck 4 green cards underneath it!

Therefore, having an equal choice, I'd still choose side B.
Halikarnassós
It probably has the biggest strength difference between side A and side B. Side B offers:

3 "take from discard pile" actions, instead of 1, on every wonder stage.
7 Victory Points less, in exchange for that.

Is it worth it? You bet it is! The 2 extra cards you'll pick from the discard pile will usually make up the difference, but you'll have more flexibility. Side A's ability is right in the middle, which is really bad, as you'd want to construct your Wonder at the last round of the game, after everybody has discarded their cards. However, you'd lose the final 7 Points if you do, with side A.
Furthermore, side A suffers from Alexandria's side A problem, by requiring a double Loom for the final stage.
Olympia
This is different. Side B gives you:

The ability to copy a neighbor's Guild.
A double trading post.
5 Victory Points less and stripes you of the ability to construct a card for free on each Age.

Is it worth it? Eh, kinda. The double trading post on the first stage is always welcome. However, the "copy a Guild" is usually pointless: a Guild giving lots of points to a neighbor will usually be worthless for you, so the final stage won't give you more than 3-4 points. Even the double trading post won't be always useful, as maybe the player across you has been hoarding all the resources, making your neighbors reluctant to take any brown cards. Furthermore, the ability to construct a building for free is a good one to have and gets very useful in Ages II and III.
3 player games are an exception, the double trading post is much more powerful, as you are able to buy resources from all your opponents and the final stage tends to give out more points. You can safely pass the Builders' Guild there!
Consequently, when I have a choice, I pick side B if I'm playing a 3-player game and side A otherwise.
Rhodes
Side B gives you:

7 coins
Only 2 stages and the two shields distributed over them.
3 Victory Points less.

Is it worth it? Definitely. Taking money into account, side B gives you 9.33 points, but 3 coins are strictly better than 1 Victory Point, as you can spend them to get a valuable card which otherwise you'd have to pass up and they count towards the tiebreaker. Therefore, the 7 Victory Points && 7 coins are borderline "strictly" better than 10 Victory Points. This money comes very handy in the game.
Furthermore, taking one shield at a time is much more useful, as you can sometimes have your first shield in Age I, which might make the difference. Not to mention that you have one wonder stage less to build, although this decreases the value of the Arena or Builder's Guild and makes Rhodes a bad anti-science Wonder.

To sum up, side B is almost always much more useful and flexible than its side A counterpart. For this very reason, in my groups I always let people chose their favorite side, than just rotating the card as suggested by the rules. Even if the group's rules are to randomly choose the sides, I'd definitely defend anyone who ended up with side A of Alexandria, Ephesos or Halikarnassós, as they are clearly underpowered.


text:
Why Does Magic Use The "Paris Mulligan" Rule?
In the early days, Magic had crazy, subpar mulligan rules: you could shuffle up and draw a new 7 if your hand had no lands or all lands, otherwise you were out of luck.  A long time ago the Paris Mulligan rule was introduced and it seems to have stuck.  If you don't like your hand, throw it back and draw 6.  If you don't like that one, try again drawing 5, and so on.
This seems like a pretty good rule and I didn't question it, but recently I've been playing "Duels of the Planeswalkers" on XBox (incidentally, if anyone else has that and wants to challenge me, let me know!).  The mulligan rule there is a little different.  You can throw your hand back and draw a new 7 once for free.  After that Paris rules are in force: every subsequent time you mulligan your hand size goes down by one.
It seems to me that this "lenient mulligan" rule is really good for the game.  Magic is a game with a large element of randomness from the shuffle.  Quite often players will draw a 7-card hand that's basically unplayable and have to throw it back, unless they're crazy gamblers with nothing left to lose.  If your opponent keeps and you mulligan to 6, you're at a significant disadvantage already.  If you have to mulligan to 5 or fewer, you'd better hope somebody up there likes you.
Does anyone know (or have an informed idea) why Magic retains a mulligan rule that allows unbalanced games to happen as often as they do?  It just feels like the XBox mulligan would allow players to play their decks "as they were meant to work" more often, and have to battle against long odds much less often.  What is the defence for the continued predominance in the Magic world of the Paris mulligan rule?

http://magic.wizards.com/en/articles/archive/news/new-mulligan-rule-starting-battle-zendikar-prereleases-2015-08-20

Any answer that is accepted or upvoted should address the fact that permissive mulligan rules allow combo decks to win faster and more consistently.  This means, especially in faster formats like vintage and legacy, that there will be more games won in 1-3 turns and fewer skill-intensive games where many decisions are made.

