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1. If the world revolves around your character, they're probably a Mary Sue. A Mary Sue is, contrary to popular belief, a state of being, not a state of appearance. A lot of people tend to try and "fix" a Mary Sue problem by having their character be ugly instead of pretty, jerkass instead of sweet, a fat, helpless blob instead of someone who can throw lightning bolts, and stupid instead of smart. (This is called an Anti-Sue, as opposed to a Purity Sue; many subspecies of Sue exist; just ask TvTropes.) But, somehow, everyone in the world manages to not comment on her ugliness, bad temper, obesity, or stupidity. In fact, they try real hard to buddy up to her, even plan their calendars around her, despite what should be obvious unlikability. Oh, and anyone who tries to call hero out on these things is a bastard, to be shunned by all, no matter how legitimate their argument might be. This is, in essence, what it means to be a Mary Sue. Believe it or not, story worlds can actually have stakes other than the hero, and in fact, SHOULD. Because your reader lives in the real world, and would like to see enough of it reflected in the story to relate to it. Namely, that since their own life doesn't serve as an axis for the globe, universe, and fate itself, the hero's shouldn't either. Especially when tragedy for the hero, an individual, can be far outclassed by things like genocide and war. (Or at least, they always are from this reader's perspective.) It only gets worse when things like relationship problems or burning one's French Toast are equated with the aforementioned, because, like, oh my God, you're talking about MARY SUE!
2. If your character has super-awesome-wow skills that have no story relevance, they're probably a Mary Sue. This goes double for when the skill in question is never even demonstrated properly, leading to Informed Ability, i.e., the author's told us it's there, but hasn't shown us. Like when Eragon is the best swordsman evar and he keeps getting knocked out in the first book? Yeah. Like that. If your character is a fantastic artist or writer, this can be awfully hard to show, but you as the author should at least work at it. When less mundane things, like ninja skills or leaping walls, come into play, it just gets worse. If someone can leap walls, I better see a wall somewhere in the climax, and the leap over it better have something to do with the characters growth arc or something, or I'm gonna be pissed. Note that something mentioned in passing, like "he rode horses on weekends" doesn't necessarily apply if it's serving some other purpose (like characterization), but most Mary Sue writers tend to gush endlessly about how great their characters are. (Even without meaning too, bless their stupid little hearts.) I don't want to see an entire scene of someone practicing ballet if ballet has nothing to do with the story or the character's journey. Mary Sues are often weighed down by their awesomeness, to the point where it makes the story screech to a halt. Double points for when a scene serves no other purpose than to demonstrate these unnecessary skills and laud the Mary Sue.
3. If everyone reacts in terms of your character, they're probably a Mary Sue. If a nuclear blast went off somewhere, and all the characters can talk about is how much fun it was to chat up your hero last night at the bar, you're in Sue territory. Believe it or not, real people tend to have multiple priorities, to the point that, yeah, they'll leave a friend in the semi-emotional lurch for the chance to play XBox. Your characters should, too. Too many supporting characters prioritize in terms of the Mary Sue: is she happy? Does she have everything she wants? Can I get her a soda, just for the chance of a sweet, sweet wink? (Or in the case of an Anti-Sue, a sweet, sweet spit on the cheek?) The cast will drop whatever they're doing just for a chance to be near the Mary Sue, twice as fast if she's in some kind of "distress". If your characters are constantly asking after the Mary Sue, always talking about her or something heavily relevant to her ("I heard that French Toast ballet competition is going to be fierce!"), you need to examine the priorities of your supporting cast. This includes villains, by the way; remember, Mary Sues are defined by an inordinate amount of undeserved attention, and the villain can sometimes be the worst offender. (My God, could it possibly be Galbatorix's offscreen villainy is due to not caring about Eragon's hissy fits!? Paolini, you got one right!?) And as always, double points for a character expressly created for a Mary Sue's well-being or what have you, usually a love interest or best friend.
4. If all obstacles and established laws don't apply to your character, they're probably a Mary Sue. Oho, is this one ever a doozy. The Mary Sue flouts all rules and authority, usually flipping a middle finger to it from atop their transforming Lamborghini dragon. Swordsmanship takes years to learn? Mary Sue doesn't just get it in two weeks, she masters it. Reading and building a vocabulary takes years? Mary Sue writes and speaks like a native overnight. Need I go on? It gets better with societal laws! We all know what rudeness is, right? Not a Mary Sue! In her presence: rudeness? What's rudeness? Or bragging, for that matter? Everything she says is beautiful truth, with a honeyed glaze of fact, even if it's a controversial subject with no easy answers! Please note that this goes for any established law, like the rules of magic (in addition to things like "don't steal" and "follow the school's dress code") that ALSO results in zero repercussions for the Sue in question. Exceptions are made, the judges all pardon, the cops look the other way without being bribed, the magic works without effort, whatever. A hero overcoming obstacles is far removed from a Mary Sue playing in her perfect world of no consequences. Double points if the author sets up these rules for the express purpose of having the Mary Sue knock them down: "No one's ever been able to Master the Staff of Fire, ride a dragon, or turn invisible all at the same time under the age of 50! Gasp!", or in the case of fanfiction, trouncing the established rules of the canon world just to show that their Mary Sue can. (We call that a Fixer Sue. Of course, if you're reading fanfiction, maybe you deserve it?)
5. If a society is perfect because everyone in it is, they're probably all Mary Sues. While the sci-fi genre tends to use this on the lesser scale of ham-fistedness, it still, for the most part, can be pretty heavy to have aliens underline and highlight all the shortcomings of humanity. We can do that just fine on our own, thanks! With fantasy, this tends to get about a thousand times worse with elves, who never tire, go hungry, or even know the meaning of either. (Right, like the bastards never had a bad harvest, blight, or storms ... I'll bet.) Most of the time, lofty attempts to make a society beyond such petty things ends up making them look racist or worse. Many Mary Sue writers will tend to make a race fantastic and wonderful by the sheer virtue of being what they are: dragon men are awesome on every level, therefore your argument is invalid. Utopian societies lack conflict, something a story desperately needs, and by introducing a Mary Suetopia into a conflict, you're going to end up making some kind of statement you didn't mean to make. Usually with hilarious results, like facist Suetopia, anti-individualist Suetopia, and my personal favorite, stagnant-civilization-with-no-adaptability Suetopia. (That one tends to get skewered by world-building logic. Even a low-IQ dose is lethal.) The point is, if people in general don't like Mary Sues, what makes you think we, humanity, as a race (and remember our tolerance and equality record, here), would appreciate a race of them? Even if they had nuclear ray guns and interstellar travel, odds are some idiot somewhere would at least try chucking a spiteful molotov cocktail. This can be something interesting to explore, and many authors have done so successfully; it's just when one gets into the slapdash type of Mary Sue races that trouble arises. Things like rape, theft, and war have biological, agricultural, economic, and societal reasons for existing, and to think that someone supposedly "solved" all that simply by being a dragon man (or, ahem, certain ten-foot-tall blue-skinned aliens) is aggravating. In fact, I'd be much more interested in a story that can explain why they don't exist! And, naturally, double points if an individual character from the race Mary Sues on all of their established laws! "We do not murder, puny human. We are far above such petty things." "Hey, I just dismembered this guy." "All right, then."