text:
Are enchantments and artifacts considered spells?
In Magic: The Gathering, are enchantments and artifacts considered spells?
More precisely, can I use Cancel (which says "Counter target spell.") on an enchantment or an artifact?

Since you asked for some citations from the official rules:

111.1. A spell is a card on the stack. As the first step of being cast (see
  rule 601, “Casting Spells”), the card
  becomes a spell and is moved to the
  top of the stack from the zone it was
  in, which is usually its owner’s hand.
  (See rule 405, “Stack.”) A spell
  remains on the stack as a spell until
  it resolves (see rule 608, “Resolving
  Spells and Abilities”), is countered
  (see rule 701.5), or otherwise leaves
  the stack. For more information, see
  section 6, “Spells, Abilities, and
  Effects.”

So basically, pretty much every card in Magic is a "spell" - or rather, becomes one, during the transitional period from being a card in your hand to  being a permanent on the battlefield.  But wait!  Why isn't a land a spell, in that case?  Well, because it has special rules making it an exceptional case:

305.1. A player who has priority may play a land card from his or her hand
  during a main phase of his or her turn
  when the stack is empty. Playing a
  land is a special action; it doesn’t
  use the stack (see rule 115). Rather,
  the player simply puts the land onto
  the battlefield. Since the land
  doesn’t go on the stack, it is never a
  spell, and players can’t respond to it
  with instants or activated abilities.

The Magic rules are pretty arcane at the best of times (appropriately enough), but I hope that clarifies things for you!
EDIT: Actually, it occurs to me that even more clarification may be necessary.  Enchantments and artifacts aren't spells when they're in play: they're permanents.  They aren't spells when they're in their hand, they're enchantment cards and artifact cards.  But they are spells while on the stack (i.e. in the process of being cast).  I'm sure you already know that "counter target spell" means "stop something on the stack from resolving and becoming a permanent, and send the card to the graveyard instead", but it wasn't 100% explicit from your question!  So, just in case there's any doubt at all, Cancel can stop an artifact or enchantment card from coming into play, but it can't send it to the graveyard from play, or anything wacky like that...

+1 You can counter a spell as it is being cast, and any non-land card is a spell at that time, but you can't use a counterspell to remove a card already in play.

FWIW the clarification is definitely necessary: if not for Elliot, then for beginners, for whom it's really not obvious than a so-called "enchantment" is not a so-called "spell" until the rules you've quoted are explained.

@Elliott: just out of curiosity, was the edit necessary in this case?

@Lohoris it wasn't necessary because it already seemed reasonable that you can't use Cancel to send something to the graveyard from play, but it was very helpful to know the exact reason (cards stop being spells after they have been resolved).

Bröda, what's the source of what you're speaking? I can't seem to find this 305.1 stuff in the good old Google. I assume this is the whole official rulebook?

Yes. Of all the card types, the only one that isn't a spell are lands. "Counter target spell" can counter sorceries, instants, enchantments, artifacts, creatures, and planeswalkers.
Tribal is also a card type, but it always shows up in conjunction with one of the other types, so it doesn't matter in this case.

Awesome, "Cancel" just got that much better.  Do you know where I can find a reference for this (e.g. in the official rules)?

If you think Cancel's good, you should check out Counterspell

A "spell" refers to a card while it's on the stack. In order:
While it's in your hand, it's a card (enchantment card, creature card, land card)
While you're casting it and it's on the stack, it's a spell. (creature spell, enchantment spell, etc.)
Once it's resolved, if it stays in play it's a permanent. (Instants and sorceries just resolve and discard.)
Lands are the exception - they don't use the stack and go directly from card to permanent.

@TheSunNeverSets has given a great rules explanation.
For a bit more of an intuitive answer, take a look at the cards Artifact Blast and Arenson's Aura.  These cards only work if artifacts and enchantments are spells while on the stack.


text:
Easy way to see changes between core sets?
Is there an easy way to see which cards were added and removed between core sets?
For example, by looking up the card Giant Growth I can see that it was in both M10 and M11, but was not included in M12.  Now on a single-card basis, it's not too hard, but I've got a lot of M11 that I want to go through and see which cards are still legal in standard, and searching Gatherer by card will get pretty tiring.

wiki.mtgsalvation.com is your friend.
http://wiki.mtgsalvation.com/article/Magic_2012/Changes, for instance, has complete lists of all the cards that were added to M12 and removed from M11.  You can find the same information for all previous core sets too.
In addition, there are useful sections on changes in rarity from previous printings, and also a running tally of cards that had (up to that date) been included in every Core Set.  Go go Giant Spider!

I mourn for Giant Growth. :(

Yes; without Giant Growth it doesn't feel like core Magic, somehow.  Ah well.  It'll be back.

I like Giant Spider better: green pump is all well and good, but the spider is the perfect creature design: not flashy, not powerful, but the crucial glue that holds the whole flyers-versus-non-flyers combat dynamic together.  I don't think it matters whether a set has a 1cc or 2cc Giant Growth analogue; but the 4-drop that blocks all common flyers and survives is irreplaceable IMO!

well, Mutagenic Growth could be considered a better replacement, because as long as you have more than 2 life you can play it.

Note that you could use the compact listing of Gatherer to show all the cards in M11 or show all the cards in M10, and then quickly visually scan which ones do not include the M12 symbol. It's easy to see because the symbols are in release order, newest first.
Alas, search limitations don't allow you to "filter" cards that are NOT in Standard or M12.
Using this method I was able to find this piece of trivia: there is only one card printed in M10 or M11, but not in M12, that is still Standard tournament legal: Shatter. It was printed in Scars of Mirrodin.

Although that could work, it's still not as easy as a list already provided, such as the one @thesunneverset posted

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text:
What strong draft decks exist for Innistrad-Innistrad-Dark Ascension draft?
Innistrad has a decent amount of archetypes that you can build.  In draft, what tier 1 deck types exist?

"Proven to win 75% of the time or more"? Er... I'd say that's a bit of a tall order, even for a good archetype.  "Decks with an unfair number of bombs in them", maybe.

@thesunneversets what should i ask then?

@DForck42 The usual terminology is "tier 1", basically referring to the "decks to beat". Here's an example of usage (also the beginnings of the answer to your question): http://www.channelfireball.com/articles/breaking-through-redefining-limited/

Loosely speaking, I think strong Dark Ascension Limited (i.e. the official format, DKA-INN-INN) decks fit into one of three strategies corresponding to the three stages of play (which I have given these semi-descriptive labels):

"Aggro": Try to win before your opponent can really get rolling.
"Midrange": Build up a crushing advantage with powerful two-for-ones.
"Control": Hope to take over the game by making your opponent's stuff irrelevant, usually with a big bomb.

Decks that don't do at least one of these -- such as a deck full of mid-sized creatures but no way to actually turn them into card advantage -- tend to be at a disadvantage. Note that there's a lot of overlap here, probably much more so than in Constructed decks. You really can't classify most cards as belonging to just one strategy. Instead, the goal here is to view the card differently, and prioritize it accordingly, based on your particular gameplan; and to adjust your gameplan and later picks based on your cards. Darkthicket Wolf is a primo aggro card and a good midrange card, for instance; your "Human Sacrifice" deck may turn into a more controlling deck if pick up a really good late-game bomb.

I've never drafted with the pros and I don't play MTGO (where you can draft til you pass out from exhaustion), so I can't speak for the subtleties of "tier 1" vs. "tier 2", but I can list what I've found effective. Names in quotes below are those that are widely used, to my knowledge; I've made the other ones up based on whatever described a deck acceptably. Colors are suggestions rather than set in stone; each archetype has good cards in some of the other colors.
Aggro
These are decks that aim to win with an explosive start. Drafting them involves careful attention to your mana curve.
I think the top-tier decks in this category are:

White-weenie-style (W/G): Attack fast with dudes; keep attacking with lots of dudes. "White weenie" is a proven strategy in any format where white has strong low-cost aggressive creatures. Thalia, Champion of the Parish, and Cloistered Youth definitely fit the bill, as do Darkthicket Wolf and Hamlet Captain. Basically this deck depends on its low curve to win with speed. In addition to the usual pillars of efficient creatures and a bit of removal, these decks can benefit greatly from Travel Preparations. I think it's good to have at least one of Faith's Shield, Spare from Evil, or Feeling of Dread in the deck -- a card you can use to either push through damage or stop an opponent's attack while creating an opening for a decisive counter-blow. Be careful with CC cards like Loyal Cathar and Strangleroot Geist, since it's not always easy to actually cast these on turn 2.
Red-Deck-Wins-style (R/B): Attack fast with dudes; sacrifice board position to push your dudes through if needed; use burn to finish the job. The big difference from white weenie here is burn, with the standout card being Brimstone Volley. Evasion is usually part of the game plan: cards like Vampire Interloper, Crossway Vampire, Wild Hunger, and Feral Ridgewolf.

Midrange
These are decks that build up card advantage Jund-style. Dark Ascension has been good for this style of deck, with many of its premier cards having some component of built-in card advantage: Huntmaster of the Fells, Strangleroot Geist, Lingering Souls.
Some examples of good decks are:

"Human Sacrifice" (W/B): Trade your value-added weenies for your opponent's strongest creatures to build an advantageous position on the field. The "ideal" version of this deck starts with Skirsdaag Flayers; you feed them cards like Doomed Traveler, Loyal Cathar, and Elder Cathar. This deck also benefits from Human-token-makers like Gather the Townsfolk, and death-trigger cards like Wakedancer and Thraben Militia. Easy access to expendable dudes also makes cards like Demonmail Hauberk and Falkenrath Torturer better than they are in other decks. I like the Hauberk in particular because it's a good "escape plan" if you've picked up the relevant white cards but black has dried up: you can turn your deck into a pretty solid W/R or W/G morbid deck with potential for really crushing attacks.
Morbid two-for-one (G/B): Pressure your opponent with mid-sized creatures; when your trade, use morbid and undying two-for-ones to come out ahead. There are two morbid Grey Ogres I absolutely love in Dark Ascension Limited: Wakedancer and Ulvenwald Bear. Both of these are easy-to-pick up cards with a lot of mid-game power. A mixture of these guys, undying beaters (basically all the Strangleroot Geists you can grab), and quality combat tricks allows you to build up card advantage and superior board position; Victim of Night and Tragic Slip are also very desirable to help answer your opponent's biggest threats. Outside of rare bombs like Vorapede, the ideal big creature for this deck is Galvanic Juggernaut. Try to avoid the bad undying creatures -- 2/2 can't-block four-drops and 3/2 flying six-drops are not what you need to win the game. And don't just focus on morbid and undying: any source of incremental advantage is great; good rares for this plan are cards like Mayor of Avabruck or Gutter Grime, for instance.
"Werewolves" (R/G): Play dudes that become better than your opponent's dudes. Werewolves are a lot better with the additional cards from Dark Ascension. The key card here actually isn't Huntmaster: he doesn't really care about the creature types or colors of the rest of your deck; it's Immerwolf. The werewolves plan has always been to punish an opponent's slow development with a crushing tide of big creatures. Immerwolf and Moonmist make it much easier to consistently ride that plan to victory. As with all decks, don't get hung up on creature types; any card that fits the gameplan will work fine for this deck. Also, be choosy! Not all the werewolves are good.

