I'm Heart, your friendly neighbourhood admin!

[michaeld]

Here is my (admittedly not very thorough) thoughts on the stone problem, by the way:

If the largest pile a player has access to has only three stones, e loses. Thus, if the largest pile a player has access to has 6, 5 or 4 stones, e can ensure that the next player has a largest pile of 3, and thus having a largest pile of 6, 5, 4 is a guaranteed win condition, provided the player plays correctly, i. e., leaves the other player with two piles of 3. It now follows that if a player has a largest pile of exactly 7, this must be subdivided into a largest pile of either of the previous, and hence a largest pile of 7 will be a loss. This is easily generalised to include [8..14] as a win condition, 15 a loss, and we see the pattern emerging: having stones 2^n - 1 is a guaranteed loss, provided the other player plays it right. Thus Kolya will not necessarily win, but Vitya can if he plays it smart, since 31 is 2^5 - 1.

This of course always requires that all the smaller piles generated as a by-product will be divided into even smaller units, which I think should be possible, though I have not really gone through it rigorously.

Sorry for coming in really late here - I've been distracted!

Yep! Amorphous has it! :cake:

The position is a second player win (with optimal play by both sides) if and only if the number of stones is one fewer than a power of 2.

My full explanation, originally posted here:

I think it's a second player win if and only if the number of stones is 1 less than a power of 2, which 31 is. So V can win.

If you start with 2^r-1 then the strategy is for V, after n moves, to leave K with one or more pile of 2^r-n-1 as the largest pile(s), and possibly some smaller piles too.

Why can that always be done? Well if K has a pile of 2^t-1 they have to divide it up into a+b = 2^t-1. Then exactly one of a,b has to be >= 2^t-1. So V can subsequently leave a pile of 2^t-1-1 and the remainder will be <= (2^t-1)-1-(2^t-1-1) = 2^t-1-1. And as for the other piles V is left with, they are all <= 2^t-3 so can be divided into piles of sizes <= 2^t-1-1 and 2^t-1-2.

Therefore V can always keep on feeding K largest piles of size one less than a power of 2. Therefore V wins.

Note that all 2 player games of this type - finite number of moves till guaranteed completion, definite winner, perfect knowledge (no dice or other unknowns) - have this form. Every position is either a first player win or a second player win.

A position is a first player win if and only if there is a move that takes it into a second player win position. (Because after their move, they will effectively become the "second player".) A position is a second player win if and only if every possible move leaves you in a first player win position. (*)

Those statements (under (*)) are enough to let you deduce by induction that every position in a finite game of this type has this property - either it's a first player win in which case the first player has an unassailable strategy that will inevitably win no matter what the second player does, or it's a second player win and the second player has such an unassailable strategy. What's more, the fact (*) is in principle enough to work out, recursively, which positions are first player wins and which are second. Of course doing that in practice may be computationally expensive. But it gives a roadmap for a solution, which is the approach I and I think Heart and Amorphous took. Start small - with positions that are close to the end (in our case with a small number of stones). Classify these positions into first player wins and second player wins, building upwards. Try and spot a pattern that tells you which position is which. Prove this pattern by induction using (*). In principle that's all there is to it!

HOWEVER... when you start taking infinite games into account things get a lot more interesting. There are perfect knowledge games (no chance) between two players that are guaranteed to result in victory for exactly one of them and yet neither has a winning strategy. What does that mean? It means that given any strategy for the first player (i.e. a function that decides what to do in each situation), there is a strategy for the second player to beat the first player's strategy and win the game. Similarly for every second player strategy there is a first player strategy that beats it. Weird huh?

The catch is that these games are infinite - they take an infinite amount of time to play and you can only see the result from the perspective of eternity. So they are not suitable for us mortals living in this universe. But it's very weird intuitively that they will inevitably end in a win for one side and yet neither side has a winning strategy. If you played two fully rational (infinite!) players against each other, assuming such a thing exists, who would win I wonder...?

Maybe in a follow up post I'll give a description of a known game with this property...

