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In the 17th century, many young girls are enlisted for national service and are subjected to weeks of harsh selection processes to weed out muggle[1] before being trained as sorceresses. Of course, some time later, the remaining/surviving sorceresses were decommissioned and replaced by the Amazons.

However, I must clarify to my wide audience, why only certain girls whom are just starting puberty can fly on a broom or twig depends on personal preference and proficiency.

Initially, I thought of using genetic mutations of the X chromosome but that failed miserably. I'm hoping for an answer that could cause a change in neurochemistry as to not only affect emotional states but also grants the user magic, similar to D&D style, after going through the rite of passage(training and trial).

The best answer would be like a scientific paper with as few, or no handwaves. By the way, I already ruled out blood type, testosterone level and diet!

[1] This is a term coined to celebrate majority of us who is intelligent enough to use the broom appropriately but not intuitive enough to abuse it.

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    $\begingroup$ You seem to be looking for an explanation completely alien to the 17th century. Are you looking for an explanation at that time or are you looking for an explanation from historians how that happened in 1700 AD in their museum exhibit in 2098? $\endgroup$ – nvoigt Nov 3 '15 at 7:04
  • $\begingroup$ @nvoigt: the latter one and because I enjoy science a lot. $\endgroup$ – user6760 Nov 3 '15 at 7:07
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    $\begingroup$ pity, you could have great fun with the former one... ;) $\endgroup$ – colmde Nov 3 '15 at 9:59
  • $\begingroup$ Nothing in magic is in a vacuum. How do they fly? The scientific explanation for how they fly may give great insight into why only some can do it. $\endgroup$ – Cort Ammon - Reinstate Monica Nov 4 '15 at 6:42
  • $\begingroup$ @CortAmmon: telekinesis. $\endgroup$ – user6760 Nov 4 '15 at 6:45
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Use Tetrachromats as an example.

In humans, two cone cell pigment genes are present on the X chromosome: the classical type 2 opsin genes OPN1MW and OPN1MW2. It has been suggested that humans with two X chromosomes could possess multiple cone cell pigments, perhaps born as full tetrachromats who have four simultaneously functioning kinds of cone cells, each type with a specific pattern of responsiveness to different wavelengths of light in the range of the visible spectrum.

Humans normally have three types of cone cells and are Trichromats. Since cones are controlled by the X-chromosome, some small percentage of women end up with four distinct cones and have better color differentiation than normal.

Your magic can work the same way - you need to have two X chromosomes to display magical abilities, and they have to be different X chromosomes in terms of the specific magical genes to fully show, so it's not just "every girl".

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    $\begingroup$ Note, however, that males with XXY chromosomes might potentially get the magic gift as well. $\endgroup$ – MichaelS Nov 3 '15 at 7:07
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    $\begingroup$ If the magicical abilities was linked to being a tetrachromat, then they can be identified by using a tetrachromatic color vision test. $\endgroup$ – Samuel Nov 3 '15 at 20:38
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    $\begingroup$ @Samuel: I'm not sure about that test, snopes seems to think that standard monitors are insufficient: snopes.com/politics/medical/tetrachromacy.asp "Unfortunately, computer screens do not provide enough colour information to be able to 'tap into' the extra dimension that tetrachromats may possess. It is therefore impossible for an online test to investigate tetrachromacy." It is possible newer monitors might be capable but that article has been updated as of Jun 2015, so probably not. $\endgroup$ – Dan Smolinske Nov 3 '15 at 21:37
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    $\begingroup$ @Dan I doubt they'd be using standard monitors in the 17th century :) $\endgroup$ – Samuel Nov 3 '15 at 22:29
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    $\begingroup$ @MichaelS That's really interesting. According to that link, it affects approximately 1 in 500-1000 males. That could make an interesting plot point. $\endgroup$ – Prinsig Dec 31 '15 at 11:38
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If you're willing to allow a few outliers, I'd recommend avoiding the quagmires of genetics and just focus on neurology and development during puberty.

There's a glorious class of neuron that's just begging for magic systems to abuse them: the Mirror Neuron.

The function of the mirror system is a subject of much speculation.

Mirror neurons have been implicated in facial recognition, language comprehension, even empathy. There are also scientists who argue that they don't actually exist, but are rather artifacts of the way we categorize neural activity. This is ripe for any quantity of hard magic we need! As an added bonus, there's observed differences in them between the genders explaining why only girls gain the ability.

