students with solutions

Problem of calissons

Couple years back I met a nice math problem, which is mainly know as the problem of calissons. Given a regular hexagon with side of length n. You can fill it with rhombuses, where every rhombus is made by gluing together two equilateral triangles. Here is an example:   

As you can see the rhombuses have three different orientations. 

Theorem 1: The number of rhombuses with a given orientation is one third of the number of the rhombuses.  

It is not hard to see that we will always use $3n^2$ rhombuses to fill the hexagon. So the theorem says that we will always have $n^2$ rhombuses of each orientation. 

Proof No. 1 

This problem became known it has a nice solution. It is a “Proof Without Words”. Just look at the following picture.  

We colored the rhombuses according to their orientation. If you try to look at them in 3D, you will see some cubes, and it is easy to feel the truth of the theorem. This proof is not too rigorous though.

When I was researching this topic I found two really interesting notes on the topic from Dijkstra (who is known for Dijkstra’s algorithm). In these notes he shows that this proof obscures a more general theorem, where we need an other proof.   

If we have a figure that is covered by rhombuses with $a$ of them in one orientation, $b$ of them in an other orientation and $c$ of them in the third orientation, then we will say that $(a,b,c)$ is the frequency of the covering. So Theorem 1 says that no mater how we cover a hexagon, the frequency is the same.    

Theorem 2: If we can cover any kind of figure with the rhombuses, then the frequency is independent of the covering.

For example we the following figure always needs 8 rhombuses of each orientation:

Proof of Theorem 2.

I will show of the proof of the second theorem using the original hexagon. First divide the figure into triangles. Color the triangles black and white such that neighbors have different colors. Assume we have a covering. This way every rhombus covers one black and one white triangle. In each rhombus we can draw an arrow from the middle of the black triangle to the middle of the white triangle. For example this is what we get from the covering above. 

  And now we must use a bit more advanced math. Consider the arrows as vectors!  We will consider the sum of these vectors in two different way. 

We have three different vector, one for each orientation, denote them by $v_1,v_2$ and $v_3$. Let $(a,b,c)$ be the frequency of the covering. This means we have $a$ of the $v_1$ vector, $v$ of the $v_3$ vector and $c$ of the $v_3$ vector. 

We can now calculate the sum of these vectors: $av_1+bv_2+cv_3$. 

 We need an other way to consider this sum. The important thing here is to show that the sum does not depend on the covering. Each vector goes from a black triangle to a white triangle. This means that we can write each vector as a difference of two vectors, one pointing from zero to the center of the white triangle minus one pointing to the center of the black triangle. If we consider the sum of all vectors now, we can write everything as a difference. So the sum of the vector is just the sum of vectors pointing to the white triangles minus the sum of vectors pointing to the black triangles. This quantity does not depend on the covering, just on the figure. We will call this vector $v_{sum}$.

So we have  $av_1+bv_2+cv_3=v_{sum}$. Also $a+b+c$ is a fixed number since the number of all rhombuses can be calculated from the area of the figure. 

Since we are in two dimensions, the equation  $av_1+bv_2+cv_3=v_{sum}$ is actually two equations, one for the x-coordinates and one for the y-coordinates. 

So all together we have three equations for $a,b$ and $c$ that has to be true independently of the covering. From linear algebra we know that three equation uniquely determines three variables. So $(a,b,c)$ is independent of the covering. QED 

Proof  No. 2.

Once I posted the original problem in a math competition. Most of the students gave the following solution. We can draw $n$ paths from one side of the hexagon to the opposite site using two orientations. 

In each path we must have $2n$ rhombuses. All rhombuses outside the paths belong to the third orientation. So we have $3n^2-2n\cdot n=n^2$ rhombuses in the third orientation. By symmetric arguments we have $n^2$ rhombuses in each orientation.         