[Heart]

Here is my (admittedly not very thorough) thoughts on the stone problem, by the way:

If the largest pile a player has access to has only three stones, e loses. Thus, if the largest pile a player has access to has 6, 5 or 4 stones, e can ensure that the next player has a largest pile of 3, and thus having a largest pile of 6, 5, 4 is a guaranteed win condition, provided the player plays correctly, i. e., leaves the other player with two piles of 3. It now follows that if a player has a largest pile of exactly 7, this must be subdivided into a largest pile of either of the previous, and hence a largest pile of 7 will be a loss. This is easily generalised to include [8..14] as a win condition, 15 a loss, and we see the pattern emerging: having stones 2^n - 1 is a guaranteed loss, provided the other player plays it right. Thus Kolya will not necessarily win, but Vitya can if he plays it smart, since 31 is 2^5 - 1.

This of course always requires that all the smaller piles generated as a by-product will be divided into even smaller units, which I think should be possible, though I have not really gone through it rigorously.

Sorry for coming in really late here - I've been distracted!

Yep! Amorphous has it! :cake:

The position is a second player win (with optimal play by both sides) if and only if the number of stones is one fewer than a power of 2.

My full explanation, originally posted here:

I think it's a second player win if and only if the number of stones is 1 less than a power of 2, which 31 is. So V can win.

If you start with 2^r-1 then the strategy is for V, after n moves, to leave K with one or more pile of 2^r-n-1 as the largest pile(s), and possibly some smaller piles too.

Why can that always be done? Well if K has a pile of 2^t-1 they have to divide it up into a+b = 2^t-1. Then exactly one of a,b has to be >= 2^t-1. So V can subsequently leave a pile of 2^t-1-1 and the remainder will be <= (2^t-1)-1-(2^t-1-1) = 2^t-1-1. And as for the other piles V is left with, they are all <= 2^t-3 so can be divided into piles of sizes <= 2^t-1-1 and 2^t-1-2.

Therefore V can always keep on feeding K largest piles of size one less than a power of 2. Therefore V wins.

Note that all 2 player games of this type - finite number of moves till guaranteed completion, definite winner, perfect knowledge (no dice or other unknowns) - have this form. Every position is either a first player win or a second player win.

A position is a first player win if and only if there is a move that takes it into a second player win position. (Because after their move, they will effectively become the "second player".) A position is a second player win if and only if every possible move leaves you in a first player win position. (*)

Those statements (under (*)) are enough to let you deduce by induction that every position in a finite game of this type has this property - either it's a first player win in which case the first player has an unassailable strategy that will inevitably win no matter what the second player does, or it's a second player win and the second player has such an unassailable strategy. What's more, the fact (*) is in principle enough to work out, recursively, which positions are first player wins and which are second. Of course doing that in practice may be computationally expensive. But it gives a roadmap for a solution, which is the approach I and I think Heart and Amorphous took. Start small - with positions that are close to the end (in our case with a small number of stones). Classify these positions into first player wins and second player wins, building upwards. Try and spot a pattern that tells you which position is which. Prove this pattern by induction using (*). In principle that's all there is to it!

HOWEVER... when you start taking infinite games into account things get a lot more interesting. There are perfect knowledge games (no chance) between two players that are guaranteed to result in victory for exactly one of them and yet neither has a winning strategy. What does that mean? It means that given any strategy for the first player (i.e. a function that decides what to do in each situation), there is a strategy for the second player to beat the first player's strategy and win the game. Similarly for every second player strategy there is a first player strategy that beats it. Weird huh?

The catch is that these games are infinite - they take an infinite amount of time to play and you can only see the result from the perspective of eternity. So they are not suitable for us mortals living in this universe. But it's very weird intuitively that they will inevitably end in a win for one side and yet neither side has a winning strategy. If you played two fully rational (infinite!) players against each other, assuming such a thing exists, who would win I wonder...?

Maybe in a follow up post I'll give a description of a known game with this property...

STORY TIME STORY TIME!!!

I can't think off hand of a real-life game like that... would it have to encompass multiple generations? Does it have to actually go on for infinity, or just have the capacity to? Is this like a game of life kind of scenario?

[michaeld]

Ah no, games of this type really do inevitably last an infinite amount of moves. The winner is not (generally) decided until both sides have made infinitely many moves.

So yeah these games cannot be played in the physical universe - at least as far as we know. But pure mathematicians enjoy this sort of thing, and such abstract scenarios have a funny habit of being eventually relevant to the physical world.

Anyway, here is the best known example of an infinite game. This is called the Banach-Mazur game...

