cartesiandaemon

Uncountable infinities in Magic

2008-07-22T14:58:45Z

Update, someone else raised a very similar question on a message board. Indeed, using much the same technique I suggested, and I had found that link once before, but forgotten about it.

Page 1
Page 2

On page 1, someone asks a rules question that's a particularly apposite example of uncountable rules:

* The defender has generated an infinite number of small blocking creatures
* (Using a combo which involves mana burning down to a negative infinite amount of life!)
* The attacker has a spell which will win the game if any creature is unblocked
* The attacker uses two Nacatl War Pride, which when it attacks copies itself for each defending creature
* And turns both Nacatl War Pride into creatures that also are doubling season ("whenever a counter is put into play, instead put twice that many into play", although I think only the first one is relevant). Thus the second one puts an infinite number of creatures into play, to which an infinite number of doubling effects apply
* And asks "Will there be any unblocked attacking creatures?"

It's a particularly good example, because the cardinality is exactly relevant: the defender is exactly trying to make a bijection between blocking creatures and attacking creatures, and the attacker wants to know if there will always be an excess attacking creature.

On the second page someone proposes an explicit bijection (or rather, absence of a bijection).
I think this is functionally equivalent to my example.

Bike light

2008-07-22T00:55:43Z

I also think I fixed by bike light, which had been nagging me for ages. I don't know that I'm ahead on things I should have done, but it's always good to make steady progress.

Thanks to alex for the best suggestion, of using tinfoil. But in the end, when I actually examined the innards in detail, I discovered the problem was not, as I supposed, that the batteries slipped sideways off the contacts. Rather than they are held in place at one end by a conducting plate, and the other, by a conducting springy tongue, and when you cycle over a bump, the battery moves laterally, temporarily compressing the insufficiently springy tongue, and momentarily losing contact with the plate.

As I'd thought, it would actually have worked better if there was a physical on-off switch, rather than an electronic one. The electronic solution is conveniently extensible, if you want to let the same button cycle through flashing modes, etc. However, the physical on-off switch would have had the advantage of remembering its state after a temporary power outage. Which shouldn't matter, but made the whole system just a little more fragile.

Fortuitously, having identified the problem, it was eminently susceptible to the most trivial of solutions. I inserted a little bit of folded paper behind each tongue, and lo, it had much less give, and the battery has no tendency to move in it's slot. It stayed on even when I tapped it, but I didn't feel confident calling it fixed until I'd ridden it an it kept working, which it now has.

ETA: Although come to think of it, that can't be the whole story, because it didn't used to just turn off: when it was tapped, it would sometimes cycle between "bright" and "faded" and "off". Anyway, the same solution seems to have cleared that up too.

Bad pun about uncountable infinity

2008-07-21T12:54:55Z

Q. OK, so what about uncountable infinities in magic?

What about it?

Q. Does it make infinite combos less degenerate? Does it even make sense?

Well, you may need some tweaks to the rules, because a lot of them are predicted on the idea of things happening one after another. But suppose, eg. you just go round an infinite loop ω₁ times.

Q. Well, that was boring.

OK, ok. Suppose you can't do that. Try this. Doubling season says "Whenever you put a counter into play, instead put two of them into play." Suppose you get an infinite number of doubling seasons into play (eg. jumping through a few hoops and an infinite amount of mana to put a copy of it into play an infinite number of times).

Then put a token into play. I think the infinite number of doubling seasons produces an uncountable number of new counters. (Either an uncountable number of new creatures, or almost cooler, one creature with enough +1/+1 counters on that it has power and toughness ℵ₁/ℵ₁ :))

Think about this either as 2^ω counters, or by counting the counters individually: number the first counter 0, and the counter produced by the first double 0.1 and the counters produced by the second double 0.01 and 0.11, and so on.

Q. Right... Is that rigorous?

I think so... You need either to be comfortable with an infinite stack, (the order things come off it might be a problem?) or to have events happen "in parallel" when it's obvious what's going to happen.

Q. Is it interesting?

I don't know. Maybe. Probably not. It does mean that if you gain uncountably infinite life, your opponent has to jump through some extra hoop, it's not enough just to have an infinite combo. You don't just need a way of gaining infinite cards, infinite mana, and infinite damage, you need an infinite number of copies of some doubling effect as well.

Q. So, specific cards?

Aforementioned doubling season, together with a card that makes it a creature (opalescence) -- or anything else if you can copy it -- and a reusable card that puts a copy of target creature or artifact (kiki-jiki "tap to put into aply a copy of target creature" plus a way to untap it, mirrorweave "all creatures become a copy of target creature" plus infinite creature tokens, spitting image "pay mana and discard a land to repeatedly put a copy of target creature into play").

Isn't there some red enchantment which does twice as much damage?

Q. I ask the questions here.

That's not a question.

Q. I ask the questions here. Right, asshole?

Guided busway

2008-07-21T11:50:43Z

The guided busway has reached the King's Hedge's Road entrance to the science park. You can see the concrete tracks stretching away on the west, and slowly creeping east along the abandoned railway behind the houses on King's Hedges Road.

The foot/cycle path entrance closed "WARNING: GIANT DIGGERS", but is open again, diverted to come out on the College road, and thence cut through the building site to the pavement on King's Hedges Road. But I normally just cycle along the road.

If you think waiting at a pseuo-level crossing for a guided bus to go past takes amusingly long, just try waiting for a guided busway to go past :)

On the other hand, approaching my building from the non-cul-de-sac end of the cul-de-sac is a revelation: the two carpark entrances being designated "entrance" and "exit" make complete sense when the entrance is the first one you reach, and the exit the second (since the carpark is at the far end and people instinctively like turning through the first turning available). It only seems arbitrary if you're used to cycling from the cycleway into the cul-de-sac end of the cul-de-sac, when the idea that you should go up to the second entrance and back again is risible :)

Weekend films

2008-07-20T22:25:58Z

I successfully went swimming, even though it looked a bit overcast beforehand, and it turned out nice when i was there.

I took Mobbsy, Sonic and Mds to see Forbidden Kingdom. It was very pretty, with lovely characters. Although I think Mobbsy was right that to say it had very little plot -- I though tthe mythology was lovely, but nothing much happened during the film. Basically, if you liked the trailer, you should see it -- it was like you'd expect, but better, very well done (and not as silly as you might fear, seeing Jackie Chan and Jet Li wrangling). I don't know enough Chinese mythology -- a variety of things come from there, and I don't know if that would make it interesting or painful if you do know, but to me it was a very pleasing blend of Western-style and Chinese-style things.

