My agent just told me that my visa has been approved. Yes, coming back to Vietnam.   But yeah, I'm about to go to sort out the COVID test So I'm doing final checks and check out what I find in my fridge Look how many packets of ketchup chili sauce and all the sauces. Oh my god. They must about 60 packets there.  I'll just leave it for the landlord. He can sort it out The day has arrived. I'm leaving Thailand. Whoo. I'm all packed. Got some boxes here Some bags here.  Oh my god This is gonna take a couple of trips to even get downstairs, but this is it.  I have a lot Perfect. Thank you. Quick as possible. Yeah. Yeah. Yeah field to van up Yes, all the things would be loaded apart from my bag. Yeah. Yeah. Yeah So I just checked in I was actually 15 kgs over But luckily the staff at Singapore Airlines allowed me to wave off the charge, which was great saved me some money The airport is relatively dead as you would expect It's got about two hours until my flight comes because I decided to check in three hours early Was that a good idea? Go to Singapore for this 12 hour layover Hopefully I can find a hotel have a good sleep and then Vietnam tomorrow the return of the prodigal son Yeah Good name for the video I've been at Singapore Airport for about confined to this area I thought originally that I could book a hotel But that's not happening and everybody in this terminal was just sleeping everywhere. I Got lucky and actually found this couch. I even saw someone pitch a tent, which is pretty cool Should have been more prepared with surviving off vending machines. I Got here at around about 9 30 and my flight is at 9 30 in the morning So there's 12 hours to kill.  Luckily? There's a smoking area Covid traveling is weird. Anyway, I think I'm gonna call it a night Should try to get some sleep Made it after more than 24 hours on that journey.  It's like he tried not to speak to people He just kind of waves you tells you to sit down because I've been in Thailand so long the mannerisms for him to just Wave me and point for me to sit down. I found that actually quite rude But I've been away for about a year and a half. So maybe I'll just need to get used to it Whatever no points there for customer service But I like how they fast-track you just sit down I'll do the paperwork and then I get you out.  They stamp your passport. It's pretty damn quick Good thing is they've already loaded all your bags off the belt onto the floor go to another line You turn the what hotel that you're staying at show them the paperwork your driver should be waiting for you there you put on your blue sort of hazmat suit shout hazmat and And yeah, they rush you on to the bus then you go into the hotel then you're gone Nothing is complained about because now I'm here 14 15 days 15 days here Shit, and now we relax. Not bad. Is it?


This study investigated the use of local fly ash from Colombia for the production of ultra-high performance concrete, UHPC. A numerical optimization process was employed to determine the optimal mix ratio for the desired properties of flow and compressive strength. The results indicated that the use of local fly ash could result in compressive strengths of up to 150 MPa without any heat treatment. This article was authored by Joaquina Belengarcia, Nancy Torres-Castellanos, Jamie Antonio Fernandez-Gomez and others.