Control
These are decks that want to waste time until they can play something that trumps your stuff, like a big Spider Spawning or a Reaper from the Abyss. Basically my standard for determining whether a deck is "control"-y in Limited is "Will this deck ever want to cast Think Twice?" Control gains some defensive power in Dark Ascension with the addition of quality defenders like Headless Skaab and Nephalia Seakite, but it loses a lot due to pack dilution. In particular, the "Mill Yourself" decks -- like Burning Vengeance or Spiders -- are weaker in Dark Ascension because their key cards are less likely to show up now that you're drafting from 2 Innistrad boosters instead of 3.
So far, I think the most consistently draftable deck in this category is actually a rather boring one:

Flyers (U/W): Clog up the ground, beat your opponent to death with evasive guys. I'm calling this a control deck for Limited purposes because it's really hoping to make most of an opponent's creatures irrelevant with some huge defenders and tricky plays, then grind out a win with evasion. Cards like Nephalia Seakite and Dungeon Geists are representative of your gameplan: they're basically flyers that double as removal. A big advantage of drafting this archetype is that you can often late-pick "big backside" cards for your defense.

is the offical format to open the DKA pack first?

@DForck42: Yes. Opening the most recent set first in a draft has been the policy since Mirrodin Besieged.

This is a great answer. My only nit is with your list of three archetypes: aggro/midrange/control describes basically every format ever, and is hardly limited to DKA.

@JSBᾶngs I think there's an absence of top-grade tempo and combo decks, at least from what I've seen so far. Consider a deck like Faeries.

A previous question covered the major archetypes for Innistrad draft. Most of the information there is still relevant, as Dark Ascension hasn't dramatically altered the draft landscape for most of the core archetypes there.
However, there were two marginal strategies in ISD draft that have become significantly better with the addition of DKA, if you open the cards for them in your first pack.
Mill
Getting a dedicated mill deck in triple-ISD was chancy, but it was playable if the pieces came together. Adding DKA has made this a tier-1 strategy, so long as you get this:

Increasing Confusion by itself can mill your opponent for 15 or more after you flash it back, and it's definitely first-pickable. If you get it in your DKA pack, you should strongly consider moving into a blue control deck with the rest of your picks. Take strong defensive creatures such as Armored Skaab, Fortress Crab, and Stormbound Geist, tempo-based cards such Silent Departure, Grasp of Phantoms, and Griptide, and additional mill components like Dream Twist and Curse of the Bloody Tome. If you get an Undead Alchemist in your ISD packs, you'll do a little dance, since that zombie is unplayable in most decks, but nuts in a dedicated mill deck.
Note that even if your U/x mill+control deck doesn't come together, you can still use Increasing Confusion as an alternate wincon in any number of other decks, or even a deck that's splashing blue.
Tokens
Intangible Virtue was a late pick in ISD, simply because there weren't often enough tokens to make it worthwhile. But throw in DKA with Lingering Souls, Gather the Townsfolk, and Increasing Devotion, and it becomes plausible to have a deck that uses tokens for its primary creature base. This deck obviously wants every copy of Intangible Virtue and Midnight Haunting that it can get.
Note again that all of the cards above are perfectly playable on their own, and you don't have to have a "mono-tokens" deck in order for them to work. But the existence of these extra token generators makes a token-heavy white aggro deck an especially attractive choice in DKA draft.

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text:
2 draw piles for convenience?
Sometimes when the table is large enough, or people don't want to reach too far, a single draw pile is split into multiple draw piles.  This has always bothered me.
Does it bother anyone else?  Are there any statistical reasons to not split the draw pile?

Aside from cases where there is additional information available about the card(s) or something has manipulated the deck, it also shouldn't matter if a player draws from the bottom instead of the top of the deck.  Getting hung up about this type of thing ranks up there with the idea that dealing to the players out of order will give someone else "your" cards.  Random is random...

Hypothetically, there's nothing wrong with splitting the draw pile. Splitting a randomized deck into multiple decks, won't affect the randomization. However, you need to carefully consider how you interact with the draw pile during the course of the game.

Do cards ever get put back on top of,
into, or under the draw pile? If so,
having multiple draw piles would
affect the game. For example, if
players can put a card on top of the
draw pile, can I choose to draw from
either pile?
What happens when the
draw pile runs out? If the normal
rule is to shuffle the discard pile
to create a new draw pile, you
probably need to wait until all draw
piles run out before doing so.
If    you have multiple draw piles,
will    you also have multiple
discard piles?    If so, you need to
consider how you    interact with the
discard pile, also.

Also, when we did this with a mixed older base set and newer expansion set in Race for the Galaxy, some players would opt to take the fresh card from either deck, which is obviously cheating.

@rob-renaud, when you say "fresh card", you mean a card that is obviously from the expansion because it is less worn/broken in?

Cards are randomized through shuffling not through picking the draw pile. Statistically you should have the exactly same chances of drawing any particular card from either pile.

When the table is inconveniently large and we are playing a game with a thick pack of cards, we sometimes spread the draw pile out into a large fan. That way, everyone can reach conveniently and anyone who suspects they may be disadvantaged can draw from any part of the pack. Myself, I can't see it makes any difference, but those who have some doubts can pick from anywhere and therefore have no grounds for suspicion.

The problem with that plan is that if (say) the ace of spades is dog-eared then anyone who knows that fact can choose to pick it (or avoid it), whereas in a 'normal' game the card goes to the person whose turn it is: this has to affect the game.

While splitting the deck is statistically irrelevant, as others have said, it does affect those games in which you might have such options as "check the first X cards in the deck and put them as they were" (say, to simulate precognitions), "check the first X cards and reorder them as you wish" etc.

We do that with some games (including tiles in Carcassonne), but you have to allow people to pull from either pile they want, and don't reshuffle until both piles have been depleted.

While I obviously agree when you say that all stacks/decks must be depleted before reshuffling, granting that each player may draw from either stack/deck is fair, polite, but not actually needed.


text:
Could I turn Mistform Dreamer into a Shrine?
Mistform Dreamer says I can turn it into any creature type until the end of turn, could I turn it into a shrine for use with any of the Hondens?

Instants and sorceries share the same subtypes. Creatures and tribals share the same subtypes. Nothing else does.

No. Alas, Shrine isn't a creature type, it's an enchantment type.
Rules reference:

205.3h Enchantments have their own unique set of subtypes; these subtypes are called enchantment types. The enchantment types are Aura (see rule 303.4), Curse, and Shrine.
205.3m Creatures and tribals share their lists of subtypes; these subtypes are called creature types. The creature types are Advisor, Ally, Angel, Anteater, Antelope, Ape, Archer, Archon, Artificer, Assassin, Assembly-Worker, Atog, Aurochs, Avatar, Badger, Barbarian, Basilisk, Bat, Bear, Beast, Beeble, Berserker, Bird, Blinkmoth, Boar, Bringer, Brushwagg, Camarid, Camel, Caribou, Carrier, Cat, Centaur, Cephalid, Chimera, Citizen, Cleric, Cockatrice, Construct, Coward, Crab, Crocodile, Cyclops, Dauthi, Demon, Deserter, Devil, Djinn, Dragon, Drake, Dreadnought, Drone, Druid, Dryad, Dwarf, Efreet, Elder, Eldrazi, Elemental, Elephant, Elf, Elk, Eye, Faerie, Ferret, Fish, Flagbearer, Fox, Frog, Fungus, Gargoyle, Germ, Giant, Gnome, Goat, Goblin, Golem, Gorgon, Graveborn, Gremlin, Griffin, Hag, Harpy, Hellion, Hippo, Hippogriff, Homarid, Homunculus, Horror, Horse, Hound, Human, Hydra, Hyena, Illusion, Imp, Incarnation, Insect, Jellyfish, Juggernaut, Kavu, Kirin, Kithkin, Knight, Kobold, Kor, Kraken, Lammasu, Leech, Leviathan, Lhurgoyf, Licid, Lizard, Manticore, Masticore, Mercenary, Merfolk, Metathran, Minion, Minotaur, Monger, Mongoose, Monk, Moonfolk, Mutant, Myr, Mystic, Nautilus, Nephilim, Nightmare, Nightstalker, Ninja, Noggle, Nomad, Octopus, Ogre, Ooze, Orb, Orc, Orgg, Ouphe, Ox, Oyster, Pegasus, Pentavite, Pest, Phelddagrif, Phoenix, Pincher, Pirate, Plant, Praetor, Prism, Rabbit, Rat, Rebel, Reflection, Rhino, Rigger, Rogue, Salamander, Samurai, Sand, Saproling, Satyr, Scarecrow, Scorpion, Scout, Serf, Serpent, Shade, Shaman, Shapeshifter, Sheep, Siren, Skeleton, Slith, Sliver, Slug, Snake, Soldier, Soltari, Spawn, Specter, Spellshaper, Sphinx, Spider, Spike, Spirit, Splinter, Sponge, Squid, Squirrel, Starfish, Surrakar, Survivor, Tetravite, Thalakos, Thopter, Thrull, Treefolk, Triskelavite, Troll, Turtle, Unicorn, Vampire, Vedalken, Viashino, Volver, Wall, Warrior, Weird, Werewolf, Whale, Wizard, Wolf, Wolverine, Wombat, Worm, Wraith, Wurm, Yeti, Zombie, and Zubera.