The game is played on the interval [0,1] between two players A and B. This set [0,1] is the collection of real numbers (or decimals if you like) that are between 0 and 1.

Before the game starts, a subset S of [0,1] is chosen and announced to both players. This set is called the "winning positions for player A". You'll see why shortly.

OK so A plays first. For their move, they have to pick an interval [a_0,b_0] inside [0,1]. The interval they pick has to have positive length. In other words they have to choose the endpoints 0 <= a_0 < b_0 <= 1.

Then it's the turn of B. They have to pick a subinterval [a_1,b_1], again of positive length, of the interval A just picked. So they must pick a_1,b_1 such that a_0 <= a_1 < b_1 <= b_0.

Back to A. They must now pick a subinterval of [a_1,b_1]. This is [a_2,b_2] again with a_1 <= a_2 < b_2 <= b_1.

And so on....

Both players continue forever.

Oh yeah one more rule. (This isn't present in some versions of the game but it makes things simpler and there's no harm in adding it.) Let's say that each interval a player picks has to be no more than 1/2 the length of the previous interval. So we have (b_1-a_1) <= (b_0-a_0)/2 and similarly for later intervals.

OK so they both make infinitely many moves. What now? Well it's pretty easy to see that these nested intervals are homing in on a unique real number. In other words their intersection is of the form {x} for a single real number x. It's unique because the intervals are shrinking in length rapidly so there can't be two different numbers in every interval. And the existence of x follows by some easy real analysis covered at first year undergraduate - I'll skip this.

So the intervals home in on a unique real x. Here is the outcome: if the real number x belongs to the set S (chosen before the game started) then A wins. That's why S was called the set of winning positions for A. If the real x does not belong to A then B wins.

Hence there is always a winner. But to know who wins... usually you have to wait till both sides have made infinitely many moves.

OK so what can we say about who wins if they play well? It depends what S is obviously. First observation. If S contains an entire interval (e.g. [0.1,0.11] say) then A can clearly win. How? Simple. On their first move, they choose [a_0,b_0] to be such an interval. It now lies entirely within S. And as all future moves lie within that interval, the real x in the intersection clearly belongs to S too. So A will win.

What about a more interesting example.. how about if S is the set of rationals in [0,1]? The rationals are dense - that is, in every positive length intervals you can find one (in fact infinitely many). And yet... despite that, A will lose if B plays properly. Why is that? Well B just has to make sure that the intersection point x is not rational. Here we can use the fact that the rationals are countable: they can be enumerated q_1,q_2,... The strategy for B is now pretty clear. They wait for A's first move (it doesn't really matter what it is). They then choose their move [a_1,b_1] to miss out the rational q_1. A bit of thought shows this is possible. They then wait for A's second move. B then chooses their second move [a_3,b_3] to miss out the rational q_2. And so on. As these shrinking intervals miss out q_1,q_2,... in turn (indeed q_n is missed out on B's nth move i.e. [a_{2n+1},b_{2n+1}]), the intersection point cannot be q_1,q_2,.... Therefore it falls outside S. Therefore B wins.

With the first example it was inevitable A was going to win after their first move. Not so the second example. If you were an observer of this game, you couldn't be sure B was going to win unless you watched infinitely many moves.

Let's generalise that last example a bit. We say a set S is of the first category if it is a countable union of nowhere dense sets. Another name for first category is meagre. What does the countable union of nowhere dense sets mean? It means it's the union of sets B_1, B_2, ... such that each B_i has the property that in every interval (positive length) you look, an entire subinterval (positive length) is missed out of B_i. The rationals are clearly first category (as is every countable set) because the singleton set {q_i} is nowhere dense.

Now it's pretty easy to see that if S is a first category set then B has a winning strategy. Why? Well if S is the union of B_1, B_2, ... where B_i are nowhere dense, then B can just do the following. We need to ensure that the intersection point x is not in any of B_1, B_2, ... Well B accomplishes this simply by making sure that their kth move [a_{2k+1},b_{2k+1}] has no points in common with B_k. This is always possible precisely because B_k is nowhere dense - as that means that within [a_{2k},b_{2k}] (or indeed any other interval), an entire subinterval is missed out. Of course we need to make sure the length is smaller than twice the previous length, but that just makes things better for B not worse...

Anyway as B's kth move misses out B_k, the intersection misses B_k for every k and therefore is outside S. Therefore B wins.