As a note, MtG no longer uses the Paris mulligan, it currently uses the Vancouver mulligan (which adds a scry at the end if you are at less than 7 cards) and is now testing the London mulligan, redrawing a full hand then sending cards to the bottom of the library based on the number of mulligans taken.

And now the London mulligan has indeed replaced the Vancouver mulligan as the standard type of mulligan.

A few points:

"Good for the game" is drastically different if you are playing competitively or casually. Competitive Magic should encourage good deck building and make it as fair as possible to each player. Paris mulligan vs. lenient paris mulligan isn't going to drastically change the fairness between the two players at the table and adds more complexity to an otherwise already complex game.
Learning when to mulligan is actually a very useful skill in Magic. Making it easier to stomach doesn't necessarily make it a better mechanic. Designing (and testing) decks with 7, 6, 5 card hands is very important. Likewise when playing against particular decks. Some decks struggle against a 1 card advantage. Some decks shrug it off and keep pounding.
Real life Magic mulligan problems can often be solved by shuffling properly. People can get lazy in their shuffles and tournament nerves can make it worse. Take the time to adequately shuffle your deck each time you draw a hand. This is especially true after a mulligan: Simply cut-shuffling a few times will result in you drawing the exact same cards you just threw back.
When playing casually, I recommend players just draw back up to 7 after a mulligan. I would rather play against a deck's best chance than roll over someone with a 5 card start. Casual games should be fun more than fair; if the lenient paris mulligan is more fun for your play group then that is how you should play.

I agree with the spirit of your point 1, but I disagree in specific with Paris mulligan vs. lenient paris mulligan isn't going to drastically change the fairness insofar as lenient mulligan rewards decks that are explosive but inconsistent (for example, two-card monte, and to a lesser extent, dredge)

Strategy bias isn't the same thing as unfairness.

I'm old and tired, and I haven't played Magic much in the last 15 years.  With that as a caveat, my recollection is that the harsh penalty for declaring a mulligan was to prevent excessive "gaming" of declaring a mulligan.  Part of the point of M:tG was to create a deck that was playable despite the fact that the order of the cards would be random.  Being able to mulligan out of a "bad" hand lets players create a less balanced deck that needs a specific start hand to succeed.
In other words, playing decks "as they were meant to work" should take into account the fact that sometimes things won't come out the way you expect.
Related link of possible interest: http://www.wizards.com/magic/magazine/Article.aspx?x=mtgcom/daily/af3

The old "you can mulligan if you don't have lands in hand" rule was killed fairly early on, because players would build decks with no lands (Moxes and other artifacts for mana instead), and then legally mulligan over and over until they drew their perfect hand.

Under Wizards' official no-land/all-land mulligan rules, you only got one mulligan; you couldn't 'legally mulligan over and over'.  I can't dig up a copy of the old floor rules ATM, but look at http://archive.wizards.com/Magic/magazine/article.aspx?x=mtgcom/daily/mr112b for confirmation of this.

Interesting. I'm sure I've heard stories of repeat mulliganing, but the ye olde days are vague in memory. Good link!

Informed idea:  The problem with a more "open" mulligan rule is striking a balance between "fair randomization" and "how a deck is supposed to work."  It's fairly easy to construct a deck in which the correct 6-8 cards being in your starting hand gives you a near-guaranteed win in 1-3 turns.  Preventing this is the reason decks have a minimum size, and individual cards have a quantity limit.  But if players could freely reshuffle and re-draw, they would.  Over and over, until they got that perfect hand which won the game.  The concept of "how a deck is supposed to work" would morph into perfect, 7-card combos.  To prevent this, and create more options in deck design, mulligans are very, very costly.