If only [mtg:Magical Hacker] had the subtype Teenage. Then you could have a Teenage Mutant Ninja Turtle.

Sadly with only 2 power...

Aww, well thanks for the answer. I should read over the comprehensive rules.


text:
In Dominion, when it says reveal a card, who must I reveal it to?
The text of Bureaucrat reads:

Gain a Silver card, put it on top of your deck. Each other player reveals a Victory card and puts it on top of his deck. (or reveals a hand with no Victory cards).

When I play the card and a player does not have any victory cards in their hand, it instructs you to reveal a hand with none. Should they only reveal their hand to me or are they required to reveal their hand to the entire group of players?

Reveal means to show their hand to the table.
From the rules, page 7.

Reveal – when a player reveals a card, he shows a card to all players and
  then returns it to wherever it came from (unless instructed specifically to
  put it elsewhere). If the player is required to reveal cards from the top of
  his Deck, and he does not have enough cards, he shuffles in order to
  reveal the required number of cards.


text:
Does Courser of the Kruphix's ability play multiple land cards?
Courser of Kruphix says:

Play with the top card of your library revealed.
  You may play the top card of your library if it's a land card.
  Whenever a land enters the battlefield under your control, you gain 1 life.

Does this mean if the top cards in my deck are a series of land cards they will all be played as they are revealed or is it still limited by the one land per turn rule?

This is a common point of confusion because people remember Oracle of Mul Daya, which is a very similar card that did have that extra clause. Courser doesn't. Watch out for this mistake, especially at tournaments.

See also: Future Sight/Magus of the Future, which have the same limitation.

It simply means you may play it as if it was in your hand.

You play the land normally, as if it were in your hand, with all the same restrictions. You are still limited by the 1 land per turn rule. In order for a card to let you play more than 1 land, it would need to say "you may play additional lands", or it could say "you may put it onto the battlefield" instead of saying "play."
See the rulings on Gatherer:

1/1/2014   Playing a land with the second ability counts as your land play for the turn. If you play a land from your hand during your turn, you won’t be able to play an additional land from the top of your library unless another effect allows you to.

And of course, the timing restrictions still apply too: you still have to play the land at a normal time, during your main phase when you have priority with an empty stack.

What would be the benefit of this card then, other than the ability to gain life when a land comes under your control? Does this mean that say, for example, you draw your card for that turn (which is not a land) and then a land happens to come next, you can play it even if you aren't allowed to draw it until next turn?

@agent154 That is correct. If at any time the top card of your library is a land, and you haven't played a land this turn, then it's like getting a free "Draw a card".


text:
When is my earliest opportunity to destroy a creature I couldn't counter?
My friend has been playing a sliver commander deck and uses Root Sliver which is a thorn in the side of my control.
Lets say I have an instant in my had that can destroy the Root Sliver and a counter creature card that I may want to use on the next sliver (and the mana to do both). I can't cast it while Root Sliver is still on the stack as it is not on the battlefield yet so I cannot target it.
As I understand it after the stack resolves however my opponent regains priority meaning they can put their next sliver spell on the empty stack before I have a chance to play my card to destroy the Root Sliver.
I was originally thinking I could put both the counter and the destroy in the stack but the counter would have to go on first so that it would resolve after the destroy, but since before the destroy resolves the sliver on the stack is not a valid target for the counter so this move is illegal.
Is it possible to kill the Root Sliver in time to counter the next sliver?

You can put the counter spell on the stack targeting the sliver by the way.  It just won't do anything when it resolves if you haven't removed root sliver by that time.  "Can't be countered" is not the same thing as "cannot be targeted by spells and abilities."  It's just like casting Terror on an indestructible creature.  The play is legal, it just won't do anything when it resolves.

Your opponent gets to put their silver on the stack first. However, you can then add your incinerate to the stack, targetting their Root Silver.

Incinerate
New Sliver

You can then pass so the Incinerate resolves, destroying the Root Sliver. Stack is now:

New Sliver

Then you cast your counterspell so the stack is

Counterspell
New Sliver

Your counterspell resolves, and the New Sliver is countered.

Oh I didn't know you had an opportunity to play cards as the stack resolves. I thought once it started resolving it was going resolve completely before anyone gained priority.

That's the way it worked in Magic 4thEdition, but since the introduction of the stack both players get a chance to add things to the stack every time something resolves.

Ah so that's probably how it was when I played as a kid, so I didn't just magically pull that out of my head somewhere. Good to know.

The play described in the original question is actually completely legal too.  Although this is a better line of play because your counterspell isn't wasted if your opponent cancels your incinerate.

This is a possible line of play, but I'd like to see @Affe's comments included in an answer - details on how counterspells work when something can't be countered, as well as how spells resolve on the stack would greatly improve the understanding of how this interaction could work.

I'll add that there are situations where an opponent's creature entering the battlefield will result in a trigger, allowing you a chance to respond to the triggered ability before your opponent can do any sorcery-speed action. (This matters against Birthing Pod decks where you can Dismember a Deceiver Exarch before your opponent can pod it into Restoration Angel.)


text:
Go - How does one count territory in these two scenarios? (9x9)
I'm learning to play Go and I was trying to develop a game in java, but I'm still not sure how to count territory in some cases.

In the scenario below, which side belongs to White, Top or Bottom?

$$W Scenario1
$$ --------------------
$$ | . . . X . . . . . |
$$ | . . . X . . . . . |
$$ | X X X X . . . . . |
$$ | . . . . O O O O O |
$$ | . . . . O . . . . |
$$ | . . . . O . . . . |
$$ | O O O O O . X X X |
$$ | . . . . . . X . . |
$$ | . . . . . . X . . |
$$ --------------------

Next one, White has the whole board divided in three pieces. Does Black get
  any territory here?

$$W Scenario2
$$ --------------------
$$ | . X X . . O . . . |
$$ | X . X . . O . . . |
$$ | X X X . . O X X X |
$$ | . . . . . O X . . |
$$ | . . . . . O X . . |
$$ | O O O O O O X . . |
$$ | X X X X . O X X X |
$$ | X X X X . O . . . |
$$ | . X . X . O . . . |
$$ --------------------

Is there a way (I don't know Go, so I can't suggest a way) to clarify the question beyond "these two scenarios" to be slightly more specific?

Neither game is over; as both players have useful moves in both scenarios - points cannot be counted (in general) until a game is over.

This is tagged [end-game], but neither of these boards are near the end of the game.  Are you instead asking about how to predict how the game will end during the middle of the game? I've played a Go app that would tell me how it would score if I passed on my turn.  Is that what you're after here?

Hi Rikku, welcome to this site! I've tried to answer your question, but if you feel like it, also check out http://playgo.to/iwtg/en/, in particular their counting lessons. I learned Go using that tutorial, hope you find it helpful too :)

@ForgetIwaseverhere, What if both players pass?

You've misunderstood the scoring principle. White is, as you explained, dividing the board. But that has no impact on scoring by itself. We often say, misleadingly, "divide the board into areas", but we should really say "stake out territory by surrounding it with walls". Only then can you count the score.
It is a bit tricky at first, but once you grasp it, it's very easy and natural.
$$W Scenario1
$$ --------------------
$$ | b b b X . . . . . |
$$ | b b b X . . . . . |
$$ | X X X X . . . . . |
$$ | . . . . O O O O O |
$$ | . . . . O . . . . |
$$ | . . . . O . . . . |
$$ | O O O O O . X X X |
$$ | . . . . . . X b b |
$$ | . . . . . . X b b |
$$ --------------------

In your first picture, player 2 has surrounded the corners (I've denoted that with 'b'). However, the other area is not yet finished, because it is not completely inside one player's walls. It is wrong to ask which side belongs to white, because nothing belongs to white yet - she did not surround any territory at all yet!
This means that this area is neutral and does not add score for any player. Instead, the players should continue playing and trying to stake out a wall around this area, naturally as large as possible. Let us try that.
$$W Possible result
$$ --------------------
$$ | b b b X b b X O w |
$$ | b b b X b X X O w |
$$ | X X X X X X O w w |
$$ | b b X . O O O O O |
$$ | X X O O O X X X O |
$$ | . O w w O X b X . |
$$ | O O O O O . X X X |
$$ | w w O X X X X b b |
$$ | w w O X b b X b b |
$$ --------------------

This could be a result. You can see that the previously neutral area has been divided, and both players got a certain share. You can see that now, white also has some area for herself (she had nothing before).
You'll also notice that after resolving, there are still some points on the board which are not controlled be either player. Because neither player can now surround these points, it is not necessary to include them. Any player could play a stone there to remove them, but he would not gain any points, because he is not surrounding new territory. (I assume you're using Japanese rules, it's a bit different in Chinese rules.)
In your second picture black has 3 solid areas (2 of them are split into eyes). White, similarly as above, has no definite territory at all yet.

mafu, I don't agree with you! It is not so clear and the question is not so stupid!! In the top image in the OP's question, both players may consider the black group in lower right corner dead, so then the whole low part belongs to white! The black group is dead actually, so the whole bottom belongs to white! But even if those black stones wouldn't be dead, but both players agreed that they are, the group would be removed without a fight and whole territory accounted to white!

It was not a stupid question, many people initially have trouble with scoring. You're right that the LR group of the first diagram is not alive (it is unsettled, the tsumego is a bit complicated as far as I can tell), but I think the asker was not referring to the L&D status of the groups rather, but to the general principle of scoring, as it seems they misunderstood how it is supposed to work. L&D is more advanced/complicated so I did not even consider it in my answer.

@mafu, This answer is for official Chinese-International rules? The same counting system used in https://deepmind.com/research/alphago/match-archive/alphago-games-english/ ?