Oh and my logo was inspired by all of this :)    

The Signs as Art School Stereotypes

Libra: Sweet Art Education student who seem traumatized by the weirdo studio majors, always smiling, super long hair

Pisces: New Age, Eccentric Sculpture student who goes to music festivals every weekend, makes projects about the Illuminati and NWO, more #woke than you will ever be

Aquarius: Reserved Drawing/Illustration Major who draws hella good anime, never opens their mouth in critique until forced, vegan, fandom tees

Leo: Hipster painting student who thinks they’re the absolute shit, brags about smoking hookah and everything they do is meant to be ironic

Scorpio: Irritated Photography student who hates being in class but is always there, favorite color is black, always looks like they need a nap, opinionated

Aries: Yuppie Visual Communications student who brags about their love of craft beer, blood made of coffee, has really strong opinions about fonts, needs to relax

Virgo: Chill Ceramics student who smokes a lot of weed, thinks they suck but they really slay, succulents and cable knit sweaters

Capricorn: Super skilled Furniture Design student who can find a solution to anything, wears flannels and steel toed boots and it actually looks natural, works 109 hours a week

Cancer: Friendly but reserved Art Therapy student who is quietly reading everyone like a book, internal dialogue is constantly sassy, pastel hair, always on the verge of mental breakdown

Taurus: Edgy Printmaking student who shops at stores no one has heard of, small clique, loves to read, super detail oriented, doesn’t accept criticism well, makes everyone uncomfortable by bickering with professor

Gemini: General Fine Arts Student who floats from clique to clique seamlessly, indecisive, friendly but values alone time, inquisitive, listens to Nirvana and also Disney Soundtracks

Sagittarius: Headstrong Double Major who can’t keep their thoughts straight, skips class a lot, obsessed with Tim Burton movies, smokes cigarettes for aesthetic, talks too much during critique


22/6/2017: Queen Rania visited Jerash and commended the Jordan Association for Human Development for its innovative community-empowerment projects and achievements.

Her Majesty toured the local community bazaar and met with the Assocation’s Board to learn about its interventions and projects focused on capacity-building in the local community of Jerash. 

Among the projects visited was the Social Innovation Labs, where students shared pioneering solutions and project ideas addressing the local community’s needs. The Queen also dropped by Makani activities, a UNICEF funded initiative offering innovatively-designed learning experiences. 

Also part of the visit was the Rural Economic Growth and Employment Project, which shared life skills trainings for Syrian and Jordanian, and a Child Safety session that involved awareness-building with an aim to spread knowledge and awareness on child safety as well as empower children to protect themselves through interactions with real life experiences and situations.

During a final stop, Her Majesty visited a training workshop conducted in collaboration with the Jordanian Economic Development Corporation where she praised the training program conducted for a group of farmers to build their capacities and skills related to building micro-business, financial literacy, connecting to financial services, credit, deposit, mobile wallets, and digital payments. 

Established in 2011, the Jordanians Association for Human Development has worked extensively on a number of programs focused on capacity building for women and youth, generating income-generating activities, early childhood development and primary health care services. (Source: Petra)

Me: Although I never got proper closure, I think I’m finally over that traumatic period of my life. Time to move on. 


How to be an Effective Tutor // by starPLATINUM

Hey guys, Star here. I’m a new studyblr, so you guys probably don’t know but I’ve been a math and physics tutor for two years now (ever since I started college!) and an engineering Statics tutor for one year. I’ve been told by a bunch of our freshmen that I’ve somewhat of a fan favorite, because I have a bunch of experience with the subjects I tutor and with tutoring in general.

Some of you bright students may be offered tutoring jobs at your schools/universities, so I thought I should offer some tips on how to tutor effectively. Alternatively, these tips apply to study groups, as well, if you have to catch any of your friends up with the material!

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What is the Nap Room Project?

The Nap Room Project is an initiative started by two two Marymount University students as a solution to combat colligate sleep deprivation, which can both impact academic performance and be incredibly dangerous. 

The solution is simple: a designated room for students to nap in. 

Many students cannot go back to their dorm or place of residency for a quick nap due to commuting, time, or a noisy dorm. 

The room would have a padded bedroll based flooring and a bring your own sleep aids (pillows, blankets, Teddy Bears, etc) policy. 

Vernon/OFC: Hey baby, I got my ion you

Genre: Fluff, university!AU

Word Count: 1572

Characters: Hansol Vernon Chwe/Original Female

Prompt: She gets a terrible (terribly cute) lab partner for the day. Cue terrible chemistry puns and back to school vibes. 

Lab was definitely not the class someone wanted to miss. Chemistry was one thing, but lab was an entirely different component that had as much weight to ones grade as any other class. Lab was instructional and hands-on, if you didn’t participate, you didn’t pass.