One advantage the xbox version has over RL is that it can shuffle the deck instantly, whereas in RL it take a bit of time to shuffle.
Other card games take alternate routes to make it work with less mulligans needed.
Universal Fighting System: To mulligan you remove your hand from the game and draw a new one. Only one Mull is allowed.
Magi-Nation: Depending on your starting Magi you search through your deck and pick out 2-3 cards and then draw the rest of your hand afterwards.
Commander (MTG Variant): Partial Mulligans. You set aside any number of cards and draw that many from your deck. You may do this again drawing one less card than you set aside, until you are happy with your hand or run out of cards. Then you shuffle all of your set aside cards into your deck. Note that Commander (or EDH) has a 100 card deck size, and no duplication rule so this might not work as well for regular magic.
Now, I'm not suggesting that time saving is the only reason for less lenient Mulligan rules, but I would posit that it is a factor.

One element that I haven't seen in any of the answers yet: a 'lenient' mulligan rule advantages some deck archetypes over others.  Some decks are built on the concept of redundancy: for a mono-Red aggro or burn deck, for instance, there may be minor differences amongst its various one-mana creatures or its burn spells — maybe this spell does an extra point of damage but only hits players, maybe creature X has a modest downside but creature Y has one less point of toughness — but for the most part, they're indistinguishable; just (e.g.) 12 copies of 'guy who costs 1 mana and has 2 power', etc.  For a deck like this, absent issues with drawing the right number of lands, one 7-card hand is going to look like any other.
On the other hand, a combo deck — let's say the Splinter Twin decks in Modern — is looking for a very specific set of cards, often to the point where almost all the rest of the deck will be some combination of card draw and selection, with just enough control elements to keep the deck alive while it assembles its combo.  This sort of deck wants a very specific set of cards in its opening hand (it would, for the most part, much rather have its combo pieces than have to find them) and so it often has to mulligan aggressively to find them; getting a 'free' seven-card mulligan is worth much, much more to a deck like this than a red aggro deck, and so decks like this become much more viable in the presence of a free-mulligan rule.
This can actually be seen in action in many casual EDH/Commander environments, where people tend to be very lenient in their mulligan rules and so wacky combo decks are often the order of the day.

So, this does explain one consequence of a free mulligan rule, but you still need to link that back to why Magic chose to use paris mulligan and not free mulligan.

@doppelgreener I thought that was implicit in the above - one reason (and there are certainly several) is that it's a game-design decision made to balance the power level of different archetypes.

@doppelgreener Paris mulligan was selected first. As a general principle, I support not changing rules unless there is something to be fixed. As this answer points out, a more lenient mulligan rule doesn't fix a problem, it just changes some aspects of the metagame.

Another way to look at this that might make it more obvious to some people: the more free the mulligans, the closer you'll get to the best possible hand with the deck. For some decks the best possible hand is incredibly good, way better than average, and so lenient mulligans favor those decks and make them "unreasonably" good.

@doppelgreener I second your concern. Stating how the metagame would be affected is not enough to answer the question. The questions was not why you would choose one over the other, but why Wizards chose one over the other. To answer that, you either need to be able to read minds or you need to read this article that does a great job explaining why Wizards settled on Paris mulligans. I am having trouble figuring out when that article was posted. It may have been after the question was asked.

There is an excellent article up by one of the Magic developers that discusses the issues with various mulligan types, and why they upgraded to the new "Vancouver" Scry 1 mulligan.
A quick summary:

The "cheaper" it is to mulligan, the more likely you are to do it. On digital, this is instantaneous. In paper, properly shuffling takes a lot longer, as does deciding whether to keep the hand.
The more control you have over your mulligan (e.g. the way Hearthstone works, where you select cards to replace), the more likely it is that certain powerful decks that are currently kept in check by their reliability issues will simply become broken
They want to have the same mulligan rules for all the major 1v1 formats; stuff that's fine in Limited can be bonkers for constructed.