I've been cleaning up and packing, with mixed success, but a lot more productive that I normally manage :)

Fishpi, his Rachel, and I went to see Wall-E. I didn't like it as much as everyone else did (not as much as the Incredibles), but I thought it was very good (one of the best other Pixar films). It's marvellously animated, Wall-E and Eve are tremendously expressive. It's sweet. And never annoying. And generally enjoyable. But it never made me laugh much, and there wasn't that much to the plot. I guess time will tell.

Sample Format: Infinite Magic II

2008-07-20T14:16:12Z

In Magic:TG, you have a deck of 60+ cards, containing at most four of any card (except basic lands, which produce mana). This is sort of an historical aberration. When Richard Garfield first invented Magic:TG he was working out what it should be from scratch, and he imagined people trading one-on-one for cards they wanted, thus part of the game is acquiring the cards you need, and part of the game is building a deck out of them, and part of the game is playing a game with those decks.

However, it turned out that the internet came along and made it very easy to trade cards with people all over the world. Thus a casual player might be limited by the cards they had to hand, but someone willing to pay, or a professional player who had to pay to get the best deck, could buy the best cards. So, instead a rule was instituted you could only have 4 of any card, and magic cards are designed with this assumption.

Most cards would be fine if you had a deck full of them. If you could start with a deck containing 50% a simple creature, 50% a simple land, and see how it did, and refine it from there, it would fulfil deck design better. (You could also do away with the 60 card deck rule, as there's no advantage by starting with only the cards you want.) However, a few cards are fine in small numbers, but completely break in large numbers.

Eg. if you relax the "four card" limit in normal magic, one very good deck would be sixty Rocket Powered Turbo Slug. (This is a card from the humorous Unhinged set. But there are similar problems in tournament-legal magic cards.) This card is a creature you can attack with and pay for the turn after. Normally this is just a cute trick and you can learn about resource management and investment. However, if you start the game with a hand of seven of these, they can all be put into play, attack, and deal 3 damage on the first turn, and your opponent is dead.

What happens if you play infinite magic, with an infinite deck?

An infinite deck

I think you actually need to have an uncountably infinite deck, since it's impossible to have countably infinitely many cards, and choose one of them with equal probability. This doesn't matter if you express the composition as percentages, so I'm going to ignore it for now, but I think is broken.

What decks are good in this format? Well, much the same as the decks that are good in normal magic without the four card limit, except that cards which let you pay life to do things are even more broken. For instance, a good deck would be:

50% black lotus (An infamously overpowered card from the first magic set, which gives you one-off three mana. If the game was guaranteed to go on several turns, this would be fair, as most cards give you mana every turn, and this gives you a short burst at the expense of having mana in the future. However, if you can use lots of mana to win on the first turn, as we're about to do, this is irrelevant.)
50% necropotence (An infamously overpowered, blah blah blah. This lets you pay life to draw cards. Which sounds ok -- who wants to lose life? However, if you pay nearly all your life, you can draw lots of cards, and the perfect combination of cards can be a win on the first turn, blah, blah, only more broken if you have infinite life, blah blah )
1 fireball (This turns any amount of mana into damage to your opponenet, and is a good card, but is not infamously overpowered.)

You could use almost any other card instead of fireball; anything that lets you turn infinite mana into infinite damage.

The strategy: get an opening hand with at least one black lotus and at least one necropotence. Play the lotus, kill it, getting three mana, use the mana to play the necropotence, pay infinity life (leaving infinity left) to draw infinitely many cards. Play an infinite number of black lotuses to get an infinite amount of mana, and kill your opponent with one infinitely large fireball.

Conclusion

That's not very interesting. Obviously you can tweak it around. For instance, there are cards that let the opponent counter any spell without paying mana (again, the idea is they pay for it later), so one of those can ruin your whole deck. You need some answers to that, probably more of the same card.

If you play Yawgamoth's Bargain instead of Necropotence, you can draw those cards on the first turn, rather than at the beginning of the second turn. That's a bit better, but there's a 5% chance of drawing 6+ bargains, and you need two black lotuses to pay to play one.

There might be other combinations which let you win even earlier. (There was a fun combo in normal magic which let you get magic before your first turn, and then you can bootstrap to one of these infinite kills.)

I'd be interested to see the metagame (which decks are good, and which beat which other ones, like rock-paper-scissors), but it doesn't look like it'll be very different to playing finite decks with no four card limit.

Next time

What if you play with the four card limit on infinite decks? (Or maybe ban cards which are very broken in this format?)

cartesiandaemon @ 2008-07-19T01:32:00

2008-07-19T00:39:50Z

Q. What do you get if you cross:

* a musical
* a videoblog style format
* a league of supervillains
* writing by Joss Whedon
* Nathon Fillon (aka Malcolm Reynolds aka Caleb) playing a smug larger-than-life jerk, Captain Hammer

A. Doctor Horrible's Sing-along Blog

I didn't like it as much as most other people I know, but it's very much worth watching! I particularly like: the balanced characterisation of the "hero" and "villain"; and the most epic moments of solo singing.

Sample format: Infinite magic 1 [1]

2008-07-18T17:48:11Z

Is there a simple example which would make use of the rules? How about infinite magic: each player starts with infinite life?

It would be a place where people who like making infinite combos could play, because those would always be the best way to win, and never beaten to the punch by someone attacking for 20 or 30 points of damage :) I doubt that would be a fun format forever, but it would be very nice to try :)

The normal methods of kills would be:

* doing infinite damage
* special cases, like "you win the game" cards, and giving ten poison counters
* running out of cards (that's a loss)
* doing damage each turn for an infinite number of turns[2]

Of course, there'd be some other differences:

* Creatures and cards which attack for a finite amount of damage don't really matter
* Cards with drawbacks like "opponent may pay X life to counter this card" are useless
* Gaining life is really useless (Unless you find some way to gain an uncountable amount of life, one lot of infinite damage can take out all of your life however much you gain.)
* Any card that has "pay life" as a cost gets a lot better. (Fortunately, most of these don't matter.)
* Any card which lets you pay life do something repeatedly is insanely good. These are pretty broken even if you only have a finite amount of life, so they're not that many.
* You'd really have to play cards that always let you put cards back into your deck after you've used them, else you'd lose when you run out of cards.
* Spells which do X damage to you and your opponent are as good as ones that do damage only to your opponent
* If you gain an infinite amount of mana, you don't burst if you've nowhere to put it (Or maybe you should?)