But she wanted to cry tears of joy when the professor said her lab partner emailed that he was sick and couldn’t come into class. Sure, Seungkwan was somewhat of a decent person, but he just talked and talked and got them in trouble more often than they received high marks. So here she was, waiting alone on her double-desk for further instruction. Once a majority of the students were in, teaching assistants passed out today’s itinerary. All of a sudden, a textbook banged against her table, causing her to jump in surprise.

“Guess we’re lab partners today.” Chwe Hansol smiled warmly, taking the seat next to hers. He wasn’t a Chemistry major, in fact he was one of the other sixty percent who was trying to get their science requirement over with as smoothly as possible.

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it must be chemistry

A/N: i didn’t even realise the pun until five seconds ago, so i’m rolling with it now and this fic thus has a really cheesy title. there you go

i may make this a chaptered fic?? i’m not sure?? let me know what you guys think please

basically a Snowbaz AU - i was trying to avoid doing homework while i was away so i decided to make up for it (kind of) (not really) by doing this instead. Snowbaz in high school, in which Simon is in on a scholarship for athletics and Baz is the tryhard valedictorian. Hope you guys like it!

disclaimer: i don’t actually know chemistry that well so if i got anything wrong i’m really sorry, and this is really unedited as well so forgive me for that!

“Simon Snow with Basilton Pitch-”

Miss Possibelf hasn’t even really finished her sentence, but Baz can already feel that all too familiar prickling on the back of his neck. He’s never believed that people can actually feel other people glaring at them, like it always says in the books, thinks it’s probably just them being paranoid, but he thinks that after five years of this always happening he might be starting to believe that they can.

He braces himself mentally – doesn’t show anything outwardly, of course, because he’s Baz Pitch and his goal is to be as emotionless and calm and collected as possible. His family all do it. They’re Pitches. They’re trained to be as still and cold as the marble walls they live within. Never lose your head, Baz, his dad always tells him. Emotions are the one weakness they can use mercilessly against you, the one weapon you’ll be utterly hopeless against.

And of course things have to be complicated, because obviously being from a well-known family and having your mother having been the previous headmistress of your boarding school before she slipped into a coma isn’t enough for one scrawny fifteen-year-old. No, he’s got to be bloody in love with his roommate, too. A roommate who hates him and squints suspiciously at him every chance he gets. A roommate who’s the scholarship student of the school and beloved by everyone (including the current headmaster. Of course.). A roommate whose name is Simon fucking Snow.

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When teachers use ableist slurs

Anonymous asked:

Today we did a spelling test in English,and when someone asked what question two was when we were on question four, the teacher shouted. “Special NEEDS!!”. Is this acceptable??!

I don’t mind teachers swearing at us,but this seems even more inappropriate.

Should I complain?

realsocialskills said:

I think there are two questions here which may have different answers:

  • Did the teacher do something significantly wrong? and
  • Should you complain?

So I’ll consider them separately. The first question is easy. The teacher definitely did something wrong. Several things, actually.

The first thing they did wrong was insult a student who was asking a question. Teachers should encourage questions. It was entirely reasonable for the student to want to have questions they’d missed repeated. Spelling, writing, and paying attention are hard for some people, and a moment of difficulty or inattention shouldn’t mean that you’re not allowed to ask what the question was. It’s really unfair to mark students as not knowing the material when the problem was actually that you refused to make the test accessible to them. That would have been wrong no matter how the teacher chose to insult the student.

It’s especially wrong that the teacher chose to use the insult they used. When they said “special NEEDS!”, they were expressing contempt for students with learning disabilities and learning difficulties. They were also threatening students by implying that if they show disability related struggles, they won’t be seen as having a legitimate place in the class. That’s a horrible kind of sentiment.

They were also showing any students with disabilities who may have been in the room that this teacher is not a safe person to discuss disability-related struggles with. That’s awful, too.

What the teacher said was mean and hateful. Teachers ought to be building their students up, not tearing them down. Teachers ought to be teaching their students to be respectful of everyone, not participating in a culture of ableist hate. Teachers ought to be actively showing their students that they will find solutions that make it possible for them to learn; not insulting them for asking for help. They ought to be actively seeking out effective accessibility and accommodations; not mocking special needs.