We've shown that B has a winning strategy if the set S is of the first category. The surprising fact is that that's all! If S is not of the first category then B does not have a winning strategy. If B does not have a winning strategy in such cases then... well there has got to be a winner as every game ends in a win for A or B. So does that mean A must have a winning strategy in such cases?

Well if you ask the same question for A you get a similar conclusion. A has a winning strategy if and only if there is a positive length interval in [0,1] in which S's complement is of the first category. In other words S is comeagre in some subinterval of [0,1]. And that's it.

It might be worth a quick interlude here to ask what does a winning strategy mean? It's nothing very complicated. A strategy for a player - let's say A - is simply a function F_n that tells A what to do on their nth move given their position (a_2n,b_2n). You could allow F_n to know all previous moves too - that really doesn't make any difference. Similar for B.

So we've said which sets give winning strategies for a player. But now here's the bizarre thing. There are many sets S that are *neither of the above*. In other words they are neither of the first category, nor do they have the property that there's an interval in which their complement is of the first category. A favourite example is the Vitali set, which you learn about in a course on measure theory and you can read about e.g. on wikipedia... but there are many others. All make use of the axiom of choice.

That means that if A plays a particular strategy F_n then B has a strategy G_n that beats it. But by the same token for every strategy G_n for B there is a strategy H_n for A that beats it. The issue is that B cannot really tell what strategy A is playing (if any) after a finite number of moves, as they've only seen the start of it. You have to look at the whole eternity to see what strategy A is really playing. If you knew that in advance then B would know to play a strategy that beats it. But in fact B doesn't know A's strategy until time eternity, and of course by then it's too late as all the moves have already been played and the winner is decided. By the same token, A cannot really tell what strategy B is playing until the game is over. Therefore there is no winning strategy for either player in such cases (though every game will have a unique winner). That's the weird and wonderful world of infinite mathematics for you!

[Heart]

Thanks Mic!!! More food for thought :D

[Dodecahedron314]

Arrr!

So Halloween will have a PI-rate theme?

...I may or may not have to steal this idea for my Halloween costume. Pirate coat and boots with my Pi/Pie T-shirt, plus getting back up to snuff with memorizing as many digits of pi as possible* and then reciting them in a pirate voice? Sounds like a plan. I mean, the dorm I live in is stereotypically full of math and physics majors, so hopefully someone will appreciate it...?

*(I'm already up to about 60-something consistently just off the top of my head, but when I've been practicing I've gotten as high as 80-something.)

So....y'all remember this thing that I said I was going to do?

Well, because I am a ridiculous human being, this is indeed a thing that I am going to do.

https://www.dropbox.com/s/r1xp263qdy9jv28/2015-10-30%2021.22.42.jpg?dl=0

In case my phone camera is too bad for you to tell, the thing on my head is a tricorn hat with a pi symbol on it (homemade out of a couple of random boxes I found in the hall this afternoon), the eye patch is a circle with pi in terms of the radius, circumference, and area written on it (it's a pi patch, get it? :lol: ), and the shirt is the one I posted on here a couple of pages ago. (Also, please ignore the super messy dorm room in the background)

Also, as if this alone didn't make me enough of a massive dork, this is what I'm wearing to the football game tomorrow because I'm in the pep band and they encourage us to come in costume. One of the songs we're playing is the theme from Pirates of the Caribbean, so it works. According to my RM, this qualifies as exceptionally nerdy even by nerd school standards. My work here is done. :D

[Kelly]

I love it. :) Awesome hat, D-314.

*joins in on the Halloween Pi theme* Arrr!

[Heart]

Dodec, THAT IS SO AMAZING!!

My halloween costume was nothing as good as that :P I couldn't think of any good puns this year, so I am only sorta ashamed to admit that I resorted to one I'd used in previous years. I made a superman logo, pinned it to my shirt, and then got a conductor's wand (like, the conductor of an orchestra). I was a super conductor ;)

...yeah. I'm just that punny.

[Dodecahedron314]

Dodec, THAT IS SO AMAZING!!

My halloween costume was nothing as good as that :P I couldn't think of any good puns this year, so I am only sorta ashamed to admit that I resorted to one I'd used in previous years. I made a superman logo, pinned it to my shirt, and then got a conductor's wand (like, the conductor of an orchestra). I was a super conductor ;)

...yeah. I'm just that punny.