On the other hand, some deck designs from finite magic could win in exactly the same way:

* By doing infinite damage (not just "much more than 20" damage, but actually infinite)
* By dealing ten poison counters.
* By running the opponent out of cards.
* By making a "lock" where the opponent can't act. You still need some way to win but either (a) opponent running out of cards (b) eventually finding your own cards that do infinite damage or (c) doing one or two damage a turn work fine
* By Phage the Untouchable "When this does combat damage to your opponent, they lose the game", * Or by Barren Glory "When you control no other... you win the game" etc

There would still be a race to see which of these got there first, just like normal magic.

[1] We're not going to run out of instance numbers soon :)
[2] I think if you both deal damage every turn it's a draw, even if you do more damage, because both limits are infinity. Does that sound right?

An infinite amount of pain

2008-07-18T16:58:54Z

Back to the example of Magic:TG. People have often pondered the most appropriate infinity rules. There's a few things it would be nice to do:

* If you gain an infinite amount of life, you can't be killed by any finite amount of damage
* If you gain an infinite amount of life, you can be killed by an infinite amount of damage
* If you gain nought life an infinite number of times, it doesn't make any difference

In jargon, "gain an infinite amount of life" refers to a cycle of moves which gains life and you could repeat without stopping. (Eg. "When X happens, gain 1 life", "When you gain life, do Y" and "When Y happens, do X".)

The normal rules approximate these with finite numbers pretty well. If you have an infinite loop, you have to choose how many times to go round it, but then to do something else. So in fact, if you gain infinite life, you think of the biggest number you can, and gain that much, which can be exceeded if someone does infinite damage, but hopefully not by any finite combination[1]. If you have a loop that doesn't have any useful effect, you can't force a draw, you have to stop[2].

First infinity rule

But it would be nice to actually have rules that could properly get to grips with infinity. The obvious interpretation is that if there's an infinite loop, the result is what you get when you take the limit of any number that changes.

For instance, if I have a loop where everything stays the same, but my life goes 20, 22, 24..., then after the sequence is complete, I obviously have ω (infinity) life.

But what happens if player 1 can play "Target creature gains flying" and player 2 can play "target creature loses flying" for free? If player 1 gives his creature flying, and player 2 takes it away, and they repeat, what's the result?

The traditional rules give the second player the last word. The analogue would seem to be that the second player can choose any subsequence to take the limit of. So if player 1's life goes 20,22,24.. then whatever player 2 chooses, the limit is omega. But if flying goes 0,1,0,1.. then player two can choose the sequence 0,0,0,0...

What if player 1's life goes 20,10,30,10,50,10,90,10,170,10...? Then if player 1 could stop at any point, he could have any large amount of life. But if goes on to infinity, player 2 could choose 10. Maybe player 2 gets the choice only if they were able to stop the loop, otherwise player 1 chooses. Is there a better description of that?

I think you can always find some convergent subsequence (ie. your favourite infinities are compact. Is this right?) Surreal numbers may provide any limit, but I don't think it's a useful limit if the sequence isn't monotonic.

Second infinity rule

However, I believe there's a more fundamental problem. As soon as you introduce an infinity rule, you have to ask, "if I gain an infinite amount of life, can someone kill me by doing an infinite amount of damage". The interpretation we have says "no", since the limit of ω, ω-1, ω-2, ... is, if anything, a surreal number infinitesimally below infinity, so still larger than any finite number, and emphatically non-zero.

This is also the answer in the only official magic rules touching on the situation, "No. Infinity minus infinity is still infinity. Stupid infinity" from this article.[3]

However, I believe our interpretation is actually flawed. It's fine for a countably infinite amount of life, but if you have a countably infinite amount of life, and someone does an uncountably infinite amount of damage, you would still be alive! Your life would be infinitesimal just smaller than infinity smaller than any infinitesimal you can reach by subtracting one from infinity any countable number of times, but that's still larger than any finite number.

But the uncountable amount of damage is greater than countably infinite amount of life by any metric, and should totally kill you. (Uncountable infinities require some slight reinterpreting of the rules that I'll come back to later, but at least conceptually we have a problem.)

My answer is that instead of being ordinals measuring "how many times", life totals should be cardinals measuring "how many things". When you play magic, you already normally represent your life total with a number of counter, the only difference here is to make that official. Thus when you deal damage, you have to specify which of the counters you want to take away. This normally makes no difference at all, except that if you're taking away infinitely many counters from an infinite number of them, you can either choose to take away each of them in turn, or to take away each other one, thus leaving an infinite number.

This also has the advantage that it matches the finite rules. If I gain an "infinite" amount of life with the finite rules, my opponent can still kill my by doing an "infinite" amount of damage, as we both chose large finite numbers, and he can choose a bigger one. I'm not allowed to retrospectively have chosen a bigger number when I went "infinite". However, if I gain an infinite amount of life, and then spend an infinite amount of life to do something, I can easily choose the first number much bigger than the second, and thus both spend and be left with an effectively "infinite" amount of life.

Turn order

The same "take a limit" logic ought to work fine for turn order without any changes to the rules at all -- I don't think the rules ever specified only a finite number of turns. If my opponent has an infinite amount of life, but can't do anything, and I deal her one damage a turn, and neither can draw any cards or anything, what happens? Obviously, after an infinite number of turns I kill her.

Conclusions

There's definitely more to do. How to deal with infinite decks. How to deal with uncountable numbers. Do we deal with "choose a number" ok? And we know there are edge cases unspecified.

However, I get the feeling the infinity rules ought to work as they are, having made only very very minor changes to the existing rule set, and let the players play on to a defined result if one of them manufactures or is trapped in an infinite loop.

Can anyone think of any broken edge cases?

Footnotes

[1] I recall a discussion somewhere about a magic deck which could generate the largest number without being able to go infinite. Does anyone remember that?

[2] Indeed, obviously, there have been many abilities you can use at will where it doesn't make any difference if you used them a second time, you can't just say "I activate this ability" forever. There are some subtleties, like if both players are counteracting each other's moves, the second player gets to play last. If the loop is mandatory, then the game is a draw, though I don't know if that's necessary.

[3] The card in question comes from the "Unhinged" set, which is a parody of normal magic, and does all sorts of crazy shit just to see if it's possible, such as a card that gains infinite mana. The official rules team, designed to stop loop-holes in sane sets, threw up their hand at a set that included this sort of crazy zany shit, and handed the official rules off to someone who used to script-write for Roseanne[4]. Thus the "unhinged" ruleset allows for infinity, though even he drew the line at "choosing a number" including ω

[4] Mark Rosewater, working at Wizards of the Coast, and author of many humorous articles.