The second question is more complicated, and I’m not sure I know the answer to it. It depends on a lot of different things, and I think it is on some level a personal choice.

Some options:

Complaining to the teacher directly:

  • I wouldn’t necessarily recommend this in your situation, but:
  • Some teachers who say this kind of hurtful thing don’t understand the implications of what they’re saying
  • Sometimes when someone points it out to them, they listen and stop doing it
  • This is risky, especially if you are in grade school rather than university.
  • I wouldn’t recommend talking to this teacher about the problem directly unless you have a generally good relationship to them and have reason to believe that they’d care what you think and listen seriously

Talking to another teacher:

  • Is there another teacher you trust to understand why this was an awful thing to say?
  • If so, it might be worth talking to them and seeing what they think is the best way to proceed
  • (But be careful about this too - some teachers in this situation might not understand that you’re vulnerable and might repeat things or  pressure you to confront the mean teacher in ways that are not in your interests)

Talking to an authority figure:

  • I know that it can sometimes be done effectively, but I don’t know how to describe how to do it
  • One thing is that you can’t assume that they will understand why this is a big deal
  • But you can sometimes insist that it is a big deal
  • It helps to be as polite as possible in every way aside from the fact that you’re pushing the issue
  • (Eg: It is helpful to refrain from shouting or swearing, dressing in a way that’s against the rules, or doing anything else they can claim is a discipline problem)
  • It also helps to be pushing for a specific solution. If there’s a built in thing they can do that would get you to stop bothering them, they’re much more likely to do something
  • (Figuring out what to ask for can be complicated. What do you want? Do you want the head teacher to tell your teacher that they can’t say things like that? Do you want a general memo going out about why you can’t say things like that? Do you want to put a letter of complaint in their file? Do you want to to be transferred into a different English class? You might be able to get one of those things to happen if you push in the right ways.)

Involving your parents:

  • If your parents are supportive and understand why this is a big deal, it might be worth talking to them about ways they might help you with this
  • Sometimes teachers and administrators who don’t listen to teenagers do listen to their parents
  • Parents can also sometimes be anti-helpful, so I don’t know whether this is a good idea or a bad idea for you. You’re the best judge of that.

Talking to other students:

  • You might be in a position to influence and/or support other students here.
  • Do you think other students think this was wrong? 
  • Do you think they know that you think it was wrong?
  • Knowing that someone else thinks it was wrong can make a huge difference to people who are vulnerable
  • There’s probably at least one other student who you could support in this way
  • (Possibly discreetly, like talk to a particular person alone at lunch and say something like: Hey, did you hear what Ms. Meanteacher said to Rina the other day during the spelling test? That was so mean/ableist! Why do teachers think that’s ok?“ Or "Why is Mr. Meanteacher always insulting us?”)

Beyond that, I’m not sure what to suggest. Do any of y'all have ideas about what might be effective in this situation? (Answers from people who are familiar with the education system in the UK would be particularly helpful.)

anonymous asked:

new doctor!jongin fics pleaseee

There’s not many recent doc au sorry ;; so here’s around 3-4 newer and what hasn’t been recced before. Make sure to check the masterlist for more doctor au though ^^

Enjoy ^^ 

- Admin J

Sar(ah)castic: Old questions & Modern Medicine

On the Sar(ah)castic Series I’m ironically writing about topics that busy myself. Don’t take them too serious … or maybe you should? Enjoy ;)

At our med school we have to take an annual exam which comprises all lectures & topics we were thought by profs in two semesters. We are all very busy during the school year & when suddenly the summer months arrive, we get stressed like hell: HOW SHOULD I MANAGE TO STUDY THIS AMOUNT?

The majority of my fellow students see a solution in old exam questions. They pass the exam, celebrate the semester ending & forget their medical “knowledge”.  Isn’t it sad? 

When I’m talking to the “grown up” doctors they tell me stories about the frightening Pathology Exam or the “Bone’s Coll” … This generation actually had to learn facts & terms and demonstrate their knowledge under the watchful eyes of the Profs. 

And us? We’re just sitting in front of old questions, trying to combine a “Finasterid” with a “5-alpha-reductase inhibitor” … well, let’s get to next old exam question


i want to understand.

Old questions may be the easiest way, but in my opinion not the right one. They may save your butt to pass the exam, but they won’t save your patient.