It's looking to the east, looking to Japan with Steve Zercher, who is a professor at Kanzai, Gadae, did I get that right, University in Kobe, Japan, and he is also a member of the faculty at the Shidler College of Business here in Hawaii, and he's an international person. He's been an international person for many years, and he can appreciate the exchange of students from the point of view of an educator between the U.S. and Japan. So today, we're going to talk about trends in international study. I guess that means exchange student study for the U.S. in Japan and for Japan in the U.S. We're calling it Japan Yes, USA No. Hi, Steve. Good morning and good afternoon. Thank you so much, Dave, for taking time to talk with you once again. This is a subject that's very near and dear to me. I originally came to Japan as an exchange to myself. I did that for one year, my sophomore year in college, and that profoundly changed the direction of the rest of my life, the arc of my life. So I believe in international exchange. I think I tell my students in Hawaii and also here the ones I understand that the fact that they're on an exchange program, they're spending the semester or two abroad is the most important thing that they can do during their undergraduate college years, but that's certainly what's true in my case. So now, as you pointed out, I'm a professor and also a dean at Kansai Gaida University, and part of my management area is watching over the exchange program at Kansai Gaida. And of course, I watch closely the trend in terms of international exchange for foreign students. This is a big business for America. It's a huge business for America. I think your listeners and certainly you know, Jay, a large portion of the students who are studying at American universities are not native born Americans. And for the university, it's one of the primary revenue courses for the university because oftentimes, local students are on scholarship or there's state regulations that allow them to go to school at reduced rates and so forth. But foreign students, they come in, you know, they cannot apply for any of that and they're not likely to get scholarship. Well, usually they pay full tuition when they're studying. Like my wife went to Berkeley, she's Japanese. It's the first year she was a non-resident. I think it was like $35,000 a year back at that time. Of course, it's much, much more. And the local people, California residents were paying probably $15,000 or $20,000, so much, much less. So this is a big business. Well, there's a difference too, is there the attitude that students have when they come over from another country. You know, the U.S. is a big deal. They've been told and they've fully integrated the idea that you can get a good education here. And they're like immigrants and immigrants are motivated and they're more motivated. They're not here to play sports necessarily. They're not here to go to fraternity parties and meet girls. They're here to work hard. They're here to get a really good education that best the U.S. can offer. And as a result, most of them seem to me anyway, you can correct me, are really good students and they get good grades and they learn more perhaps than the average American student in the same school. Yeah, I agree with that, Jay. I have to say that there is an exception that's like $20,000. When it comes to my own experience, looking for girls was an important part of my foreign exchange experience. But anyway, in general, I think you're absolutely right. And that's just one of the students that I have at Consigai there. They're from all over the world. And Disney professors that come in, they were Mark and I have the same impression that these students, they're kind of self-selecting students that go abroad, are generally more motivated, seem to be more on the ball, maybe a little bit smarter, more kind of aware of what's going on in the grander scheme of things. And as a result, they end up being better students. So that's my experience, my direct experience each year over the last eight years. I really enjoy teaching foreign exchange to them. So does it work the same way? Does it work the same way in reverse? How do you find that American students who come to Japan, how do they compete with the Japanese students in the same school? Again, I think it applies. The American students that come are self-selecting. They're a little bit more courageous than the standard American students. It's still, if you look at the overall population of college students in America, the number of students that go abroad is about 10% through the course of four years. And this is the short-term trips as well. So it really is a minority of the general American students that do go abroad. And I find that the American students that are studied here in Japan, accounts like I did, to be good students and open-minded, curious, intellectually skilled in many respects and generally higher performers when I compare them to the domestic Japanese students who are sometimes in my classes as well. Well, let me make a short rant, if you don't mind. My rant is this. I've been suggesting to the parents of high school graduates or college graduates here in Hawaii for years and years and years that the best thing they can do is urge their kid to go to Asia. That means Japan or China learn the language in those countries and become a citizen of the world. You know, have the special globalist kind of education you can get if you study in another country. And I wouldn't put down studying in Europe either, but I think it's really important for somebody in Hawaii who has so much contact with Asia simply because the way