To infinity and beyond and beyond

2008-07-17T19:04:38Z

Infinity rules in go

Any go rule is generally either universal, or divides rulesets into three groups[1] which play the same in practice: one which ignores it; one which uses a big ban-hammer to make it deterministic, but proponents of the third interpretation think is anaesthetic; and one which defines it terms you can only understand if you already have a good go player's intuition about the game.

1. The "ko" rule This says that you cannot immediately repeat a board position. If black places a stone at a, capturing a white stone at b, white cannot immediately place a stone at b, if this captures the black stone right back again.

This is a venerable part of the game of go, and any alternative interpretation has to be equivalent to it.

Although it would be interesting to see if any of the alternatives would make sense. How would go change if the "ko" rule said that the first player could not repeat the ko position? That would favour a defender rather than an attacker. Or the chess rule allowing a player to claim a draw (or equivalently, letting the ko repeat forever). That would mean the player in the stronger position would be unable to play in the ko.

2. The "superko" rule

There are a few specific situations that are like a ko, except a position can repeat after a larger number of moves. These are really rare, so the rules don't make much difference to the game, but there are a few different ways of treating them: Basic rules ignore this. Traditional rules said the game was cancelled if the most common sort of multiple ko came up. Western rulesets tend to use a superko rule: forbid the repeating of any board position[2]. Some use a complicated set of special cases to capture a good player's intuition about what ought to happen.

I'm not sure if everyone's happy with the superko rule? It seemed simple to me, and obviously does the job of ensuring a game produces a result, and is an obvious generalisation of the ko rule. I get the impression some rulesets aren't satisfied with it -- I don't know if it's just hard to enforce without a computer, or if it fails to capture some subtleties more complicated rules strive to encapsulate.

For elegance, it would seem good if a more complicated ko situation had the same effect as a ko. This would basically come down to the superko rule, I think? However, traditional rules made a triple ko a nullified game if neither player wanted to give it up. I don't know fits better what "ought" to happen. A normal ko is resolved by the players making threats elsewhere of greater value than the ko: when these resolve, one player wins the ko because the other player must respond to a greater threat elsewhere. Is this applicable to multiple ko? [Insert: talk about kos as infinitely repeatable threats.]

It would be interesting to see if any infinity rule did simplify any answers. (Everyone's rules agree in practically every case, but there's a trend in go to look for simpler rules that work the same as the old rules for people who know how to play, but are cleaner to describe and but let you play everything out until the result is completely incontrovertible if you liked.)

Might it also apply to seki? (Groups of adjacent stones where neither can capture the other) Traditional rules differ very slightly in how to score this. The result is determined -- either player could play in the group if they wanted, but would lose it if they did, so the result is up to the players. But if the seki involved a ko, would it make it more obvious what the result was?

However, I don't know enough go to yet see if an infinity rule actually would be at all applicable.

[1] "Every Arabic word has a basic meaning, a second meaning which is the exact opposite of the first, a third meaning which refers to either a camel or horse, and a fourth meaning that is so obscene that you'll have to look it up for yourself."
[2] There's a subtlety whether this takes into account whose turn it is to play or not.

To infinity and beyond

2008-07-17T17:49:40Z

I've spent all day thinking about this now, and not got that far, because go is a game of unsurpassed depth and subtlety, and my head hurts :) In any game where each player acts in turn, and it's possible for a sequence of moves to repeat, the rules have to face the question of what to do if the players get stuck in a loop.

In Magic:TG there are complicated but well-defined set of rules which invite you to compare them to either conceptualised rules "what would happen if you could repeat the sequence infinitely many times" or "what happens if it went on and on, but one player had to break out of the loop eventually". (These are often discussed, eg. on toothywiki.)

In magic the situation is complicated by it sometimes mattering how many times you went round the loop (eg. if you can repeatedly put a new creature into play, can you end up with infinite creatures? Or an arbitrarily large amount?) But it occurred to me, the rules are essentially doing the same job as the ko rule in go or the three-repeats-or-fifty-reversible-moves-is-a-draw rules in chess.

In chess or go, going round the loop multiple times is the same as going round it once, so if you ever break out of the loop, it's the same as doing so at once.

How do the actual rules in place compare to what would happen if you could endlessly repeat?

Chess

In an idealised non-timed game of chess, the three-repeats and the fifty-reversible-moves rules are actually equivalent. If you repeat endlessly, you will eventually make fifty reversible moves. If you make make an unlimited number of non-reversible moves, you must eventually (a very, very long time) repeat a board position.

In fact, to prevent a losing or stubborn player dragging a game out, the limits are set at three repeats or fifty moves, which are supposed to be long enough that they should normally only occur if the game has stalled[1]. What would happen in an endlessly repeating cycle of moves if the rules didn't address it?

If one player has an advantage, he ought to be able to force a win by doing something else, so it's in the other player's interest to repeat the loop, and the first player will eventually do something else, unless he has no choice.

If neither player can gain by ending the loop -- eg. if white is a queen down and repeatedly threatened black's king or queen, then white only has a chance by prolonging the loop, and if black cannot move the threatened piece without it being threatened again, (and would be losing without his queen) then black cannot break out either, without losing -- then the loop would go on forever.

In this case, like analysing a chaotic system, we've found a final resting place that isn't static (like a checkmate), but is a repeating sequence. The rules consider starting this sequence to be a draw (since it should only happen if both players are forced to choose it). This is pretty much the only reasonable interpretation in most situations (one exception below).

One exception

What if the two kings are in separate situations on far sides of the board, unable to interfere with each other, and white can repeatedly check black's king, and black can force a checkmate of white's king?

Under the rules, if white moves first, white will force an endless loop, and if black moves first, black will win.

But what would happen if white moved first, but actually considered an infinite number of moves, and then an ω'th move, and an ω+1'th move, etc? One side of the board would end up in a superposition, with the black king and the checking piece in a fuzzy circle.

And whoever moved first afterwards on the other side of the board (if you assume they can't move the fuzzy pieces), black would win, since the white king can't escape.

I think this is equivalent to letting the two independent situations evolve simultaneously.

It doesn't give the same result as in chess, but I think it's equally reasonable -- white has obviously managed to force some equivalence, but black is in a stronger position, so you could justify calling the situation either a draw, or a win for black.

I don't think this sheds any light on chess, but I wonder if it does on go. That had better be a separate post.