@Pacerier This answer builds on Japanese rules mainly. Usually, there is little point difference of results between Japanese and Chinese rules (I believe +/- 1 point, unless seki is involved), so you might as well disregarded it completely, unless the game is quite close.

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text:
Can You Buy a Catan Border That Fits?
"The border of the 4th edition Settlers of Catan is too small for the board hexes.  To be exact, each hexagonal tile is about 79 mm wide.  Five of them, end to end, form the longest stretch of land requiring 395 mm of width within the shore tiles. And now, the distance between the shore tiles? About 393 mm. Yup, 2 mm short."  http://www.tinysci.com/2010/07/04/the-settlers-of-catan-something-completely-different/
Lots of custom boards out there because of this problem.
Is there an inexpensive commercially-available border that actually fits?

This is an old question that has recently reappeared on the main page. I'm voting to close it because, as written, it is looking for a specific product, which sounds pretty similar to game recommendation questions. While at one time, those questions were allowed, they are currently banned. Closing seems to be the best way to not attract new answers to this question.

Get a crafty friend to cut a set of interlocking tiles for you.
http://www.thingiverse.com/thing:6681
Or print a set of border pieces.
http://www.thingiverse.com/thing:6436


text:
What are the "help suit" game try methods?
You open one of a major. Partner raises you to two, showing support with 6-10 high card points. You are a rebidder with an intermediate hand, 15-17 high card points. Opposite an average of "eight" points, you have enough to try for game, but not enough to insist on it as you would with an 18-count.
So you rebid three of a lower ranking suit (usually a minor, but even hearts if your suit is spades) as a "help suit" try to see if there is "something extra" to propel you to game.
In the version I was taught, a three level bid in a new suit shows side suit controls, typically a king or ace, with 3-4 cards in the suit, and a potential for a double fit and several side suit tricks. Let's say a partnership has a double fit of two eight card suits and 17-18 high card points. That leaves ten cards in two more suits, hopefully divided 2-3 opposite 3-2, that is, four losers counting ruffing values. But there will be an ace or king left over (22 minus 18) to reduce the losers to three.
Here's where I am confused. Many players in my club use what I call a "short" help suit convention. Most of the time, this happens in the context of an "implied" raise (not an actual one), such as a Jacoby 2NT over one of a major. Under these circumstances, a bid of a new suit at the three level shows a singleton or void. The idea is to identify (cross) ruffing values that will allow a game to be made with 22-23 high card points. There's even a third version in use at my club, where the three level side suit bid shows controls: either a singleton or void, an ace, or a KQ combo.
Any system is subject to partnership agreement. But which (or all) of these systems is in general use and how do they work? In any case, have I confused any alleged system with something else?

After 1M-2M, I'm almost always making my decision as to whether to pass, invite, or bid game based on losing trick count, not high card points.

@AlexanderWoo: And my comments about points refer to their "efficiency" in generating a good losing trick count. My experience is that 21-22 high card points is enough for game if you have two suits under complete control, a side stopper in a third suit, and enough "shortness" in your two weak suits, or possibly three suits well controlled and shortness in the fourth. The reason you need 24-26 is if you have "wasted values. In many hands of the latter sort, you can through out one or two of your worst honors to get down to 21-22 without changing the result.

I should capitalize Losing Trick Count - this is actually a formal method of hand evaluation.  In a suit with 3 or more cards, you have one loser for every one of A/K/Q you don't have; in a suit with 2 cards, you have on loser for every one of A/K you don't have; a singleton is a loser if it's not the A.  Over 1M-2M, hands with 5 losers force game, hands with 6 losers invite, and hands with 7 or more losers pass.

My sense is that the most common agreement in the US is "help suit" game tries. In help suit game tries, you name a suit in which you could use some help; usually your suit is something like Kxx or Axx, and partner will be able to help with some honors or else shortness to ruff losers. Partner is generally expected to sign off with a bad 7 or worse, go with a good 9 or better, and with an intermediate hand to look at the named suit and decide if they have help. Without help, they might name another suit below 3 of your major in which they do have help, in case help in that suit is good for your hand. This convention is described in more detail in many places accessible through your favorite search engine.
Alternatives to the help suit game try include the natural game try (name a suit with length in search of a double fit) and the short suit game try (name a suit with shortness to help partner identify a hand without wasted values). There are also more exotic methods, like two-way game tries (these allow you to make either a long-suit or a short-suit game try, depending on what is suitable for your hand).

Accurate, but I would argue the "if minimum, stop, maximum go" part.  As our local expert said to my partner once, after 3H+1, "I didn't ask if you had extras, I asked if you had help in diamonds."

I suppose either way is playable, but I think "if minimum, stop, maximum go" is more accurate more often, unless you have a way to more narrowly limit the response range. After 1H-2H-3D, if you could hold AKxx QTx xx Jxxx or QJxx QTx xx Jxxx, it seems to me that the first hand is often two tricks better than the second, and must be enough to go even without diamond help.

+1 for giving the IMHO crucial statement. The point of a short suit trial is not to check for the possibility of a cross ruff (as the OP indicated). Instead, it is about diagnosing wasted strength opposite a singleton.

I think you’d just describe game tries broadly, as a shape-based invite.
As Alexander says ~6 losers, and with a shapely hand. ( With 17+ HCP you can make a strength-based invite in NT or 3 of your suit). So probably less points but with useful shape but there is a lot of overlap. The shape shown is  down to specific partnership agreement.
Some find short suit trials most useful (I dont). Thats akin to making an invitational splinter (and alertable since not a natural suit).
Some still play long suit trials, a weak suit with 3+ sure losers in the suit ( can you cover some losers partner?).
In modern Acol the trial simply shows 5 cards in the 1st suit bid and invitational strength ( in contrast to 4 cards for 2NT or 6 card with 3X).
GIB seems to use a modern and typical broad Help try, promising just 3 cards including Q+, but not denying more strength in the suit though inevitably short elsewhere. (So partner, you work out how your hand can fit with mine to reduce your 9 losers to 8. Can you cover or ruff my losers or make a double fit? etc.)
Any shape agreement is theoretically possible though typically longer or shorter than average shape which is useful to know. Control bids are generally kept for slam try situations rather than game tries (such as after a jump raise 1H 3H 3S)
(NB: And, one shouldn’t confuse game tries with the strong invite after opponents compete in another suit 1H 2H [2S] 3D, which is a different context)

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text:
Were my actions over a takeout double correct under either actual or different circumstances?
At matchpoints, my left hand opponent opened one diamond. Partner doubled (takeout). Right hand opponent stuck in a one heart bid, which theoretically "took me off the hook."
I elected to bid one spade with ♠QTxxx ♥JT ♦Jx ♣Txxx. This was now a "free" bid, which usually entails eight points (partner had expected 10). Factors in my decision to bid light included the fifth spade, three tens, and two doubletons in the opponents' suits, which would make "transportation" easy, despite my low point count.
Partner raised me directly to four spades with an 18 count consisting of  ♠AKJ ♥AKxx ♦xxx ♣QJx, leading to a result of down two. In partner's shoes, I would raised to only three spades , holding only three spades. According to these tables of data an eight card fit with a 18-4 high card point distribution is a decided favorite to make nine tricks, while a nine card fit is slightly favored to make 10 tricks.
Given that we were vulnerable, our down two led to a "shared" bottom (with others). I would think twice about doing this again when vulnerable. But if we were not not vulnerable, or partner had bid only three spades, -100 would have gotten us a shared top against opposing part scores.
So was vulnerability the deciding factor in how I should have responded, or were other considerations?

I agree with the answerers that nowadays your 1S call does not promise 8 points or so. I would bid 1S with your cards, but if your partner expects more, then you must pass. But I really don't like the 4S bid. The opponents have not found a fit, so the "promised" diamond shortage they don't have is a big minus. 4333s generally play badly.

This depends on your agreements with your partner.
If you've agreed with partner that your 1S bid promises 8 points, then you have nowhere near it and shouldn't bid.
Now, even if your 1S bid does promise strength, I don't like partner's 4S bid, or a 3S bid, because you haven't promised 5 spades for your bid.  I much prefer 2D, artificial and forcing (promising at least around the kind of strength your partner has), giving you a chance to bid something other than spades if you only have 4.
Personally, I prefer to agree that a free bid, especially at the 1 level, doesn't promise very much strength, just enough distribution to encourage partner to compete further, even if they have a minimum.  Since -100 is better than -110 and disrupting their bidding can have good effects even if they have game (or make it harder for them to find their slam), I think it's worth bidding 1S on almost all hands with 5 spades.  (Almost: I wouldn't even think about bidding with xxxxx QJx xxx xx.)  Think of it this way - even if your partner has the flattest almost minimum like KJx AQx xxx Kxxx, wouldn't you want to be playing 2S rather than letting the opponents play 2H?  The opponents are quite unlikely to double.  (And if your partner has an even weaker hand, it won't cost to try 2S at some point, because opponents have a game and 2SX is unlikely to be down 3.)
But if your partner isn't willing to play this, your choices are to pass and accept a bad score when your partnership should have competed, to bid and accept a bad score (and maybe an angry partner) when your partner bids expecting you to have something, or to find a new partner.  (For serious games, I would go the find a new partner route; for less serious games, I would do what partner prefers, keeping in mind some weaker partners do want me to mastermind and hand hog.)

I am in a "social" bridge club where people are assigned parts (unless they sign up for games as a pair). "Rejecting" a partner assignment is saved for the very worst cases, like when a partner yells at you repeatedly.

In that case, the right thing to do is always whatever your partner wants, not whatever is the best bridge action.

Also - if you're actually interested in learning how to play good bridge, you need a better game.

I'd argue that I have "stretchy" eight points, though not 8 HCPS. I count 4 HCPS, a fifth point for my three tens, two points for the fifth spade (partner only has the right to expect four) and one point for the doubleton in diamonds. The club standard is "eight," and while I'm rounding up, I am doing so within normal club limits. I hadn't realized that my (occasional) partner is ultra conservative, even by club standards.