One interesting point in the article is that it agrees with you that it would make casual games a lot more fun. So if you want to casually agree to do it outside of ranked events like FNM, go for it! But be aware that it's not how cards are balanced, and so as you progress, certain decks may be more powerful than they should be.

In a well-designed deck, the possibility of a truly unplayable hand is small.  The possibility becomes vanishingly small after a single mulligan.
Thus, there really should never be a need to mulligan more than once (unless you're fishing for your golden starting hand).
Being down a single card is NOT a "significant disadvantage" - it's relatively minor.  The current mulligan rules are established in such a way to discourage fishing for gold while not preventing a relatively fair start for each player.

Actually, I'd argue this isn't true. For example, in a 40 card deck with 17 land (above average) there is about a 10% chance of drawing only one land or less. Assuming a single mulligan back to 7 cards you've got a 1% chance per game of pretty much losing at the start. In a 16 round tournament you'd play between 32 - 48 games. So basically half the players would be hosed with a single mulligan at least once. Without a mulligan 10% loses due to poor luck is unacceptable.

First - this question and the answers to it are out of date. The Paris mulligan was first replaced with the Vancouver mulligan (Paris, but anyone who mulligans to six or less gets a scry 1 before the first turn starts), which has since been replaced with the London mulligan (Draw 7 cards and put 1 on the bottom of the deck for each time you took a mulligan when you accept a hand).
The digital versions of Magic the Gathering and the formats that have a free mulligan in paper have a big difference from paper tournament play that no one has commented on, these are games and formats where a match is played "best-of-one" where events in other formats are played "best-of-three".
Commander is a multiplayer best-of-one format that allows a free mulligan. Two-Headed Giant is a best-of-one two-on-two format that allows a free mulligan. Originally, MtG Arena, like the Duels of the Planeswalkers games before it, was a best-of-one only game, this game does something very different* with the starting hands, but does not allow a free mulligan. The more common formats, Modern, Standard, Draft, Sealed, Vintage, Legacy are all meant to be played in a best-of-three match setting, and these do not allow for a free mulligan, one single bad hand in one of the three games doesn't have near as much impact as a bad hand in a best-of-one format.

*Magic the Gathering: Arena shuffles up 2 copies of each player's deck and draws 2 starting hands then shows only the deck which has the better balanced starting hand to the player - the other copy is discarded. This effectively is an automatic "free mulligan" where the game, not the player, decides.

In any card game, a person who draws a bad hand will always be at a disadvantage against someone who draws good cards. That's just the nature of the game. An exception might be someone who draws a "mixed" hand (a technically inferior hand with good chances of improving). Such a hand might be four to a flush or straight versus one or two pair in draw poker.
If you get a "redraw," you actually have an advantage, maybe not against the good-drawing player, but against a "random" player. Magic tries to compensate for this fact with the Paris Mulligan rule; the second draw costs you a card, the third, two cards, etc. Then it's up to you to determine whether your starting hand is so bad that it's worth paying the penalty for a redraw.

The best solution would be to increase starting hand size.  The % of games that are decided by initial bad draws/mulligans is very high, especially when considering how much meticulous thought and planning deck building requires.  This is why having an 'auto-lose' from bad luck is frustrating to people, because most of the effort is acquiring cards and building the deck, and then they get to the 'actual game' and someone's fate in a match or tournament can (and will, given enough games) be completely sealed by luck of the draw.  Even the pros do not have that high of a win%(although still higher than everyone else by at least several %), what they do have is the patience and temperament to keep chugging along despite the bad beats that no one can avoid no matter how clever or skilled they at this card game.
There is only a very small number of competitive decks in standard at any given time, this is because deck options are extremely limited due to the incredibly high level of consistency needed.  This also makes competitive magic very expensive, as the need for consistency leads to only several % of the most powerful, efficient cards worth playing, and it also makes mana bases extremely expensive as every 'free' card that gives you more consistency will lead to more wins.
In summation, Mulligan-ing well isn't just about hand analysis or deck building skill.  Avoiding the pitfalls of bad starting hands is more about having the small clump of cards that will actually give consistent, powerful results given that a deck size is pretty large and any one non-basic land can only make up 6.7% of the cards in the deck(note how powerful tutors of almost all kinds are, just because they let you bypass randomness for that single instance).