[1] They can happen by accident, but the idea is that three repeats normally only occur if a player is committed to repeating forever, and fifty moves should be enough for any sort of checkmate. Apparently some awkward combinations of pieces -- three bishops versus a bishop and the like -- can take more than fifty moves even with perfect play. But the idea is that in any normal situation, more than fifty moves is just taking the piss and not getting anywhere, especially if someone's time is running out.

[2] Ending a game when a king is captured is equivalent to the current rules, if you replace "must move out of check" with "must remind player, when not moving out of check, that he is doing so."

Robhu: in response to atreic's post

2008-07-17T10:56:08Z

Rob, I assumed you wouldn't mind me cross-posting this. It's a continuation of what I asked when you realised you wanted to be Christian, inspired by atreic's post. I didn't want to assume what you did think, but wanted to ask (and sorry for putting you on the spot).

I said to rob in his 'I have become Christian post' that I thought it was probably good that he is Christian, but that if that was based on conviction and observation, then I didn't see it need change his mind on, eg. when having meaningful non-marital sex is ok, or what has a soul. I know some people have very good reasons for some or all of those things, but they're not a necessary part of Christianity. Indeed, I should probably ask him directly *crossposts*.

ETA: some or all of what I thought rob thought was garbled and incorrect, I apologise for not checking first!

I get the impression you have changed your mind on several similar issues, for instance the post-fertilisation contraception atreic linked to. Do you think that's right? Obviously believing in God could make you consider the question more closely, but I get the impression you accepted things as part and parcel of believing in God maybe you didn't need to. Do you know what I mean?

Recently this week:

2008-07-16T01:35:22Z

My bike returned safe from university cycles, tuned up and raring to go.

Work continuing interesting and productive.

Failed to go to Magic prerelease event, but played magic at Alex's, which was fun: most credit goes to Alex for building the decks played.

Played bridge at Ralph's, and failed to bid two consecutive heart slams, but I think correctly.

Watched QI again, twice, which really is very funny, and educational in spirit, so I apologise for the criticisms when I first saw it being over-harsh.

Have downloaded most of Reboot ever from youtube, at low resolution, but enough to catch up and find out what happens after it fell from the air in UK terrestrial TV.

Watched ~~Hitch~~ Hancock, where Will Smith plays, rather well, someone failing to be a superhero. It wasn't one of my favourites, but it was good. The trailer sets it up as a "Hancock is a superhero, but rather than being a larger-than-life American Hero, is a drunken screw-up jerk" joke, which it is, but it doesn't just stop there. It's funny, but I was pleasantly surprised that it actually has interesting story. Many superhero stories oscillate randomly between "all-American hero" and "tortured angst"; Hancock actually shows someone kind of screwed up you can really believe.

In other news, "mare", "nightmare" and "mare (as in sea)" are all unrelated, (and so fair game for puns). "Phallus" and "fellatio" aren't either. Well, my friendslist probably knows all that, but spread the knowledge little by little.

Meta-quiz Meme

2008-07-13T00:26:22Z

Related to several places, most recently stolen from God Plays Dice here, from a book. The point being not what people know, but how good they are at knowing what they know.

"For each of the following ten questions, give a range that you are 90 percent confident contains the correct answer. Your goal is to get exactly nine of these right[1]. Yes, I know that sounds weird! But the point is that if you get all ten right, you're proabably underestimating your own abilities to predict things. If you get eight or less, you're probably overestimating them."

Assign a range to each question in a comment. Look up the answers and see how many you got right. Post it if you like. GodPlaysDice said to repost it if you liked, and to email him the answers (izzycat AT gmail DOT com) if you like; I assume he wishes to informally gauge something.

Here are the questions:
1. How old was Martin Luther King, Jr. at death?
2. What is the length of the Nile River?
3. How many countries belong to OPEC?
4. How many books are there in the Old Testament?
5. What is the diameter of the moon?
6. What is the weight of an empty Boeing 747-400?
7. In what year was Mozart born?
8. What is the gestation period of an Asian elephant?
9. What is the air distance from London to Tokyo?
10. What is the depth of the deepest known point in the ocean?

Although what interested me was that it simply meant you could have a quiz where people who don't know much about it (or who know too much about it) can play too. I'm curious to see how big the ranges are -- mine are embarrassingly wide, generally between a factor of two to a factor of ten, though of course, I know several much more precisely now.

[1] It would be more precise to say "and not know which one you got wrong". The idea being you should be pretty certain about all of them, not guess "0-1000,000" on nine and "-315.17" on the last one :)

Doctor who series finale

2008-07-12T17:26:44Z

I really enjoyed the finale. I didn't think it was as good as the first half of the two-parter, but still very good. (As someone said, it can be easier to set up something insanely dramatic than to live up to it.) In fact, I looking back, I see every episode of season #4 was good (I think the weakest was the first, although in retrospect, I'm not sure, it was a bit flat, but it also had quite a few classic moments), as was every episode of season #1.

Great things

* German Daleks!
* When Jack winks, and when he tumbles exhausted out of the furnace. They make really good use of his invulnerability -- it's obviously never easy for him to exploit it, but he's just so gosh-darn cool he can make use of it when he needs to.
* "Who invented that? Well, someone called Osterhagen, obviously, but..."
* When Micky and Jack meet. "Captain Cheesecake!"
* When Jackie teleports to safety. Yes, I'd forgotten that could be possible.
* Prophecies!
* "Donna was somehow drawn to the doctor", ah, cool, it all made sense at the time, but it makes more sense now! That is how you set up a revelation!
* Jack "What I'm thinking..."
* The daleks exploding to music, and the ending music
* Jack: "Three doctors? I can't tell you what I'm thinking right now!"
* "He has the biggest family on earth! Awww."
* Micky so happy at the end! Awww!
* Rose and Doctor 2! Awwww!
* Apparently the doctor has gone through a complete emotional development arc over four seasons. I'm not saying it was perfect (it may have been, I don't know), but I love the suggestion that he did. It's often the sort of thing that books hint did happen later, but was not evident from the episodes at the time.
* Donna leaving was sad.

Nitpicks -- general

* I won't even bother pointing out flaws in the plot, but mention a few specific things.
* Why did Davros leave Donna free?
* I'm glad there was no reset button to put everything back how it was. However, thinking there might be took a small piece of the excitement out of everything that happened :(

Nitpicks -- Osterhagen key

Hundreds of nuclear weapons buried round the world, waiting to blow the earth up... Yeah, right. Interesting idea. But there is a good guide on the subject of blowing up the earth, readily available, that I've linked to before.