I am absolutely of the opinion that the best partnership agreement is to allow a 1S bid with Qxxxx x xxx xxxx (unless you want to agree that jump bids are weak after responder bids, in which case you have a 2S bid).  Your Jx and JT aren't worth more than xx and xx here.  Being able to get to -100 rather than -110 is important; it's a frequent source of matchpoints.  But - as I have said - if your partner doesn't agree that -100 is better than -110, you don't have any good options.

As usual, agreeing with everything @AlexanderWoo says.  I'd probably be using 2♥ as the force rather than 2♦ but that's more likely to be useful exposing a psychic, so he's probably right.  Not that anybody's psyched this auction outside absolute flight A in 15 years...
My "rule #1": if I choose to bend or violate system, I own the result.  Even if partner should have figured it out, if they don't, it's my fault if it's wrong.  Of course, if it was right, I get to own at least 50% of the good score!
You chose to violate system, for good reason (I think this agreement is silly.  "Free bids showing strength" have mostly gone the way of the dodo, for very very good reasons involving "winning the partscore", "finding our fit", and "finding the right lead against their contract".  Of course, pass meaning "I have nothing interesting to say" or "feel free to describe your hand further" in many of these "free bid" auctions is often the best call; but if you do have something interesting to say, say it.  And telling partner your side has a fit in the boss suit is "something interesting".)  Partner made a call that wasn't best, based partly on not knowing how to investigate, and partly on "knowing you have stuff" (where he thought the 1♥ bidder's points were, given he can count 12+18+10 in the other three hands, is a question.  But if you "know" partner has to have it, it is difficult to think past that "knowledge" to "that's impossible").
There is fault going to partner for their handling of this hand; but the bad result is yours to own; you don't have what partner expected.
You have two (well, three) choices. You can suck it up and bid partner's style, and eat all the bad results you get from passing where others are finding their fits, or you can take flyers and own the results, good or bad.
Or you can have discussions with your partners and try to convince them their bidding style is (deservedly) old-fashioned, and join the 21st century.  Which will also lead to poor results as they learn how, even if they agree to try; but eventually better.

Its easier looking at things with hindsight, but i’d say you were both bidding very optimistically, whether vulnerable or not.
First, your 4 points includes two worthless jack doubletons and no singletons, looks worth more like 2-3 points although good enough for a 2nd round bid, while partners horribly flat 18 points 3 spade hand is more like 16 and not really worth a rebid.
I think your Partner is more defensive and shouldn't really get above 2S. Let opponents bid 3D, partner should leave them there and you will probably get them down 1 (using law of total tricks) for the best positive score. If they stopped in 1NT then leave them there, you’re laughing with 7 defensive tricks off the top (what does double dummy show is best achievable?!)
I’d estimate optimum bidding as 1D X 1H P 2D P P 2S P3D (all pass).
Partners defensive hand on lead looks worth HAK, AS, and couple more trick somewhere from a 2nd spade, a club, or a heart ruff or perhaps trump promotion.
But even 3D making is cheaper than 3Sx-1 vulnerable. Dont feel bad when you get low score because other pairs weren’t good enough to penalise their opponents properly.

I'll grant you that the jack of diamonds is probably worthless, but no the JT of hearts. With the takeout double, partner has promised Qss or better in hearts, and JT plays well opposite such a holding; I'd argue that the JT sequence in hearts is worth two points.

@TomAu. The JT might have some value in defense but none in offense (as is generally the case for side suits lead by quacks). So, all the more reason to defend. In offence, you could only ruff your hearts with long trumps which were already winners.

If I wanted to reach 3S when not vulnerable, i’d jump  to 3S after passing on my first bid. Although i prefer my chance of a + score by defending this hand.


text:
How should I have avoided getting my ace "buried" on defense?
With neither vulnerable at duplicate, my opponents arrived at a four spade contract after a nondescript auction that disguised their strength (28 high card points between them). It was one spade (RHO), 1NT (forcing, LHO), 3 spades, 4 spades. (Defenders pass throughout.) I was the opening leader with ♠T9 ♥JT83 ♦T76 ♣AQ83.
Most people in my club would lead the ace, and probably a small club, but I elected to lead the J of hearts try to develop the suit and save the ace of clubs for re-entry. Declarer won with dummy's queen, drew trumps and gave my partner her sole trick, the ace of diamonds Meanwhile, I followed low to discourage. (Partner could see all smaller cards than my six in her hand or dummy). She still returned a diamond, allowing declarer to run dummy's diamonds, and "burying" my ace of clubs.
Should I have opened with the ace of clubs, as most club members do? Should I have relied on partner lead back a club, after she saw no future in the red suits? Or was this just an instance where a reasonable play produced a bad result? Was it more urgent to lead the ace of clubs against, say, a six spade contract, which would imply that partner might have no tricks?

Where was the king of clubs?  What was dummy's distribution?  Based on your prior questions, I strongly suspect that every single person at your club is by my standards incompetent at defense, so it's really hard to draw conclusions about correct defense in this situation.  "What is the correct defense given my partner is incompetent at defense?" is not really an answerable question.

Did declarer draw 2 rounds of trump or 3?  If it was 3, your discard on the third round should've been an important signal.  Also, I don't play it because I find it almost always just tells partner something they can already figure out, but you might want to play trump suit preference.

One more addendum: I'm sure any actual expert would consider ME incompetent at defense.  After all, at least 2 or 3 boards a session I defend as if some suit had 14 (or 12) cards in it.

Give. The. Auction! Making opening lead decisions without the auction is tantamount to sticking one's hand, blindfolded, in a bowl of piranha- unlikely to provide a pleasant result.

Confucius say: "Defender who lead same suit as Declarer, be partner of partner of fool."

@ForgetIwaseverhere: The auction was one spade (RHO), 1NT (forcing, LHO), 3 spades, 4 spades. (Defenders pass throughout.) I added this line to the question. Opposing clubs were 3-2, with  KJX in dummy. After partner took the ace of diamonds, and returned one, dummy ran diamonds, allowing declarer to sluff both clubs.

What was dummy?  Should it have been obvious that dummy could run diamonds?  If there was a third round of trumps drawn, what signal did you give when you discarded?

You don't give the auction, so it's difficult to judge the lead. You don't give the dummy or the number of rounds of trump pulled, so it's difficult to judge partner's play.
Giving me this hand without the auction, I find little fault with the lead of the heart jack. The ace of clubs seems likely to lose more often than it wins (again, without knowing anything about the auction) and I don't see any particular reason to play it.
However, your play of a low diamond does not show negative attitude towards diamonds. We don't generally give attitude when opponent leads a suit because (a) it is assumed that we do not want to continue the suit, since declarer is playing on it and (b) telling declarer the location of missing honors in a suit that they are playing on is much more helpful to declarer than partner. In standard signaling, you are indicating that you started with an odd number of diamonds (which happens to be true).
When declarer gives you a trick in a side suit, it is not often correct to continue playing that suit. There are exceptions, but think about why, when you are declarer, you play on a side suit. It's often to set up winners in that suit so that you can pitch losers in another. Indeed, that seems to be the case in this hand: once declarer knocked out the diamond ace, they were able to pitch away club losers (I assume). Partner was almost certainly wrong to continue a diamond, even without knowing about your club ace.

Auction analysis
Some inferences can be made from the auction. I have never played (or played regularly against) the forcing 1NT and/or 2/1-flavored systems. I will go with the internet’s description; in a real situation, make sure to ask opponents (once the auction is over) about the precise meaning of their bids.

1♠: 5 spades, 11+HCP
1NT: 6-12HCP, 2 or fewer spades, kind-of regular hand, forcing for one round
3♠: game proposal - 6 spades (at least) and 15-18HCP

without game force (<14HCP) they would bid 2♠
with game force but no hope of slam (something like 18-20HCP without good shape?) they would jump to 4♠
with a hope of slam (20+ points counting shape) they would/should investigate at the lower levels

4♠: to play: 10-12HCP and probably 2 spades.

That auction is "nondescript" in the sense that it gives no obvious hints at what side colors the declarer will use, but it is rather precise in terms of strength of either opponent. I grant you that you have no real idea of the shape of the hands; a fifth-card side color in dummy is possible, though unlikely.
Scoring system
A previous version of this answer assumed the scoring system is IMP. In that case, beating a beatable contract is much more important than limiting overtricks of a cold contract: compared to a line that ensures =, giving an overtrick is -1 IMP, beating a NV game is +10 IMP. Hence a line that gives a 10% chance of beating and a 90% chance of giving an overtrick is worth taking.
In MP, things are very different: if everyone in the field makes =, taking a non-standard line ensures a top or bottom, so you should only take it if it is above 50%.
The competition situation matters too. If you scored 30% in the first half of a MP event, or -30IMP in the first half of an MP event, and you try to end up at least on average, you will need to take some pretty crazy decisions in the second half; sound, safe bridge in a mediocre field might score you 55% / +3 IMP on every board but that is not enough to climb back. (In statistics terms: you switch to a lower-mean, higher-variance strategy - some math here). Here, as the competition situation is not given, I shall ignore it.
Opening lead
We see 7HCP in our hands, from the auction partner must have between 4 and 7.
First of all, there is a rule of thumb to lead away from your honors (unless those honors are a sequence). The reason for that rule is to avoid giving a free finesse to your opponents. The obvious application here is if declarer has the ♣K: leading any club allows them to take a trick with it, when they would never be able to do so otherwise. But it also applies if dummy has the ♣K. Sure, if declarer takes the club finesse (after another lead) it will work, but maybe they must then choose between the club finesse (which you know works) and another line (which may or may not work; for instance, they might try to finesse the ♥K in your hand). Leading clubs makes the choice for them (even if you lead a small one, they have no other choice but to try the king if they want to not lose two tricks in the color).
It seems players at your club did not apply that rule of thumb. Let us look at the situation more specifically.
Consider first the active / aggressive lead hypothesis. Something needs to be done urgently, else declarer might discard the problematic cards on a side color. What could this be? I see two possibilities:

partner holds the ♣K, and both opponents hold at least three clubs, and the opponents can discard a club on a red suit if they are in hand. In this case, leading clubs is urgent.
declarer holds the ♣K, and both opponents hold at least two clubs, and we can establish a red honor in partner's hand if we lead the appropriate color. (Additionally, partner needs to play well and can go through declarer for two club tricks.) In that case leading clubs just gives a trick and possibly two (it allows declarer to take a trick with the ♣K and loses a tempo to establish partner’s red suit).