Aw, thanks. :3

As a punny, physicsy band geek, I approve of your costume. That's one I might have to steal at some point if I run out of ideas... :P

[Heart]

Steal away! It's too good an idea to hide away ;)

[Kelly]

Did Einstein first present his General Theory of Relativity 100 years ago today?

[Heart]

Did Einstein first present his General Theory of Relativity 100 years ago today?

I don't know off the top of my head, but if so, let's do a toast (or few) to that!

Cheers! ;)

[Kelly]

*toasts Einstein*

[Calligraphette_Coe]

*toasts Einstein*

I'm still awed every time I think of Euler's Identity. And I often remind myself of the contribution to art, architectual and mechanical design that came from the discovery of the Golden Ratio. I often think that humans are most god-like when they discover and explain these mysteries or use disparate technologies to invent new things the universe may not have constructed randomly on its own. That it's NOT the exercise of arbitrarily inherited power, but the power of the will of the intellect to conquer the harshness of the naked universe.

[Heart]

I love that. And I too will never lose my awe of the fact that exponentials and trigonometric ratios are so... intimately linked. It always makes me wonder what other intimate and secret links are out there in the universe that we just don't see yet? If Euclidean triangles and the growth of bacteria are so alike, what else could possibly have shared secrets? Is there a universe somewhere where nonEucidean triangles give rise to different continuous growths? What would that even look like?

Would a universe with larger curvature than our own (for example, a highly hyperbolic universe) have bacteria that grows faster or slower? Radioactive decays that happen with different probabilities and activity levels?

*mind blows up while wandering remote paths*

These are fun little things to think about ^_^ If someone knows more about this subject, please do speak up!

[Amorphous]

I don't think "highly curved" really captures the essence of what it means to live in a space with a significant negative curvature on the micro/nanometer scale. ^_^ Baseball would certainly be a different sport, since you would have to make very sharp turns at every corner, unless they just added more corners to it.

I'm more curious about whether such a universe would even support the same stable materials, since the relative angles between chemical bonds would be different!

[Dodecahedron314]

On the topic of how different the universe would be if its fundamental parameters were ever so slightly off from what they are, I highly recommend Martin Rees's book Just Six Numbers. I personally haven't managed to get my hands on a copy in years, but it's one of the first physics books I ever read, all the way back in grade school when I made it my mission to read the entire (admittedly rather small) physics section of the city library, and it's all about the really profound effects of even the most minute changes in physical constants like G.

[Calligraphette_Coe]

On the topic of how different the universe would be if its fundamental parameters were ever so slightly off from what they are, I highly recommend Martin Rees's book Just Six Numbers. I personally haven't managed to get my hands on a copy in years, but it's one of the first physics books I ever read, all the way back in grade school when I made it my mission to read the entire (admittedly rather small) physics section of the city library, and it's all about the really profound effects of even the most minute changes in physical constants like G.

Or if there is somewhere/when a cosmic Atlas whose job it is to keep all the Eulerian constants in balance? Maybe he makes minute adjustments in Planck Time to do the magic? That that is the reason the constants are transcendental numbers-- a sort of parallax in reality that we can't divine because we can't function on a cosmic scale, so we can't see what the Moving Hand has Writ before it moves on, we just (eventually) see *what* was written, blurry as it is from the constant change.

And maybe that's why people like us exist-- to provide balance and counterpoint to a small variable in the googleplex of universal variables that make up reality that keep it from blowing up or falling into its own black hole.

[Heart]

Wow. I'm so happy how this thread responded to my musings!

I never thought of playing baseball in a hyperbolic universe... And I wish I was a chemist so I could think in terms of bond angles! Chemistry is so cool, I find myself in a perpetual state of wishing I had more time to learn about it.

And thank you Dodec for the book suggestion, I'll have to add it to my reading list :D :D

Coe (can I call you that?), I think we all are so much more valuable than "just" balancing a universal constant, just saying ;) But that's a really neat concept. I wonder if the numbers like e and pi are still being written. I wonder if an Atlas out there is watching me type, thinking of what the next number in the sequence of decimal places should be...

If there is such a being, then I vote for 5. I've always kinda liked the number 5.