The headline is using something like 25 teratonnes of antimatter. And I don't mean antimatter having a yield equivalent to 25 teratonnes of TNT. I mean twenty-five trillion tonnes of antimatter. (And you do not want to bury antimatter in your crust. And you're not going to achieve this with conventional nuclear weapons, either.)

More than one German woman would have noticed.

Of course, presumably this is a Doctor-Strangelove-style[1] mutually-assured-destruction device. But that means it's not actually guaranteed to make the planet useless to the Daleks.

[1] I said there were spoilers :)

Nipicks -- Earth flying though space

Like the last episode, all the characters working together was really enjoyable and uplifting. And what they did was cool, but fairly stupid. Let's see if I can pick the top ten reasons wouldn't work:

1. They lassood the earth with a beam of energy. Pity the poor buggers under where it drew tight, eh?
2. The stars were totally different, so they were obviously in a significantly different place in the galaxy (if in the same galaxy at all!), so it would N years to get home, where N is somewhere between 4 and millions, even if they accelerated to close to the speed of light.
3. They must have had to accelerate at much less than G or everything (and I mean everything) would have fallen down.
4. Funnily enough, not being in orbit, doesn't make your atmosphere fly off. But flying through space at a significant fraction of the speed of light probably does. Not to mention the ionisation, radiation, mini-asteroids, etc.
5. The moon was conveniently there waiting for them when they came back?
6. Oh yes, that reminds me. You can probably survive without any tides when you're in the Cascade, but boy howdy, half the earth is probably going to get swamped if you accelerate too fast, not to mention if you go near something.
7. And people didn't notice. Is everyone now finally going to admit that aliens exist? Or not?[1]
8. What about the other planets? Was someone rescuing them?
9. OK, it seems I don't quite have ten. Still, eight reasons why something is impossible it pretty good going.

[1] Like the grammar issue, this is unfair of me. The show (correctly) doesn't pretend to make this serious or consistent. But it certainly fits on a list of "why flying the earth about like that would be a stupid idea" :)

Doctor #2

I flip-flopped several times on this. I thought implying the doctor could choose to regenerate into himself (even if, as I assume, he was joking) was a cop-out. And it also missed a marvellous opportunity to increase the sense of loss for the doctor by having his companions have to deal with a new regeneration instead (even if that's reverted at the end of the episode).

But I really, really liked how it turned out. I thought the contrast between the two (and three) doctors was really good.

I had a nagging feeling they could have shuffled round all the plot elements and come up with something that used all the best bits, but not the cop-outs, and hung together perfectly, but wasn't up to actually constructing something like that, it may be what they had was nearly the best.

Themes

The Doctor vs Davros as a villain. The Doctor's emotional growth since the time war. The doctor using people as weapons. All handled more or less well, but generally interesting ideas.

German Daleks

Daleks speaking German is good, as they're both (a) funny (b) ominous and (c) advance the plot. And (d) I think everyone understands the intent of what was shown.

I'm not sure why it seems incongruous. I think that we expect Daleks to say "exterminate", but this scene implies they naturally use a local language to communicate, and those aren't consistent. (Which isn't a flaw, I just think it's interesting.) After all, it would make sense if they always used a local language. But we probably also have the idea they say "exterminate" because they're not very articulate, so it's strange to imagine they can express themselves well in a variety of languages. (In fact, the question of language is unanswerable, fun science fiction can't always be bound by it. But that doesn't stop me thinking.)

In fact, it's not clear why they would say "exterminate" in anything other than their native language -- after all, they're more talking to themselves than their victim at that point, aren't they?

Magic: the Gathering: Prerelease (Alternative title: eaters of lives)

2008-07-07T01:40:05Z

There are a few geeky things that float around amongst my friends that I've enjoyed hearing about, but avoided getting sucked into: professional mathematics, World of Warcraft, nethack, database/network administrating, starcraft, chess, etc.

Either because they seem seductive, and would EAT MY LIFE (TM), or because they seem TOO SAD EVEN FOR ME (TM) :)

One of these is Magic: the Gathering, one of the original trading card games. I'm embarrassed there's something really fascinating about it, but there really are many fascinating ideas: a large ruleset you can enjoy hacking if you enjoy that sort of thing, both within and without the intent; pretty pictures cleverly welded to rules design; low effort open-ended socialising.

The point being, there is apparently a prerelease of the next set in Cambridge. So I'm inclined to start doing magic things for a bit, until I get it out of my system. And at a prerelease, obviously (I say with a carefully straight face), is the perfect environment to join, as no-one else knows what the new cards do either :) And with a few cards, I can go along to Alex's sometimes.

Does anyone know Inner Sanctum Collectables? Do I know anyone intending on going?

Random Media

2008-07-05T13:51:58Z

Transformers Movie

This really was a good movie. Funny, and exciting. It does a wonderful job of making transforming robots seem both natural and awe inspiring. The only complaint is that the big robot battles near the end are a bit repetitive when it's hard to see which robot is which.

We should totally tell our kids the tooth fairy is a giant alien robot! :)

"...until we were betrayed by the 'Decepticons'". I think I said this the first time I saw it. Seriously dudes, you didn't see that coming? For the SECOND time?

Erfworld: The Battle for Gobwin Knob

Erfworld is a webcomic by Rob Balder of PartiallyClips and hosted by Rich Burlew along with his Order of the Stick[1]. (Both of those are awesome.) When I first saw it, I was not drawn in: it's inevitably compared to Order of the Stick, but is harder to get into, because it tries to build up an ongoing whole, but this means each strip by itself doesn't necessarily pack as convenient a nugget of humour, while it tries to squeeze plot in. And some people are put off by how gratuitously sexy some of the characters are.

[1] And illustrated by Jamie Noguchi, who I haven't heard of, but draws beautifully. So setting it apart from most other webcomics I like, which start off as crude stick-figures until the author learns to draw better ones[2].

[2] This is gentle irony in the case of most comics, but high praise in the case of OotS, which has really good stick-figures. Really. Really.

However, I just reread through the story, and it is very good. An awful lot of the humour is worked in sideways over a few strips, so when you first see it, it's just odd, and then later on it suddenly dawns on you what atrocious pun you've been reading past, and totally crack up. And the characterisation seemed patchy seeing each strip individually, but when you see the whole, it has half a dozen characters plus who I really, really like. (My favourite is Vinnie Doombats! But also Parson, Lord Stanley, Wanda, Jillian, Bogroll, Sizemore, Misty, Charlie.)