The former is much less likely than the latter with regards to the layout of clubs; even taking into account that you need to guess the correct red color in the latter, it is still a superior play, under the hypothesis that an active lead is needed.
What about a passive lead? Well, the rule of thumb applies here. Leading away from ♣AQ will give a free finesse for the ♣K in declarer’s hand, which is moderately likely (they have the hand with most HCP). Leading in another suit might give a trick, but it is more speculative - it needs to be a color where the declarer can finesse partner, and where would have not done so on another lead. Imagine for instance declarer has ♥AQ ♦xx opposite dummy's ♥xx ♦AKxx with three losers in the black suits (three clubs, or two clubs and the ♠Q offside). Sure, leading hearts "gives" the finesse, but the declarer had no choice but to try it anyway.
Finally, what about a trump lead? As an active lead, that does not seem indicated (again, check this with your local 2/1 expert, but the auction does not scream "ruff in dummy" to me). As a passive lead, this fails if partner holds ♠Qxx. Surely that is similar to the situation in the red suits? Well, not really. Declarer can bury a red honor by running the other red suit, but they cannot bury a trump honor. Hence, playing a red suit is better.
Which red suit? All other things being equal, we would rather lead a suit that is less likely to be held by the opponents. This means our longest suit (mechanically, if we have more, they have less of it), and majors (which are more likely to be named during auction if held by them).
Hence, I agree with your heart lead (which card exactly would depend on defense conventions; in my regular partnership the jack is indicated). Leading a small diamond would be acceptable at my level. A trump is dubious. The ♣A is incorrect, and a small club is atrocious (this might give one trick and no losers to declarer with the ♣K).
Partner gets in hand
After losing one heart and three spade tricks, declarer leads diamonds. You play low, dummy plays (say) the jack. (If you played low when dummy has no honor, that is a mistake, you should have played the ten.) Partner takes the ♦A; can she beat the contract?
All three remaining tricks need to come from your hand. Partner started with 5HCP (the ♦A and a black jack); by the same auction analysis as above, and removing the ♥J that you played, she can infer you have 5-8HCP among ♥K ♦KQ ♣AQ:

spades have been played and you had nothing;
you would not lead the ♥J with the queen or ace;
she can see ♣KJ in dummy.

At that point, she can see the contract is unbeatable. The only way to put three "natural" tricks in your hand, given that clubs are offside, is ♥K ♦KQ in your hand. This is the maximum that you can have in terms of points, but even so, that requires that hearts be played once and diamond twice without allowing declarer to establish a ruff in either color, when they have all club honors. This requires declarer’s hand to be three clubs (else they pitch something on dummy’s third club), two hearts (else they take the ace and ruff the next round) and three diamonds (else they ruff the third round); this is impossible given that they have shown six spades at auction (3+2+3+6=14).
Given that, she must play to limit declarer to =. What exactly this means depends on what she sees in dummy, but:

if declarer had been able to quickly discard diamonds, they would have been able to do so rather than play diamonds, so we can assume there is no urgency in diamonds;
the only important honors to "score" are therefore the ♥K and ♣A.

establishing the ♥K from your hand requires two rounds of the color, with the first one not being played from your hand
establishing the ♣A from your hand requires one round of the color, from any hand.

Therefore I think the correct card is likely to be hearts. This assumes that (from partner’s perspective) the ♣A in your hand is not much more probable than the ♥K. I believe this to be the case, but a better player than me might show otherwise.
In practice this would have buried your hand all the same, i.e. "a reasonable play produced a bad result". Your suggestion to return a club, while inspired by frustration at your hand, is in my opinion incorrect. (Whether diamonds or clubs is the "more incorrect" play, I leave it to others to discuss.)
Against 6♠ (or 7♠)
The auction would be entirely different, and as a consequence, so would be the information you have.
For instance:

did partner double a conventional bid to request a certain lead? Or did they have the opportunity to do so, but failed to take it?
did either opponent name a control in clubs? If so, first-round (i.e. a void) or second-round? Which opponent? If you know the ♣K is onside, you may have a chance to beat the contract (if your partner gets in hand via a queen and goes through RHO, or if declarer needs to establish a trick in clubs), but you need to not lead the color.
can you tell partner is club singleton or void in a red suit? 1♣ - 1♠ - 2♠ - 3♣ (etc.) shows both opponents 4-4 in black colors, which means your partner is club singleton. This almost never happens but when it does you must pay attention.
opponents might announce whether a trump lead is safe (one of the replies to RKC Blackwood is "I have three cards among the aces and the king of trumps, plus the queen of trumps"; if their partner bids six, you know that they have AKQ of trumps). You might still need an aggressive lead to beat the contract, but you know trump is safe.
does the auction call for an aggressive lead? That is a vast subject and I am by no means an expert, but compare those extremes:

1♠ - 4♦ (Splinter) - etc.:  There is a high risk that the contract makes through ruffs, discards etc. An aggressive lead is called for.
1NT - 2♣ - 2♠ - etc.: Regular hands, calling for a safe lead is called for.

The comment " However, given the scoring system, beating a beatable contract is much more important than limiting overtricks of a cold contract. Hence, let us assume the contract is beatable." is completely false. Unless stated otherwise, one must assume a Duplicate game is matchpoints, where it is vital to not make avoidable errors, such as over-aggressive opening leads.

Re: 4S. Since partner has shown ~15-18, 4S can't be 10-12; if that were the case, you would stop in 3S in many making games. I'd say it's more like 8-12.

A thorough answer. I agree with @ForgetIwaseverhere's comment regarding scoring. Your analysis of partner's return decision is good but omits any discussion about signaling to previous tricks that may help partner here. For example, many pairs play trump suit preference; even giving diamond count might help partner better picture the hand. Of course all of this is made more difficult because the dummy is not given in the question.

@ForgetIwaseverhere My bad, I think that’s a translation/culture issue on my part. In France, the vast majority of tournaments are either 4-person teams with IMP scoring ("duplicate"), or 2-person teams with MP scoring ("pairs"). I incorrectly assumed IMP scoring. I edited the post (it does not change the decisions though).

@ruds I disagree on the auction: assuming the player who bid 1NT has no distributional values (again, I might be wrong, as I do not play against 2/1), they should not bid game with 8-9 HCP opposite a maximum of 18. (I agree on the importance of reading partner’s signals.)

@UJM: In 2/1, the 1N bid is 6-12, usually 0-2 spades (but could be 11-12 with 3 spades), and the 3S bid is more like 17-18, (since responder can invite over a 2S rebid with 2+ spades and 10-12).  1N is not always balanced - it could contain a long suit other than spades without enough strength to make a game forcing bid (since 2C/D/H responses are all game forcing)

In your "Partner gets the hand" analysis -- this is presumably after 3 rounds of trump, so the querent has had a discard, which should have been a club to indicate that he held clubs rather than hearts/diamonds.


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How few cards in a suit would still allow a responder to make a "transfer" bid?
Both sides vulnerable at matchpoints.  Partner deals and bid 1NT (15-17). Right hand opponent passes.
In a dialog surrounding an earlier version of this question, it was established that it made sense for responder to bid two spades, asking for a transfer to three clubs (up two levels) with a hand as weak as ♠Q85  ♡ 8762 ♢J, ♣97432 that had five clubs.
Suppose responder's hand had a four card major. Would it then make sense for responder to bid two hearts, asking for a spade transfer with ♠Q852  ♡ 8762 ♢J, ♣9743? You have one fewer "trump" but the bidding is also one level lower. There is a small chance that you will end up in a six card fit if opener has only two, but most of the time it will be seven, or even eight cards.

You have it backwards.  The worse your hand, and the worse your club suit (in terms of high card strength, not length), the more likely you are to want to transfer to clubs.  Suppose your partner has a solid 1N opener like Axx Axxx Axx Kxx.  If your hand is xxx xx xxx AQxxx you are likely to make 8 tricks in notrump and in clubs, so you want to play 1N.  On the other hand, if your hand is xxx xx xxx xxxxx then you are looking at 3 tricks in notrump (can't get to the weak hand for the finesse) and 5.5 in clubs, and it's worth it to transfer.

@AlexanderWoo: The way you present it, we'll make 5.5 tricks at three clubs, going down 3.5. If I leave partner in 1NT, we'll make 3.5 and go down 3.5 tricks there. Looks to me like '"six of one, half a dozen of the other."

See the edit - you really only make 3 at 1N.

@AlexanderWoo: Based on your comments, I "turned the question around the other way."

I think the question's still wrong. Transfers are for finding good contracts. When you haven't got enough cards for a transfer (or crawling stayman) you'll pass 1NT and wait until opponents double you. Only then will you consider falling back to natural suits (2c, 2d, etc) using your chosen escape method. Why gamble and force yourself up to 3C (vulnerable, doubled!!) when you can play in just 2C, etc. knowing you have a 7+ card fit?

The point of playing in a suit contract is so that your little cards in the trump suit become tricks, if that hand has more trumps than anyone else.  With a 4 card trump suit, even in a 7 card fit, it's likely that one of the defenders has 4 in the trump suit and so the little cards in that suit are no longer your tricks (because defenders can draw trump)

@AlexanderWoo: If you are correct on all counts, that fifth trump may make a difference of TWO tricks, not one.( I had hypothesized a linear relationship; one more trump, one more level.) Based on your observations, the transfer works relatively well for a five card suit, but its efficacy "falls off" when the number of trumps drops to four

There's nothing special about "transfer" bids over 1NT that differentiate them from bidding the suit directly, as far as safety is concerned.
Obviously, transfers allow a second bid after the transfer is accepted, which leads to more sequences; transfers allow the NT opener to play the hand (which, with a strong NT, is frequently an advantage),... but as far as "why would I play in the suit instead of 1NT", it doesn't matter whether you're bidding 2♦ to play 2♥ or if you're bidding 2♥ which partner must pass.
So, the question is: "how weak can your suit be to still prefer playing in the suit to playing 1NT?"  And also, "are there better ways to get to a 'better contract than 1NT'?"
Normally, a 5-2 major fit (that could be 5-3 or better, but assume the worst) will play better in 2 of that suit than in 1NT, more often than not.  There are two reasons for this:

long trumps always take tricks; in 1NT you have to have an entry to them after establishing them.
long trumps also are stoppers for their long suits; in 1NT only high cards count.