[Calligraphette_Coe]

Coe (can I call you that?), I think we all are so much more valuable than "just" balancing a universal constant, just saying ;) But that's a really neat concept. I wonder if the numbers like e and pi are still being written. I wonder if an Atlas out there is watching me type, thinking of what the next number in the sequence of decimal places should be...

If there is such a being, then I vote for 5. I've always kinda liked the number 5.

People used to always shorten it to 'Calli'. It's kind of funny the way I hide things in my pseudonyms, but the start of coming up with 'Calligraphette Coe' was the word calico. All my life I felt like the 1 in a million calico cats that was XY instead of the usual XX female. It's often like that in 3D space.... they see me and see 'female', not XY. Always being one to repeal the laws of nature by being innately female, it seemed *perfect* with being teamed up with calligraphy, which to me meant 'beautiful writing'. And just like the universe, I like to hide mysteries inside enigmas and disguise them with paradoxes.

As for paradoxes, what if pi eventually *does* resolve, and ends in '141592'? Or if the number of digits is the cardinality of another mysterious number?

I SOOO identified with Jodie Foster's character in 'Contact'. That through my transness and my NDE, I was given something I can't explain. That when questioned about it:

Michael Kitz: [standing, angrily] Then why don't you simply withdraw your testimony, and concede that this "journey to the center of the galaxy," in fact, never took place!

Ellie Arroway: Because I can't. I... had an experience... I can't prove it, I can't even explain it, but everything that I know as a human being, everything that I am tells me that it was real! I was given something wonderful, something that changed me forever... A vision... of the universe, that tells us, undeniably, how tiny, and insignificant and how... rare, and precious we all are! A vision that tells us that we belong to something that is greater then ourselves, that we are *not*, that none of us are alone! I wish... I... could share that... I wish, that everyone, if only for one... moment, could feel... that awe, and humility, and hope. But... That continues to be my wish.

[Heart]

I just got shivers reading that and remembering the scene. I remember it well.

Also, that's such a beautiful story behind your name. Calli it is from now on. So beautiful <3

[Calligraphette_Coe]

I just got shivers reading that and remembering the scene. I remember it well.

Also, that's such a beautiful story behind your name. Calli it is from now on. So beautiful <3

It's one of my all-time favorite movies! Just so many things about it resonated with me, like growing up saying "CQ, CQ, CQ....." into an Astatic D-104.

Or this line between Jodie Foster's and Angela Basset's characters:

Ellie Arroway: Mrs. Constantine? May I have a word with you?

Rachel Constantine: Certainly.

Ellie Arroway: Um, I have a big problem.

Rachel Constantine: Yes?

Ellie Arroway: Uh, do you know where I can find like a really great dress?

:)

[Heart]

It's one of my all-time favorite movies! Just so many things about it resonated with me, like growing up saying "CQ, CQ, CQ....." into an Astatic D-104.

Or this line between Jodie Foster's and Angela Basset's characters:

Ellie Arroway: Mrs. Constantine? May I have a word with you?

Rachel Constantine: Certainly.

Ellie Arroway: Um, I have a big problem.

Rachel Constantine: Yes?

Ellie Arroway: Uh, do you know where I can find like a really great dress?

:)

I remember watching that exact scene and thinking to myself "Oh no! I would have no idea what to do about that..." I sympathise with her so much in that scene :P

[Dodecahedron314]

EEEEEEEEEEEEEEEEEE Contact is my favorite book ever :D It's the only book I've ever been tempted to steal. Although, in all fairness, my freshman year English teacher did have two copies...but I was a good person, and got it at a used book shop for cheap a few years later instead. Wasn't a big fan of the movie, because the book spoiled me for character diversity and a non-simplified plot. That's why I always tell people to watch it and then read it.

Coincidentally enough, I might just end up helping build a radio telescope sometime soon...

[Kelly]

Awesome. :) I hope that you can help build a radio telescope.

I read the book and then saw the movie. I actually liked the movie.

A religion explained with math:

http://www.scq.ubc.ca/fsm-theologebra-church-of-the-flying-spaghetti-monster-algebra/

[Heart]

Ooooo radio telescope! That's so cool! And thank Kelly for the reading, I shall have good bedtime reading this evening :D

[Amorphous]

So ... don't know if any of you have heard of this, but apparently, according to a new comprehensive study in neuroscience, there really aren't any actual differences between the brains of "men" and "women", but is much better described by a spectrum!

http://news.sciencemag.org/brain-behavior/2015/11/brains-men-and-women-aren-t-really-different-study-finds

(Not sure where to post this, but it seemed relevant to share.)