-1: Increasing the hand size actually improves the chances of extremely strong situational cards, which can in fact reduce the number of competitive decks (because once you remove inconsistency, fast decks that work because they take advantage of their opponent's inconsistency go away, and slower decks become stronger). For example, with 10 cards, many combo decks have the reliability to break the format, so you end up with just those decks in the format.

I agree with deworde.  With larger starting hand sizes, you will see combo decks winning more consistently and faster.  The only thing Storm decks need to win is a critical mass of solid cards in hand.  If you give them an extra card or two (and therefore eliminate the games where they take mulligans, so they get a secondary increase in hand size), you're going to see a lot of turn 2-3 combo kills that can't necessarily be stopped by a Duress.  That makes it harder for any other deck to compete.  New players with weak decks would get blown-out in extraordinary, almost traumatic fashion.

Truly the best solution to poor opening hands is creating a balanced "deck within a deck."
Every card that searches for other cards or draws a card(s) can be considered a subtraction from the deck count if the mana cost is low enough. The card itself, mathematically, is replaced with the card(s) it draws/searches for, thus creating the external deck. Now, the internal deck should consist of the cards you need to play, such as Mana sources (basic lands, for example) or versatile, useful cards.
Imagine, now, that you have a 60 card deck. Opening hand, you draw a fetch land. Play it and pass your turn. On your next turn, it will untap and allow you to search for a land of the appropriate color. That means the fetch has been replaced by a basic land, and a basic land has been removed from your deck, lowering your chances of drawing it. If you run 12 fetch lands (pretty high number, but this is the extreme example), 20% of your deck count has been shaved. This correlates to 1 fifth of your deck, leaving you with 48 cards left to amass. Now, 4 in 48 (1 in 12) chance of drawing any given card, as opposed to 4 in 60 (1 in 15) you would have had without the fetches, and that doesn't include all the other searchers you might fit in there. It is possible in Modern format to make a deck with 30 searchers/draw cards, 20 basic lands and 10 cards that produce various desired effects and be an efficient killer.

This answers "What is deck thinning?" The question was "Why does Magic: the Gathering use the Paris mulligan?" To stay on topic, your answer needs to find a way to relate deck thinning to the Paris mulligan. I can kind of see where you were going with it, but I think you stopped writing before you established your point.

It was meant to be in response to what was said previously. People say that decks are 60 cards, and that large number can validate a more generous mulligan, but with the large number of cards that allow for low cost searching, at least in modern format, the mulligan would be over powered if given more generosity. I suppose if I were to add on to my answer, I would add the following: "And this is why the Paris Mulligan was chosen over the Free Mulligan."

You're overvaluing the importance of deck thinning.  This is almost on-par with saying that the Hindenburg blew-up partially because of a butterfly beating its wings one town over.  One fetchland would reduce the landiness of each of your draw steps by about 2%.  More negligible in faster formats than slower ones, because fewer draw steps.  A tiny benefit when you don't want to mana flood, and a tiny liability when you don't want to mana screw.  Add the benefit and the liability together; the overall net benefit is microscopic.  The Paris mulligan also predates the Onslaught fetches.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/2.5/


text:
Escape From Colditz (2 player): How do you deal with all your POWs being in Solitary?
In the game Escape from Colditz, once a POW player has acquired an item, it seems like the guards can pretty much round all of that player's team up from the inner courtyard and dump them in Solitary.
From there, they only get released one at a time, and they still have the items, so relatively few guards can repeatedly keep them locked down.
I can see how you might manage this in a multiplayer game, with one player holding the items, and the others getting them back when they escape. But in a 2 player game, it just seems like you can keep restraining the POWs forever.
Do people have a house rule to cope with this? Or am I misreading the rules?