As you can see, in both of those cases, we're concerned about the hands where responder is not strong (so no guaranteed outside entries, and few stoppers/help stoppers).  Hands where responder has some strength, we will transfer and invite (or bid game) in NT with only 5, because with values that expect to make 8 or 9 tricks, you will really only see better results (most often) with the magic 5-3 fit.  Note: this is a major blind spot for many weak(er) players - they transfer and invite in their suit with only 5, and do not understand why that's wrong.
Now, remember also that standard systems(*) can not play in 2 of a minor directly; the cheapest place to play in a minor is the 3 level.  Now, your suit has to be worth 2 more tricks than in NT just to break even.  Usually, with no outside cards, that would be a 6-card suit, but the hand in the OP is so bad that even the 5-and-a-singleton expects to take 2+ clubs if trumps, and zero in NT.
But that is an exception - with even ♠Kxx instead of Qxx, or ♠QJx ♦x, there's a chance that the clubs might set up and be runnable, or at least, helps partner's suit run or blocks the opponents' second suit.  If it does, 1NT may make, or at worst be off 1, but 3♣ only makes one more trick and scores worse.
Having said that, there's a better call with your 4=4=1=4 than transferring; Stayman.  Yes, if partner bids 2♦ you'll not be happy (and you'll probably fake being 4=5 in the majors and bid 2♥), but you will get to a 4-4 fit if you have one.
In the comments, we have "wait for the opponents to double, and run then".  And yes, that frequently works - you get to play the 4-3 or 4-4 fit with the ruffs from the singleton.  In 2♠ doubled rather than 1NT doubled.  And that's 200, vulnerable, even off 1.  If you Stayman right away, they might bid, they might not double you - they might not be able to double Stayman with "cards" the way they can double 1NT.
Remember, the best result with this bad a hand opposite 1NT is that they play the 20-20 misfit; after that is playing a fit and hopefully making it; then playing a fit that goes down; then playing 1NT that certainly goes down.  200 is called the Matchpoint Death Score for a reason.
(*)Having said that, the worst suit I have ever transferred into was ♦8xx.  Now, there were several special situations that applied:

We were playing a 12-14 NT.
I had a 2 count.
I could make a 2♣ call that "may want to play 2♦, or may have one of several invitational or GF hands, with or without diamonds."
It was in a Swiss Teams match against either the best or second best pair in the room, who I knew played penalty doubles of 1NT in balance; but again, might not have a "cards" double of our Keri 2♣ call.

It could have gone for numbers, sure; but I knew we were -800 or more wherever  we played it after LHO got to double 1NT; it was possible we could get out undoubled if I bid now.  (LHO decided to overcall my 2♣, and got to 3NT for -1 when she played it better than our teammates did.)

With Stayman, you need eight or more high card points to make it viable, because it'shard to stop at the two level. With a transfer, such a stop is much easier, and you cn do it with zero high card points.

1NT-2C; 2D-2H; pass.  Standard way to show a weak 4=5 in the majors for ever (by that I mean that the change was done (but still asked about) when I was learning in 1995).  Yes, originally there was the concept of "strong stayman" where the 2H call was forcing, but I don't remember the last pair who played it that way.


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How Closely Did Afrika Korps simulate the actual campaign conditions?
My understanding is that the Afrika Korps game parameters give the Germans and British about equal chances to win, that is, 50-50. Were the relative chances actually about 50-50 in real life? If not, what adjustments were made to the game to balance it? For instance, was Malta more or less disruptive to Axis supply convoys in the game compared to real life?
Example: Using Midway, which I know better, a number of adjustments were made in favor of the Japanese. That's because the large number of planes on Midway made it a "fourth" carrier, with all the American carriers larger than any of the four Japanese heavy carriers. That is to say that the Americans actually had the advantage historically.
So Midway, in the game, starts with perhaps one third of its historical complement of aircraft. Also, the Japanese carrier Kaga starts with three more toredo squadrons than she had historically (this is critical for game play). "Alternate" rules also prevent the American player from deploying B-17 bombers (which existed, but had no historical impact, but is a "random" factor in favor of the Americans). I believe these changes largely balance the game.

Not very well at all, but good enough for a board game.
The impact of Malta is reflected in-game as the difficulty of the Germans getting reinforcements and resupply as the game goes on. (I'd get the rules out and cite a detail if only my brother didn't end up with the game; I got Waterloo and D Day, he got Afrika Korps and Panzer Blitz; the local games club got the rest when we moved out).
Of particular interest is that the air element of the campaign is utterly missing.
Balance with "historical" verisimilitude is a very tricky thing.  Afrika Korps is not one of the more complex AH games (I played them all from about 1970 - mid 1980's).  A wide variety of decisions made outside of that theater left both Rommel and Monty with constraints on their resources, and how to make that work "in game" can be tricky.  The 'trickle in' gamism of reinforcing units is reasonably close to how things played out. The relative ineffectiveness of most Italian Divisions is also "close enough" to work in the game.
I've seen games of Afrika Korps won using zones of control and a Recce Company choaking off supply trains at the left edge of the map when Rommel is near to El Alamein.  :-)   IRL, that's not gonna work.
I did not study that campaign in detail until after college, and played it mostly in high school and college.  With the game as an introduction, it was nice to be familiar with the topic, but as one reads the history one begins to see how a game has to take a lot of short cuts with verisimilitude to arrive at a playable state.  (Midway was kind of clunky).
Something you may want to appreciate is that the sales brochures that AH put out back in the 60's and 70's had a "complexity" assessment for each game.  Afrika Korps was one of the easier games; 1914 was a nightmare of complexity.

I think I know you from another site. Welcome to this one, even though the question is largely in line with the other one.

@TomAu Yeah, I owe you an answer on Napoleon; hopefully, this weekend will allow a bit more digging.  Great to "see" you here Tom.  :-)


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Are there "established" systems where "Five Card Majors" isn't a strict rule?
In "Standard American," I must open a major with 65432 while refraining from opening a major with AKQJ.
IMHO, there are five card majors that are too weak to be opened and there four card majors that are too strong NOT to be opened.
I've created a system where "An extra trump is worth a king," and the minimum opening standard is Qxxxx. Thus, xxxxxx "maps" into Kxxxx, and qualifies. xxxxx < Qxxxx and doesn't qualify. KQxx "maps" into Qxxxx and qualifies, but not KJxx, which "maps" into Jxxxx.
I'm not sure Standard American is right for me. Are there any established systems (other than Standard American) that operate along principles similar to mine?

I don't understand this question. It is similar to: Do I kill a kitten with a gun or a knife? The answer is: don't! You have a system, stick to it. There are no judgement calls involved when deciding you have 5+ cards in the major... If you don't like the system, start playing 4 card majors. Of course, your question is quite valid when talking about overcalling...

I've taken the liberty of editing your question to focus it on what I think the essential question is.  If you disagree, you can roll it back.  Your solution to the problem as you see it should go into your answer.

@Tom: Please take this as friendly advice: You are not (yet) a good enough bridge player to invent new bidding systems. You will get into systemic trouble regularly, in ways that will prevent you from effectively improving. Please, if you are serious about improving your game, choose an established and well published system and play it. I like 4-card majors, don't get me wrong. But you will need to play a system that is in common use by the better players at your club in order to improve at a decent pace.

@Tom: Backgammon and novel writing are solitary pursuits; bridge is a partnership/team game, and bidding is the most pronounced partnership activity of the game. You cannot become significantly better at bridge bidding in isolation from your partner.

@PieterGeerkens: You have a point. I need to find/hire the "right" partner to use "my" system. And FWIW it's worth, I am one of the "better players at my club." That's not much, "big fish in a little pond, etc." and maybe that's why I've hit my ceiling. But I'm at least asking these questions, and the others aren't.

@TomAu: If your play is not yet at a level of comfort with standard openings, look around for a system similar to what you desire. Study it fervently. Play it fervently. Then learn how to tinker with it without destroying its harmony. Only when you have done this and win regularly at you local club (I mean win, not 2nd or 3rd or some placing. Win!) is there a possibility that your understanding of the game is sufficient to invent new systems.

P.S. Tinkering with the conventional add-ons is fine; that's part of the learning process, and every partner will have slightly different favourites. But the fundamental structure needs to retain its harmony or your partnership will be lost at sea. I have read that Meckwell's book runs to more than a thousand type-written pages.

When you open a five card major, you're not promising anything about the strength of your suit, you are simply telling your partner that you have five cards in that suit.
Remember that you do not win tricks only on strength, you win them on length as well.  Even if you open a raggedy five card suit, and end up playing in it after getting you're partner's support, you will win extra tricks because no one else will have trumps after you draw them. On the other hand, if you wind up in a 4-3 fit because partner raised your four card open, things can be difficult if one of your opponents also has 4 or more trump.
Also remember that just because you can't open AKQJ of a major doesn't mean you can never mention that suit.  Open a minor, and rebid your major.  That tells your partner that you have four of the major, and if he also has four then you've found your fit.
This can be hard to swallow, so I recommend building some example hands. Give yourself five lousy trump and an opening hand, give your partner three random trump and enough to raise you to game, and see how the hand plays out with random opponent hands.  Likewise, give yourself AKQJ of a major, your partner the same three random trump and game values, and get a feel for which hands are easier and which hands are harder to play.

If your partner must have Qxx to support you, won't you run into problems in the case where your partner won't support you if you open with AKQJx?