A few other humorous images and links

Simple diagram of the anatomy of an american news story, all the way from "unsubstantiated internet rumour" to "bad movie made for tv about it"
Cat macro
Other movies which can be made cooler by digitally insertinglightsabres!
Kitty pic

Doctor Who: the Stolen Earth

2008-07-04T22:34:29Z

General comments

I thought it was really, really good. It did a good job of tying together everything, but mostly not being stupid about it: every time some old friend popped up, it didn't feel obligatory, but rather awesome. I'm going to mention several times how much I like Harriet Jones still :)

I now really wish I'd seen it without any spoilers. Everyone I know is ever so good about avoiding them, but I chose to not watch it on Saturday and not worry about it too much, not knowing it was going to be good, so a few things slipped by:

* Someone mentioned the title being more literal than you might think
* NTL preview says "Save the earth from the daleks". Thanks guys. That's what I suspected from the teaser last week, but spelling it out really ruins the suspense
* Someone mentioned in passing that they'd heard RTD want to kill the doctor permanently, this implied he was going to maybe die
* That Tennant is expected to act again, and River Soong probably recognises him, implying he won't regenerate. Although I didn't actually think of that at the time. (And in theory, he could. The christmas special could be set in the past in his personal timeline, and maybe River does know him in a future incarnation, but knew not to say anything "you look so young".)

Best funny moments

It's good there were so many of these!:[2]

* When the doctor opens the door of the Tardis onto empty space
* "We're receiving a message." "What does it say?" "EXTERMINATE!"
* "A soldier in a bar?" "Don't worry, it was strictly professional!" That doesn't make it sound better, Jack :)
* When the Judoon appeared!
* "I voted for her." "No you didn't!"
* Harriet Jones, former prime minister: "Harriet Jones. Former prime minister." Dalek: "WE KNOW WHO YOU ARE."

Really, really cool moments

* When Donna mentioned the bees.
* When Harriet appeared. I hadn't realised how much I loved her until then. It says something good about the show how much I can cheer for her.
* Davros!

Really cool moments

* When the Judoon appeared!
* The shadow proclamation, the bees disappearing, several planets disappearing, the medusa cascade all turning out to have been relevant, but having not been forced down our throats, meaning this episode was "Wow! Now I see!", not "Sigh. At last." Compare to "bad wolf".
* Dalek Khan.
* Torchwood, Martha, Rose, SJ all appearing.
* Good music

Nitpicks

* The planet has a magic net to keep the ~~gravity~~ air in. Right. I agree they need something to explain why the earth is ok there, but the air? Doesn't that stay on cos of gravity? Which it should still have?
* Now we've seen a reset button used when the Master took over the earth, we know the possibility, so when bad things happen in an episode like this, you're distracted from the drama by counting the main characters in danger/killed, and mentally calculating if it's going to matter or not before you start caring.
* Dalek: "The males, the females..." why would it distinguish? Surely it's impassity could have been pointed out better if it had, eg. ordered a child to do something and not understood why it didn't.
* "Every telephone exchange" "Through torchwood itself". It was dramatic, which is good. But sounds stupid -- isn't Torchwood an organisation, not a building?
* Martha knows those guns won't hurt Daleks. But this is torchwood. It's literally full of powerful, dangerous, unpredictable technology. Why not use some of that? If it blows up, or warps the building into another dimension, so much the better. Where's the pterodactyl gone, for that matter? Where are the scavenged dalek guns from the last time we fought the daleks, or the time before?
* Unnecessarily blatant drama. Jack: "OMG! We are so doomed." OK, daleks. Bad. The last time you fought daleks, you became immortal. That's not you, that's the scriptwriters breaking characterisation in order to counterproductively try to make it scary.
* Unnecessarily slow cliff-hanger. The show makes this mistake again and again, I sympathise. It's hard. But it really kills the drama when it goes: "SJ skids to a halt. Dalek threatens her. Dalek threatens her again. Dalek says it's going to shoot. Long pause." Why not start the sequence earlier, and finish it with a quick "Dalek appears, brief beg, dalek gun glows white, bullet time, cut to black."
* Many more awkward moments, due to squeezing so many people into one episode. I won't bother to make the list exhaustive, you can see past them.

[1] Typo: soilers.
[2] In formal writing I would avoid a "!:" juxtaposition. In informal writing, I actively seek it out :)[3]
[3] Similarly with other overly generalised grammatical rules, I like experimenting, but not condoning, to me they're funny because they're over the top :) Eg. distribution over and: "The building and the front side of it are the one with "broadcom" written on it in two foot high letters". :)

Independence Day

2008-07-04T14:26:38Z

<Ozy and Millie>Happy Patriotic Explosions Day!&/lt;Ozy and Millie>

Links

2008-07-02T22:31:34Z

"A proof of the Riemann hypothesis"

Linked from God plays dice blog, a putative paper on the Riemann hypothesis. I can't read it, I never was plugged into the mathematical grapevine. But it would feel remiss in the extreme not to pass on the comment.

For non-mathematicians, the Riemann hypothesis refers to a simple formula f(z), where z is a complex number. It is zero at z=-2, z=-4, z=-6, etc. It is also zero for many complex numbers z=0.5+it, ie. with real part 1/2, and some imaginary part. The hypothesis is that these are the only complex numbers with f(z)=0.

This is explained in more detail in the comments below, directly after "unnecessarily confusing, you should have..." and before "not entirely accurate because...".

The Riemann hypothesis is interesting because it ties into all sorts of different maths (eg. primes), and everyone's sure it's true, and people have had computers checking millions of solutions and seeing that they do conform, and there's lots of maths proved on the assumption that it's true. If it were proved, it would be the most famous result in maths in the last, um, ten years :)

ETA: The result is from a stable that does work on the Riemann Hypothesis, but has had several flawed proofs published before. This was flawed, and maybe patched already.

Bacteria make major evolutionary shift in the lab

Evolution , with the magic of stop-motion fridges. (Also: Dumbasses put hands over their ears and say "la la la, can't hear you". Original scientist sends long, thoughtful reply saying "Pull your fucking heads out of your asses, already. Duh.") Disclaimer: people who believe in the absence of any sort of evolution are not automatically dumbasses. Really.

God hates FAQs

A friend recently linked to a service which, when you're raptured, can send a last message to a loved one. There are two basic approaches. You've been left behind .com is run by Christians, and employs an ingenious dead-man's-handle.

Post Rapture Post .com is run by atheists. Post Rapture Post is a lot funnier.