Hawaii is set up to go to Asia. But what I find over the past few years anyway, it seems to be not happening. Despite my advice to all these people, their kids do not go to Asia. Their kids do not go to school in Asia. And they wind up going to the mainland. They wind up, I don't know where they wind up, but not in Hawaii and not in Asia. And I feel that's really a loss. I don't think we encourage the kids here in Hawaii to do that enough. Maybe we don't express the value of doing that to them and they don't incorporate that value in their own planning. So my rant is I would like to see more kids from Hawaii study in Asia. And since China is having human rights issues and oppression issues lately and surveillance and what not, social quota issues, probably a better place right now would be Japan. So if some parent came to me right now and asked me what should he advise his kid, I would say go to Japan. But it's not happening, is it? Couple of things, Jay. It's as usual an excellent question, excellent observation. I have the data here from the open doors organization. They study international exchange and every year they produce the report on the trends in international exchange that applies to U.S. universities and your students. So to your point about where Americans are going, if I look at the top 10 destinations by country, China is number seven and Japan is number 10. So if they do go abroad, American students, number one country is United Kingdom, about 11%, 11.5% in Italy, Spain, France, Germany, Ireland, so the top six are all European countries. And I agree with you. I'm sure they have a good time, but if you think about the future of world economics and where the growth is, it's here. It's all in Asia. So the students who decide to go to Asia are investing more broadly, I guess I should say, in their future. Europe is not going to be growing. UK is having tremendous problems, but yet students go there. It's probably Jay because they think the part of the challenge you're going to Asia is that psychologically, the greater distance in their head and then also they think, oh, I have to study Chinese and I have to study Japanese. So that's a part of the issue. And then to your point about going to Japan, the good news is that Japan is cool. American students just seem to have an interest in it over the last three years, according to this report, the number of American students year over year has increased three years ago, it increased 18%. And then last year, 2018, it increased 5%. And this year, according to this recent report, Japan number of Americans increased year over year 12.4%. So the raw numbers total still, yeah, that's good. It does seem to be a preferred destination among Americans in terms of the growth rate. The raw numbers are still far below the day. The number of Americans in the country is about 8,500 last year. And the United Kingdom is 40,000, so it's five times higher. Well, you know, one thing that I have noticed is that if you go to say Harrow or Cambridge in the United Kingdom, you are not only getting a good education, I believe they give good education there, but you're rubbing shoulders with people who will be your friends for life. And it's not just social. Those people are going to go into business, who knows what, foreign service could be anything, and you will know them, and they will know you, and you will have deep relationships with people far away. It's the same thing in Asia, I think. And I might add one other point is that here in Hawaii, there's a thing called the Asia Pacific Center for Security Studies, which is run by DOD. It's a Dan Inouye project in Waikiki right across from the military hotel in Waikiki and Darusey, Fort Darusey. And what happens is they invite and pay for various management people, both business and political and military, to come from their respective countries in Asia and go to school right there, Fort Darusey and the APCSS. And they house them in condos that are in Waikiki, and for months they get to live with each other, people from different countries. And so by the time it's over, these people know each other from the classroom setting and from living in a condo together, spending a lot of time together, and they're friends for life, and they're from different countries. And so you have a built-in diplomatic connection and business connection between these various people from various countries. This is invaluable to them and to the United States because they like being here, they like the United States. So the result is that everyone involved in these programs is a citizen diplomat for life. And that is a very valuable asset to have. And I would certainly like to see that expanded. But when you have isolationist government in Washington who wants to pull back on things like that, then you lose the benefit of this citizen diplomat result. What do you think about all of that? Yeah, yes. So just a quick example of how true this is. I went to Consigai Day as a foreign exchange student. I mentioned my sophomore year, and I went back to the states and completed my studies. And then out of the blue, this is one of my friends, one of my colleagues, a student from America as well who's on the space program with me, that I'm starting a spa for business in San Francisco. And would you like to join me? And I didn't even know what the software was when he called me, but I said yes, I ended up working there. And that's how I created my career in the software industry, was to do that connection that I had made while being a foreign exchange student in Japan. So that's how it worked out for me, because again, why I'm such an advocate of this, you're