That is a known problem for which players usually agree to home rules.
I would suggest two such home rules:

Prisoners in confinement are automatically released after a few turns (this copes with the rolling doubles problem). Just put each prisoner piece on one side of the cell and move it each turn to the other side, when it should go through the wall release it.
When releasing prisoners in this way from Solitary, place them directly in the roll call spaces, where they cannot be taken again.

This way the Security Officer cannot mount a "capturing machine" that effectively blocks the prisoners actions. Also remember that the guards cannot block rooms and alleys entrances.

I just read In the published rules #39 that once a prisoner is released from solitary confinement that a guard can not arrest the POW as long as the released POW heads directly to the Prisoners Courtyard.

Whilst returning to the prisoners courtyard he is not liable to arrest. He may, however, at any time make an escape attempt, and is liable to arrest once it is apparent that he is not moving directly back to the prisoners courtyard.

The problem is that you just station someone in the prisoners' courtyard and pick them up as they enter.


text:
Are "three way" trades allowed in Catan?
My basic understanding is that all trades must involve the player whose turn it is.
Player A, whose turn it is, has ore and wants wheat. Player B wants ore and offers brick, which is not acceptable to Player A. Player C wants brick, and is willing to give wheat for it. So a three way trade would have Player A sending ore to Player B, Player B sending brick to Player C, and Player C sending wheat to Player A. Put another way, Player A gives ore to Player B and receives wheat from Player C.
Would such a trade be permitted under the rules? Looked at one way, Player A has dealt with both Player B and Player C. Looked at another way, there was a Player B to Player C leg (Player B sends brick to Player C) that did not involve Player A.
That is unless Player A first trades ore for brick with Player B, then brick for wheat with Player C. Can these two trades be made "simultaneous" through an "escrow" type arrangement? Or is this feature only reserved for say, "house rules?"

No (or yes, depending on how you define a single trade).
The person whose turn it is can trade. Each trade is with that person and one other person.
If you can accomplish what you need in a series of trades, you're welcome to try. But any promise of future trade is non-binding: your later trading partner may legally decline the trade when it comes time.
Rules:

Important: Players may only trade with the player whose turn it is.

Also note that each trade has to involve a card from each player in the trade. You can't just give a card away for nothing in return.
So in your example:

A trades ore to B in exchange for brick.
A trades brick to C in exchange for wheat.

This is legal, but it is two separate trades.
There are no escrow rules: C is free to decline the trade after trade 1 has happened.

Hmm, indeed the rules here explicitly say giving cards away is forbidden. I wonder if it's always been like that. Also you can't trade the same for same, so no trading wheat+sheep for sheep to get around the prohibition on giving away cards.

It was an odd edit. No improvement. Plus it was an incomplete edit in the case of the question, so rather than edit it further to complete it, I simply rolled it back to the OP's usage. If you'd like to ask the OP to change his terminology, that's fine, of course.

@ilkkachu Of course, if one wishes to give away a wheat,  it does seem like the rules allow the loophole of trading two wheat for a sheep, then trading a sheep for a wheat.

For completeness: A is also free to decline the trade with C after 1. has happened.

@Acccumulation That's non-atomic, though, so your trade partner / opponent can always refuse the second half. If the two players have any resource in common you can make it atomic: "I'll trade you a brick and a sheep for a brick".

@amalloy, yes, but that's exactly what the rules currently online at catan.com forbid: "However, you cannot give away cards, or trade matching resources ("trade" 3 ore for 1 ore, for example)". Game Rules & Almanac 3/4 Players, p. 7, Domestic Trade.

@BenjaminCosman I've asked on meta. Let's delete our comments here.

@ilkkachu TIL. I haven't read the rules in a long time, and I guess that's the kind of edge case one can easily forget.

While I don't agree with SQB's original edits - once the asker accepted that edit and made it themselves to the question, the answer should be edited to match terms used.