Having everything occur right around puberty is a difficult trick. However, it's worth noting that the brain does most of its rewiring in two key phases: infancy, and puberty. During those times, vast numbers of synapses are created and destroyed.

I propose a skill that involves a neuro-chemical balance, like riding a bicycle for your mind. When we are young, we cannot manage it, so our brain reacts in a way which protects us from the forces that will later be telekenesis for us. We put up walls against it, to maintain our own sanity. During puberty, those walls are shook by the rapid synapse generation, and during a short window, we have a chance to try to learn to ride that bicycle again. Miss the window, and we rebuild all those walls, and the human body lacks a 3rd major rewiring phase to try again.

A result of this would be that some may be able to learn at the wrong age, and some men may be able to do it. The brain is far too flexible to outright prevent it from occurring (although social structures might suppress it). This is a blessing in disguise -- trying to chemically define the exact circumstances where you can learn is difficult, but if you allow some slop in the process, you don't have to draw as hard of lines between can and can't. It also suggests that some rare individuals might even retain the ability from their infancy, never quite building up the neural walls to protect themselves from the strangeness that empowers the telekinetic forces.

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genetics is a good start. A certain mix of genes together determines whether the body has the means to perform magic (and one of those genes is recessive on the X chromosome).
Diet may have something to do with it as well. Maybe the planet grows a plant which contains a chemical that when consumed suppresses magical ability (or conversely, enhances it). You can play with that in several scenarios, see what happens.
For example a person needs to be genetically inclined towards magic while also having a rare genetic disorder making her allergic to that plant so she can't eat it (while it is staple food, and thus consumed by the vast majority of the population on a near daily basis).
But even if the latent power is there, training is also needed to unlock it, and a long standing taboo on performing magic in many rural communities (where the genetic makeup needed is most prevalent) makes practicing magic something most people will never consider.
Your recruiters will have to scour the land, visiting villages and looking over the young girls for signs (who knows what those are, maybe the rare allergy leaves them with discoloured eyes).

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You can explain it in the same way people explain why one person has a higher IQ than another, and why another thinks faster than another. It is related to genetics.

Just like genetics can determine whether a person is color-blind, red-green color-blind, "normal", or can actually see more colors than the average person (like this woman), genetics can determine what powers a person can tap into.

Then consider that we only use a small percentage of our brains. Those with magic would simply be more intelligent and be able to use more of their brain power, potentially unlocking abilities that normal humans can't do.

Then you add training on top of that, and you can make someone truly magical. For example, some monks have so much control of their body, they can control their blood pressure, body temperature and heart rate through thought alone. Take this to the next level, and people can start controlling additional things with their mind, even external things.

Then you can add in the metaphysical. Some say that we shape our world with our words and thoughts. Someone who is powerful with their words and mind can will things to happen, and they happen.

So you could explain that a certain level of intelligence is needed to fully take advantage of the power of the mind, and this power of the mind is often looked upon as magic by the rest of the population. Their mind is so strong that they can literally will into existence what they want, including the ability to fly on a broomstick.

Puberty might be an important cut off point, because before puberty, humans are usually dependent and curious. Some studies even suggest that children ages 0 to 7 are actually in a hypnotic/dream state where dreams and reality are mixed, and they are very open to suggestion and training. As they reach puberty, they start becoming more independent, thinking for themselves instead of absorbing everything around them. This transition from a being that focuses on absorbing information & learning to one that thinks for itself and thinks independently could lend itself to puberty being the time of selection and screening.

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    $\begingroup$ «consider that we only use a small percentage of our brains» no, that’s not true. $\endgroup$ – JDługosz Oct 14 '16 at 18:00
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Consider the option that the powers develop instead of being granted.

A granted power is one that is given by some extrapersonal force, such as from divinity, genetic mutation, radioactive insect bite, etc.

A developed power is earned through effort or understanding. Examples include: studying a magic book to learn it's secrets, or reaching a higher state of consciousness.

Magic could be a latent ability which only manifests when an adolescent possesses a receptive mental state. Perhaps some emotions inhibits people from awakening their latent power. Thoughts of aggression or glory keep males bound to the confines of physical laws. Emotions such as depression, excessive worry, and anxiety are obstructions which prevent most adolescent females from awakening. After adolescence the person is firmly planted in what they know to be reality and that adult mentality prevents them from accessing their hidden power. Then there are the few of the right age, mind set and a questioning nature (or any virtue the author decides) who break away from the mundane and learn to manipulate reality

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