absolutely right. We often have the consul generals from Australia and the United States come and give presentations to our students at the closing ceremony, or when we start the academic year. And they emphasize the same thing. Don't represent your country. You have a wonderful opportunity here. Make connections, and these connections will be valuable to you over your lifetime, no matter what you do, whether you go into pharmacy or business or teaching or whatever it may be. And that's absolutely correct. But the tagline for this session today was, and yes, so we see that the numbers are up for American students going to Japan. Unfortunately, in the other direction, that is Japanese students going to America, and actually more general foreign students going to America, this report sadly indicates that the numbers are down. So the year on year for Japanese students, the number of students who went abroad to America was reduced by 3.5% year over year. And then overall, international students going to America was slightly down about 1%. The trend over the last couple of years is a negative trend. So the NASA organization, which is the trade conference, the organization that supports international exchange, is indicating that this is having a significant material, financial effect on universities in America that they're losing approximately $5 billion because of a drop-off in international students going to America. So why is that happening? I mean, I know that's so when you have a school, HPU here downtown, HPU relies on foreign students. And if their visas are not forthcoming, if the immigration service is sitting on their application of a visa or turning them down, that has a direct effect on the school that relies on foreign students. Is that happening around the country? Yes. So the numbers are the last academic year. The number of students coming from all sources to study in America is down approximately 1%. And then compared to 2015-16, which was the peak, it's down 10%. So yes, US universities are suffering now because the number of international students going to study there is down 10% from a few years ago. Now, that 2015 year, what happened then, Jay? Ah, we know what happened, Jay. You know, that's one indication. Yeah, exactly. The message out of the United States now is not a welcoming test. You know, we're going to make this harder, use the process to support difficult now. And students who are in America often are under the bed, or they feel that their visa may be revoked as well for various reasons. And this happens to some foreign students. Of course, that gets out into the broader news. But that's one thing that's not the only thing. There are other factors. The cost of education in America is bureaucracy. So American education now is the most expensive in the world. So parents who have to pay for those out-of-state tuition are going, oh my god, you know, I have to pay $50,000 in America. And if I send my students to Canada, it's only going to be 20, or I send them to Australia, it's only going to be 20. Oh, by the way, the number of international students going to other countries is a tag rock. So the increase in the UK over this period of time is on 30, 40 percent. Australia is just booming. The Consul General for Australia, facing globally, whenever he sees me, he comes up to me and he says, thank you so much, Steve, for electing your current president, because it's making my education business boom. He just says that too. Thank you so much. Well, we shouldn't be thanking him because he's hurting us. He's hurting the schools. And, you know, my point earlier, he's hurting the relationships, the future relationships, the diplomatic and business relationships that our country has with the countries that would send students over here.  And all those students go somewhere else. And when you said, for example, that the Japan students go to the UK, I think some of them go to the UK because they can't get into the US. And the UK benefits, they have the money and the connections, and we lose in that same vein. Where else are they going? Aside from the UK, where are the Japanese students going? I don't have any information on that. Since this report is kind of American, that trick, I'd have to get that from the Japanese side. I know from my own interaction with the countryside guide, the majority of them still want to study in America, but Europe is the second location. Some want to study in South America as well, because we have some connections with sister schools there. So it's spread out. But I would say the majority of, and this maybe just because of the historic nature of our program, we started in the 1970s and they're really focused quite strongly on American universities. So within the student body that goes abroad from countryside guide, the preference still is for American school. But maybe I can look that up for you and find out. I'm just wondering also about the cost. You talked about the economics and I think that's really important. And it's remarkable how much more it costs to go to college now in the US on average than it costs back in the day. I went to Queen's College of the City University in New York City. It was a very good school and its graduates are all over the country doing well. And it cost me $12 a semester for tuition. $12. So now it costs $10,000. And I just wonder how, and then what happens is the students get student loans in the US to pay these extraordinary charges. And then now Trump is trying to, and then it was a program under Obama to forgive the student loans. But Trump is not respecting that. So then the students have to find a way to pay the loans back, which