It seems like that's a bit circuitous and would miss some 6-2 fits, but I have a feeling we could go on like this forever.  Bottom line is, as long as your partner and you have the understanding of this system, try it out and see how it works in practice.  From the wording of your original question, it sounded as if you were unwilling to open xxxxx and would open AKQJ while playing five card majors, which would probably earn you a few dirty looks from the other side of the table.

Fair warning: if you are going to play in any ACBL (or similarly sponsored) events, confusing your opponents by hiding the true meanings of bids is not allowed.  If your partner could expect you to open with a four card major, then your opponents need to know that as well.  Make sure your convention card is filled out properly.

@Tom: You can't just talk about suits in isolation. You might have AKJx opposite Qxx, but you might have an x (in hand with AKJx) opposite xxx in a side suit, and you are pretty soon going to be forced to ruff and lose control on 4-2 trump break.

@Tom: Debating if one is worse than the other, without the full hand and contract is a pointless exercise. IMO, the most important part of any bidding is that you and partner are on the same page, no matter how inferior you think the system is. If partner expects 5+ cards when I open 1H, then even with AKQJ, I will not open 1H. You seem to be giving too much importance to high cards at the cost of distorting the shape of your hand. For example, consider the law of totat tricks :-) If your partner has no clue if you have 7 or 8 or 9 trumps, how can he apply the law properly when needed?

@aryabhatta: The point is, I believe in "figurative" truth over "literal truth." If I'm expecting five cards, I'm going to be more disappointed over 65432 (literal truth) than AKQJ (figurative truth). There is SO MUCH extra strength that it makes up for it.

@Tom: Again, you are looking at the suit in isolation. The fact you have so much strength in trumps would imply you have lesser strength in the side suits. For instance, if responder held Axx, xxx, Axxx, Axx he will bid game (say you opened 1H). And across Kx, xxxxx, Kx, KQxx this is a much better 4H (even with 11 points in your hand), than across xxx, AKQJ, xx, Kxxx. Of course one hand does not mean anything and what we are discussing is just opinions without any backing data (and that is really hard to obtain, as there are too many factors).

@Tom: In the first example, replace a small club with the jack of clubs. Even without the J of clubs, the first hand has much better chances of making 4H than the second. So I don't really see your point. As to discussing merits of your system, I am afraid that is too broad a question and not suitable for this forum.

Folks, please take this discussion to our chat room.  Thanks!

Acol and standard Goren are two well-known systems that use 4 card majors.
A main reason for the popularity of five card majors at the present time is the popularity of IMP contests. Five card majors is superior for slam bidding, which is emphasized in IMPS, while four cad majors is superior at matchpoints. However the team games are the most prestigious events, so almost everyone practices their team system at matchpoints.
Barry Crane was a strong advocate of 4 card majors, and used it very successfully in both team and pairs events for 30 years.
I have occasionally encountered hybrid systems that play 1H promising 5, but 1S only promising 4. Not recommended for casual partnerships, but retains the pre-emptive value of a 1S opening. Some partnerships play this unofficially in 3rd seat, hoping to not get into too much trouble because the hand is less likely to be a slam possibility.
P.S. By unofficialy above I don't mean undisclosed; I mean not completely discussed.

My "standard" is what I call a "four and a half" card major. Meaning "something extra" (like 5 HCP), if you open with four. xxxxx isn't even "four and a half," especially using my "23" standard (trump suit strength is sum of cards and HCP). xxxxx is only 5 (cards) out of 23 or less than one-FOURTH. KQxx is 9 (five for HCP, four for trumps). Qxxxx is 7 (5+2) and "barely" qualifies. Using "mapping algorithms."

Systems built around a weak no trump may fit your ideas. 1NT is "catchall" bid for balanced hands with opening values, that is 12-14 high card points. It has the advantage of shutting out opposing one level bids that hands with comparable strength might make.
One needs five of a suit to open it with 12-13. So hands of this strength and four card suits that are unsuitable for 1NT (e.g. a 4-4-4-1 distribution) are passed. One can, however, bid four card suits with 14 or more points, because the extra points compensate for the shortness of the suit.


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How was "komi" re-set in the past 50 years?
"komi" represents a number of points added to White's score at the end of a game of Go to compensate White for the fact that Black moves first. It is usually X.5 points (e.g. 7.5) to prevent a tie (points on the board are always integer values).
When i started learning the game forty years ago, komi had been "recently" raised from 4.5 to 5.5 points. The latter was probably too little, based on the fact that White won less than 50% of the games, and 4.5 was clearly too few.
Today, komi is set at 7.5 points, at least in games monitored by artificial intelligence, and all the computers reckon Black's chances at about 43%, at the start of the game, given an adjustment of that size.
The intermediate number which probably gives the most balanced game, appears to be 6.5, but that is a number that hasn't seen much use. Why is that, and how does the governing body (e.g. in Japan) set komi?

Have you read the article (https://senseis.xmp.net/?Komi) in Sensei’s Library? It gives quite a lot of information, but includes the statement “Today the standard komi in Japan is 6.5 points, introduced September 2002” (with a footnote “The Nihon Ki-in looked at about 15,000 tournament games from 1996 to 2001 and found that Black won 51.86 % of all games, a margin of nearly 4 percent over White. The directors voted to change to 6.5 komi and negotiate with the Kansai Ki-in and tournament sponsors”), which seems to disagree with your post.

@PJTraill: Years ago, I did a (curve-fitting) model that suggested that the value of each handicap stone, x, was (2/3)x**2+12x-6. That suggests that komi should be 6 2/3. that is, (2/3)+12-6., It's closer to 7 than 6, but closer to 6.5 than 7.5. I think part of the confusion arises from the fact that we first  identify the integer number, 7, then decide which way to go on the "halves," rather estimating x.5 directly.

Interesting. At first sight there is no difference between 6⅔ and 6½, but one could use a random event to give White either  6 or 7, though that makes jigo possible again, which some may not like. There is also the question whether fair jigo depends on the level of the players.

Afaik 7.5 is used for Chinese-style rules only. Japanese rules use 6.5. Area vs stone counting means that these two numbers are more similar than they seem.

Achieving the statistically perfect komi is interesting, but generally the difference between winning by 1/2 a point or winning by 1/3 a point does not change a game result.  We play in units of one stone (i.e., one point in area scoring), not 1/3 stone.
I don't have my copy of Go Players Almanac so I can't readily check dates -- I'll try to update later -- but my memory is that before komi was introduced, jigo (a draw) was possible; but in the Japanese pro tournaments at least, a jigo counted in White's favor (though not as strongly as a win), since it was known that Black had a significant advantage.
The Japanese introduced komi in the early 1900s, at a value of 2.  (Komi wasn't standardized for a time, so there are arguments for "introduced" at earlier dates -- see John Fairbairn's discussion at https://senseis.xmp.net/?HistoryOfKomi for examples -- but I think "early 1900s" is good enough for our purpose here.)  2 was pretty quickly seen as insufficient, and by 1934, komi had been raised to 4.5, where it stayed until the late 1950s / mid 1960s, when it was raised to 5.5.
When I began playing in the late 1980s, players indeed talked about how komi had "recently" been increased to 5.5.  (Well, when the game is 2000 years old, I guess 20 or 25 years seems "recent".)  When I began playing in amateur tournaments in the late 1990s, the komi could vary, depending on the rules in use.  AGA tournaments tended to follow the lead of the Nihon Ki-in, and used komi of 5.5; but there was strong material support from the Ing Foundation for tournaments, with the caveat that the tournament had to be played using Ing's idiosyncratic rules (intended to make the game more accessible to children and beginners) -- one of which was komi set at effectively 7.5.
Sometime in the early 2000s, Chinese rules switched komi from 5.5 to 7.5.  About the same time, the Nihon Ki-in switched from 5.5 to 6.5.  The AGA switched from 5.5 to 7.5 effective 2005.
There was discussion about whether the AGA should make it 6.5 instead, but changing from 5.5 to 6.5 would affect only the margin of victory in the vast majority of AGA games, not the game result.  (It's theoretically possible to have a game with weird seki formations where that may not be true, but it's very rare in actual play.)  So for example, if Black with 5.5 komi wins by 1.5 points, then with 6.5 komi Black still wins, but by 0.5 points.  If the reason for komi is to give both players an even chance, and 5.5 isn't enough, then 6.5 also isn't enough.  So the AGA changed to 7.5 not 6.5.
You may be thinking, "what about the games that Black with 5.5 komi would win by 0.5 points?  Doesn't shifting to 6.5 komi mean that instead White would win by 0.5?"  The answer is that, with 5.5 komi, Black would not win by 0.5 (except in the unusual seki situations that rarely happen).  This AGA article describes why:
https://www.usgo.org/news/2013/09/only-a-passing-matter/
(I read a better explanation, but I'm not finding it now.  Will update if I find it.)
A reasonable question is, why did the Japanese go with 6.5?  I don't have a definite answer, but I believe it's mostly because they have a slightly different ruleset.  In AGA rules, White must play last, which would penalize White under the Japanese rules.  In Chinese rules, you get a point for each intersection you surround, and you get a point for each stone you have alive at the end of the game -- there is no penalty for filling in a point of your own territory, as there is with Japanese "territory" scoring.  (There are also differences in how prisoners are handled -- they matter in Japanese "territory" scoring, but not so much in Chinese "area" scoring.)
The mathematician Elwyn Berlekamp proposed a system of bidding for komi.  For example, on the first move, Black could play and pay 20 points komi, or pass.  Next, White could play and pay 19 points, or pass.  Eventually, one of the players moves, and pays the komi to the other player.  Berlekamp arranged a short series of games between two pros living in America.  IIRC, the komi paid tended be surprisingly high -- 9 or 10 points.  (Thanks for reminding me of this -- I'll have to dig up the details.)

In Chinese rules (thus also AGA rules), every intersection is either a point for Black or a point for White, unless there remain unfilled dame. Thus the difference in score between two games without unfilled dame must be even. The only situation where Black can win by 0.5 if if there is an odd number of unfilled dame; either a dame was forgotten by both players, or there is a seki where each player has an eye and there is exactly one common liberty. In Japanese rules it is false that "every intersection is worth a point to White or Black"; dame are not played out; every score is possible.