That would then require comments on the answer to be edited. Better to just leave the spurious edits out as much as possible. In any event, the answer is clear in its meaning and relation to the question, even after the question  was edited.


text:
What happens when a commander is exiled?
Let's say you have a commander on the field, and an opponent uses Sisters Of Stone Death to force your commander to block the sisters and then exiles your commander?  Does it remain exiled, or does it go back to your commander zone?

The rule is

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

If a commander would be put into its owner’s hand or library from anywhere, its owner may put it into the command zone instead. This replacement effect may apply more than once to the same event.

(See mtgcommander.net)
So, in this case, you get to choose whether it goes into exile or the command zone. If you put it in the command zone, then your opponent can't use the Sisters' other ability to bring it back, of course.
Several important nuances:

You don't have to put your commander back in the command zone. This is handy if, for instance, you can't afford to hardcast your commander, but you've got a reanimation spell in your hand.
As of the June 2020 rules update, your commander will hit the graveyard or the exile zone before going to the command zone, and will trigger "on exile" or "on graveyard" triggers. E.g. if you are playing Keiga as your commander, you can steal a creature if you put Keiga in the command zone when it dies.
Owner, not controller. Even if someone else Mind-Controls your commander, you still get to pick whether it goes back into your command zone. If they use Mindslaver on you, however, they can choose.
You only get a choice when your commander has just reached the new zone, in the case of the graveyard or exile, or just before it goes there, in the case of your hand or library. So, if you put it into the graveyard or exile, it'll stay there just long enough to trigger any relevant abilities, but there is no opportunity for anyone to gain priority and start playing spells or activating abilities before it moves to the command zone. If someone exiles it from your graveyard, you can scoop your commander back into the command zone. Mimeoplasm, for example, can copy a dead commander like any other creature card in your graveyard, but at least you can put that commander back in your command zone since it gets exiled (Mimeoplasm still comes into play as the copy).
Recasting your commander from your hand doesn't cost extra like bringing it back from the command zone does; nor does it count for figuring out how many times you've cast it from the command zone already.
Rare card interactions can cause a card to move from one zone to the same zone. If that happens to your commander, fear not: you do still get the option to put it in the command zone. See this answer for a detailed explanation.

The note about "tuck" effects recently became reversed. Rule 11 (under "Play") now includes hand and library as candidates for the owner-initiated replacement effect.

@Mr.Zero Thanks for remembering this question. Updated!

Should this be updated to include the embalm mechanic? This seems like a special case of a commander moving between zones.

@SocioMatt which emblem mechanic?

From Amonkhet, like on Temmet, Vizier of Naktamun. Since the player exiles the creature from the graveyard while gaining a token, it seems like it might warrant a bullet point in your "Nuances" section. Or if you think that use-case warrants a new question, I will happily ask it.

Was there a ruling where the commander had to go to such a zone from a public zone, such as battlefield, exile or graveyard at any given time? Or did I just play it wrong the whole time?

Commanders' on-death triggers still work when moving them to the command zone since the June 2020 rules update, so you can now steal a creature when Keiga dies.

@user3490 I appreciate the update! Do you mind making the changes to the text? — I haven't been keeping up with MTG rules for a little while so I don't want to butcher it myself.

Whenever a Commander is exiled or enters the graveyard, they are returned to toe command zone.
The only zones your commander remains in are:

Command zone
Battlefield
Hand
Library

A good strategy for commander removal is to put them on the bottom of their owner's library. This is a bit mean, however, and may not make you many friends :)

-1: Not "whenever". You have the option of sending your commander into the graveyard or exile zones, if you want to.

@AlexP - Good to know. We've been playing incorrectly :)

Not only that, but you can also do this when it would go to the library, so you can't return it to the library to get rid of it.

I believe the option to return the Commander to the Command Zone instead of putting it in one's Library is a rule change since this was originally written. I'm fairly sure that tucking a Commander into it's owner's Library was recommended by many when I gave this answer initially.

Yes it was a rule change after this was written, hand and deck were always forced before, exile and grave gave the options, and it was still an option back in 2011 to let it resolve and send the commander to exile or the graveyard in 2011. Now that apply to any zone changes.