can be very, very difficult because of the amounts involved. I wonder if the student loan program taken together with the cost of American education is affecting the students who would come from Japan. Can Japanese students make student loans under the American student loan policy? Are they affected by this? I, generally the way it works in Japan, and I would imagine this would extend Japanese students who are studying abroad, the contract is that the parents pay for their children's undergraduate education. Japan doesn't have the same kind of student loan or deed program that America does. In fact, when I talk about state financing for students, Japanese administrators at my university don't know what I'm talking about. So probably the way it works is the child wants to study in America and they're not under some kind of exchange program. The parents are paying for that out of pocket. So the increase in the tuition at American universities would have a negative effect if the parents do the research. I mean, if the son is the daughter of state, I have to study in America because that's where, you know, I want to, then the parent will probably say, okay, at least that's my impression of countryside guidance. And we're kind of a slightly richer school, maybe average, not super rich, but higher than the middle class, I'd say. That's how it would work. So I think that the tuition increases in America is probably driving their look at alternatives. And as I mentioned, Australia, UK, those number of, their number of enrollments at those universities is going up. So some other factors to our prime. I just read this recently in the Chronicle of Higher Education, which is the Bible for us academics, in terms of what's going on in the university transit business. So, you know, every other month or so, we have a horrible news coming out in the United States that gets well reported here in Japan. So there's also a little bit of an element of fear about studying in the United States as opposed to Canada or Australia or other places that are more. So that's not a major factor, but I think that's a factor that's driving the numbers down for us for Japanese students to study in America. And that would include the gun control issue, too. I mean, security on the campus for the women and for kids who would be concerned about gun episodes, which do happen on American campuses as well. I think, yeah, that's unheard of here in Japan. We just, the gun laws here are very, very strict and that type of crime doesn't exist. But parents who are going to study the students, their children abroad in America, they do read or hear about this, because it does get reported when it happens. But unfortunately, you know, it happens on a periodic basis. Well, let me let me ask you this, Steve. I'd like to bring on. Go ahead. Go ahead, Jay. Well, I was, I was gonna, you know, you're an international person and you know both systems. In fact, you, you know, the international education in general. And you've done research on it and all that. And if I, if I was the president of the United States, or Congress assuming that Congress becomes functional again. And I asked you, Steve, what, what kind of legislation should we adopt here to regain, enhance the benefits of student exchange, say between the U.S. and Japan or other countries in Asia? What would you recommend that I do in order to take full advantage of the fact that people in Asia like to study in the U.S. but are being held back for one reason or another? What should I do? What legislation should I pass? What programs should I adopt? Yeah, I think there's two things. One is non-relative. One is messaging that you're welcome to be here. We want you here. So I think that's very important message to send out. That's not being sent out now. So that's, I think, encouraging students. And then through the legislative aspect of this, I think making it easier for the students to go through the visa process instead of making it more difficult and the government controls the levers on that without question would be helpful. And then beyond that, one of the reasons I think also why the number of applicants for two American universities is going down is because they think that when they finish their degree, it will be highly unlikely that they can work in America. So the H-1B visa program is being produced, is being made more difficult. I'm primed from the Bay Area. I grew up in Silicon Valley. The number of entrepreneurs there is about a third or so. There's the Chinese group and the Korean group without those immigrants. Silicon Valley would not be what it is today. Oh, sure.  Yes. So many examples. Yes. So you should be more welcoming to people who are educated and talented and do as we talked about before, have proven themselves by self-collecting and are willing to leave their home country be one of the 10% and take the risk to go to some other location and study. Those are the people we want to stay. So I think that would be a very important thing, Jay, when you become president, that you change. And that will encourage, I think, more applications in a much more positive environment at university level and much more growth down the road as you graduate and integrate themselves into DACA or wherever they want to be. You can count on that, Steve. And that's why we have to keep talking. We have to talk about this and other issues that affect the U.S. and Japan and Asia. And I look forward to our next discussion. It's always great to talk to you, Steve. Steve Zercher at Kansai Gadai University in Kobe, Japan. Thank you so much, Jay. An international educator and philosopher. Thank you so much.