If anyone doesn't have access to the PNAS and wants the research article, you can send me a PM.

[Kelly]

This seems like the right thread to post it. I was going to post a similar article:

http://www.livescience.com/52941-brain-is-mix-male-and-female.html

Our brains are a mix of male and female:

...men and women's brains are an unpredictable mishmash of malelike and femalelike features...

The researchers combed through more than 1,400 magnetic resonance images (MRI) from multiple studies of male and female brains, focusing on regions with the largest gender differences. In the first analysis, using brain scans from 169 men and 112 women, the researchers defined "malelike" and "femalelike" as the 33 percent most extreme gender-difference scores on gray matter from 10 regions. Even with this generous designation of "male" and "female" scores, the researchers found little evidence of the consistency they would need to prove brain dimorphism. Only 6 percent of brains were internally consistent as male or female, meaning all 10 regions were either femalelike or malelike, the researchers found. Another analysis of more than 600 brains from 18- to 26-year-olds found that only 2.4 percent were internally consistent as male or female, while substantial variability was the rule for more than half (52 percent).

With 52 percent of people having a brain that is a mix of male and female, genderqueer is more the norm than the exception.

[Calligraphette_Coe]

This seems like the right thread to post it. I was going to post a similar article:

http://www.livescience.com/52941-brain-is-mix-male-and-female.html

Our brains are a mix of male and female:

...men and women's brains are an unpredictable mishmash of malelike and femalelike features...

The researchers combed through more than 1,400 magnetic resonance images (MRI) from multiple studies of male and female brains, focusing on regions with the largest gender differences. In the first analysis, using brain scans from 169 men and 112 women, the researchers defined "malelike" and "femalelike" as the 33 percent most extreme gender-difference scores on gray matter from 10 regions. Even with this generous designation of "male" and "female" scores, the researchers found little evidence of the consistency they would need to prove brain dimorphism. Only 6 percent of brains were internally consistent as male or female, meaning all 10 regions were either femalelike or malelike, the researchers found. Another analysis of more than 600 brains from 18- to 26-year-olds found that only 2.4 percent were internally consistent as male or female, while substantial variability was the rule for more than half (52 percent).

With 52 percent of people having a brain that is a mix of male and female, genderqueer is more the norm than the exception.

Remember when Jessel and Moir's book 'Brain Sex' came out in the 90s? And many in the transgender community seized upon it as definitive proof of identities? And how much trouble Lawrence Summers got in for saying that male brains on average had a built in advantage in STEM careers? If that book and/or what Summers said was one end of the spectrum, maybe this research is the other?

Me, I think the truth lies, as usual, somewhere in the middle. That even MRIs of brains are like pictures from space-- the map may look the same, but the map is NOT the territory and at ground zero there's a whole lot of something else going on. Such as maybe it is sex hormones that act as a tipping mechanism to tip the 'territory' in the direction of Mars or Venus.

I read a lot of stuff about neuroplasticity because I have a lot of unintentional experience with it from suffering brain trauma from multiple CVAs. ( BTW, there is a pretty good book about this out now from Norman Doidge, M.D. called 'The Brain that Changes Itself', about how people with my difficulties manage to recover against long odds.) It really makes me wonder if it's less brain structure and more 'software' tipped over by testosterone or estrogen. Or a mix.

Does HRT re-draw the maps? Or change the software? Or is it dependent on the 52% mix being tipped one way or the other? Or keeping to the genderqueer/androgynous middle?

Either way, fascinating stuff!

[Heart]

Wait, so they're saying I'm NORMAL?!? Pssssshhhh. I've never been normal in my life, I'm always extraordinary ;)

But in all seriousness, I'll have to look up that article when I'm done exams (only three more days left!). *bookmarks it for further investigation*

Thanks Kelly and Amorphous!!

[Heart]

aaaand I'm back. For those that didn't notice, I disappeared for a week and a half there, to do finals things.

I think I passed ;)

But the best result is that I can rejoin my AVEN community :D

If anyone's up for giving me a summary of gossip, I'll take it. Until then, I shall continue to read through what I missed :P