Q. How Do We Know that You Will Not Ascend To Heaven with Us?
A. The Bible says that only those that repent of their sins and accept Jesus as the True Son of God will be saved. We do neither. Some of our personal sins include: drunkenness, heresy, sacrilige/blasphemy, gluttony, laciviousness, and sloth. There is no way we are going to disappear into Heaven any time soon.

Q. Aren't You Afraid of God's Wrath?
A. We don't believe in God, remember? In the event that the Rapture actually occurs, we will go to Plan B: "Lifetime of Sin Followed by Deathbed Repentance."

FWIW, I think both sites, whatever their personal beliefs, genuinely offer the service, but don't hold me to that.

Advance Cycle Boxes

2008-07-01T22:43:49Z

Just before traffic lights, there is a cycle-lane like box the width of the main lane, in which cyclists can stop at lights without being rear-ended by cars.

0. Is there an official name for them? "Advance cycle box" is in my head, is is equally likely to be something else entirely, or made up.

1. What is the intended use if the cyclist approaches the traffic lights behind three cars? I feel it should be obvious what they're supposed to do, but admit I can't tell. Obviously if you can accelerate briskly to 20mph, there's no problem, but if you can't, or don't want to?

Undertake if there is a clear cycle lane, else wait in the queue? Always undertake if you can? (But it's not clear when the highway code permits undertaking.) Overtake if you can? (But this is unlikely to be possible.) Always wait? (But that leaves a frustrated driver behind you.) Dismount and cross the intersection from the pavement? (But that's annoying and takes several times as long.)

2. If it were safe to do either, which would delay the drivers least: moving ahead to the cycle box, or waiting in turn. One way, the drivers ahead go past, but the one behind probably misses the lights. The other, all the drivers are delayed until the road is wide enough to overtake safely again.

3. What do you do?

I feel silly for not knowing, but most of the time, it doesn't come up, either because there's not enough of a queue, or the road is wide enough to permit cars overtaking cycles safely. And then when it does, I don't actually know.

Stockholm

2008-07-01T20:33:17Z

Bike appointment ✔
Bank talked to ✔
Work guardedly successful ✔
Flights to Stockholm Thu Jul 24th - Fri Aug 1st for an unprecedented two weeks with livredor ✔✔✔✔✔

:)

Week before's Doctor Who

2008-06-30T10:30:00Z

It was well done, I think. Unfortunately, I wasn't very interested in it, but it evoked the feeling of an earth grittily doomed by aliens quite well. When the Italian family was trucked away, I was actually on the point of crying.

"All the universes there are are in danger" is a bit melodramatic. On the other hand, the stars going out, while cliched, was creepy.

I thought having the doctor die immediately was a bit of a cop-out, it would have been a lot cooler to see him regenerate, and then die again, or something. However, in 40 minutes, I understand the quicker route.

My first impression was that it was also too pat that the earth was poised so close to disaster, saved only by Donna. If every disaster was that close, so many 50/50 chances over 50 years, one would have occurred by now. If a companion is so important to the doctor, he would have found one, as he usually does even when he doesn't want to. (All adventure programs have this problem — it's supposed to be so risky, but everything always turns out ok.)

But then I realised, maybe it's right. Maybe the Earth always is balanced as close to disaster as this episode showed. But people always have to constantly nudge it back on track using dangerous time-travel shenanigans. Maybe there's lots of such interventions, but we don't normally see them, living in the universe which results.

Optical Behaviour of Vampires

2008-06-27T12:51:17Z

This came up in the pub last night, a physical theory for why vampires don't show up in mirrors, and the military applications of it. If you don't know simont, you should be made aware of his write-up: http://simont.livejournal.com/205240.html

Bisexual kisses

2008-06-26T18:46:00Z

The backstory

http://www.guardian.co.uk/media/2008/jun/24/asa.advertising

Heinz showed an advert where the mother of a family was replaced by a (male) New York deli sandwich maker. At the end, the husband is running of to work, and the sandwich maker says "Hey, aren't you forgetting something" in a humorously New York gruff way, and the husband gives him a little kiss.

The digressions

There's obviously lots of other things you might observe. For instance:

* It follows all the assumptions of having a wife at home cooking, two men having a career, etc.
* I'm sure really good deli's give really good sandwiches, however, averagely good amateurs seem to make better sandwiches than average deli's, so the basic message "as good as if you had a professional sandwich maker" didn't resonate

The point

The advert was removed when several people complained it was inappropriate for kids. (Presumably because it showed a gay kiss, although this wasn't actually stated in the article I read.) (People wishing to complain about that, someone linked to contact details here )

However, it's not showing an actual gay couple. Several people verbalised what was nagging me, atreic: "so I find it very nudge-nudge-wink-wink men kissing that's _funny_, and I'm quite glad the damn thing has been pulled.",

foreverdirt summed it up "If it hadn't been for Heinz's actions, I would be torn between quiet cheer that the ad features a same-sex kiss that isn't treated with disgust, and equally quiet fuming about how sexist and heteronormative the ad is. However, I am much, much more offended by the ad being pulled for featuring a same-sex kiss than I am by the ad itself -- it's not that the ad was a great leap forward, but the reasons that it was pulled are a great leap back."

It's a joke about gay kisses, rather than portraying gay kisses as normal. Which could be nasty, but on reflection, I think can be a positive thing.

It's not a nasty joke. There seems a sequence of societal acceptance that starts with jokes because they show [thing that makes people uncomfortable], and have a good marketing reason for showing it, without being as much of a leap as actually showing [thing] positively would be.

I've often thought this about bisexuality, which seemed to lag a number of years behind gay in getting any tv acceptence. Gay portrayal seems to have gone through the sequence -- there's obviously a long way to go, but I think it started with jokes about gay people, and now there are every so often normal gay characters on TV. TV acceptance of Bi seems to be just starting: it's generally only shown where it's funny. But I think that, counter-intuitively, it is a step on the way to becoming accepted.

Gratuitous quote from Jonathan–Ross-kissing fantasy author[1]

I can't find the exact quote, but Neil Gaiman said something like he was worried about the portrayal of the captain in the film Neverwhere, because he thought it was great, but he knew it could be seen as perpetuating a whole bunch of stereotypes. But then he got an award from a big gay and lesbian society for "best film portrayal of a (probably) gay character evar" or similar, and felt happy again, and treasures the memory to bring out if anyone ever criticises it.[2]

[1] "Jonathon–Ross-kissing" modifies author, not fantasy :)
[2] Agh, anyone got the actual quote?