Hello and welcome to a summary of what you need to know about the Lemon Orchard by Alex Lagouma. I'll explain the meaning of this story as it appears in stories of ourselves, the University of Cambridge International Examination's anthology of short stories in English. I'll begin with some context about the author of this story before explaining the plot in a nutshell. I'll explain the characters you should be aware of in the story, then our highlight important themes to consider when studying this text. Now bear in mind that we have a stories of ourselves course that goes into depth on all of these stories in the anthology so do make sure you sign up for our course. So let's get started. Now let's first begin with some context around Alex Lagouma. Now Alex Lagouma is a South African novelist who was born in 1924 and he died in 1985. He was the leader of the South African Colored People's Organization and from when he was a very young man, he took part in strikes organised to protest against working conditions of Africans and factories. For his efforts, he was banned from all political activity in South Africa as well as writing and he was kept under house arrest. He could not receive visitors or communicate with his friends and co-workers. Later on, he and his wife were kept under solitary confinement in prison and on their release they were returned to house arrest. Under a negotiated deal, he and his family were given permanent exit visas from South Africa and sent to London on permanent exile. After spending many years in London, he and his family moved to Cuba as their permanent representative of the African National Congress. Most of Lagouma's works were written and published in London as well as Cuba as he was not permitted to write whilst he was in South Africa. Now let's have a look at the plot of The Lament Orchard. Now this story is set in 1962 in South Africa where four men are walking down The Lament Orchard on a winter night. There are four white men and one African man. The men are a group consisting of one leader and three followers. One of the white people had a battery powered lantern and was in a cycle ahead of everyone else. The other three white people were apparently talking and treating the African man as a pulpit. The leader who had a gun kept him for viking, taunting and insulting this African man referred to as a coloured man in the story for being uncivilised with a minister of the church. It is said that this man abused a white minister which under apartheid construed a crime. Now this prisoner, this African man, this coloured man is an educated schoolteacher. He was a teacher at a school where the white men sent their children. The poor man was afraid of them but also had a stubbornness within him which was against them. The four men refer to his earlier crimes and even if some of the men are apprehensive about killing this man it's not difficult to imagine that the mock questioning will end in a death. The educated schoolteacher is a prey in the hands of these white predators and his body will simply turn up as another one of those unidentified dead bodies that dot South Africa. Now when it comes to the characters of this story the first is the coloured man, the African man. So he's woken up in the middle of the night bound and taken captive by four white men. He's a well educated teacher at a school in the white men's community who allegedly disrespected a white minister at their church and was whipped as punishment. In the story the men are leading him through a lemon orchard to give him a second whipping in secret. They view the coloured man and likely all non-white people in South Africa as savages and hope to drive him out of town by intimidating him through violence. Yet this African man is notably dignified and level-headed through all of this.  Now this coloured man only gives in to the leader's demands to refer to him as Bars Master when Andrus, another man in the group, punches him in the face and knocks him to the ground. The story, the lemon orchard, ends just as the white men stop in a clearing to whip the coloured man and it's implied that they're going to carry out the punishment as planned. Not much is known about the coloured man other than his job as a teacher. He only speaks once and the narrative only refers to him by his racial category. His character could thus be interpreted as an allegory for the broader experience of all non-white people under the South African apartheid system, which was essentially a period of legally enforced racial separation and racial segregation, as so many like him were dehumanised and brutalised during this time and also killed. The other character is the leader. So the leader of the group of four white men take the coloured man captive in the middle of the night to march him through the lemon orchard. The white men plan to whip him in secret as a punishment for disrespecting a white man to set their church and the leader walks at the back of the party despite being in charge and he carries a loaded shotgun.  Though all four of the white men accrue towards the coloured man, the leader is the most vicious of all.  The leader is also the most overtly racist of the bunch. He characterises African and multiracial people as barbarians, makes it clear that he won't tolerate any disrespect from non-white people and demands that the coloured men address him as Bas, which is South African term for master. The leader's behaviour thus exposes the illogic and baselessness of apartheid South Africa's racial hierarchy, since he's clearly far more barbaric towards the coloured man than the coloured man has been towards white people. Yet, he still views himself as inherently superior to non-white people purely on the basis of a skin colour. The other character is the man with the lantern, so he walks at the front of the group guiding them through the lemon orchard at night.  Yet, despite this, the man with the lantern is just as complicit in the group's violent abuse as the rest. At one point as they walk through the orchard, he hears a watchdog barking in the distance and comments that he would take care of a pet like that. Clearly then, the man with the lantern has the capacity to empathise and care for others. He simply chooses not to extend that courtesy towards the coloured man. His character thus demonstrates how discriminatory beliefs are often rooted in hypocrisy and then denial of one's moral conscience. The other character is Andres, so he's part of this group and he and one other of the men both carry whips whilst the leader carries a loaded shotgun. He commits the most severe act of violence in the story as he punches the coloured man in the face when he refuses to respond to the leader's racist provocations. His character quickly escalates from being a passive member of the group to an actor of violence demonstrating how a mob mentality like that of the white men can reinforce and heighten abusive behaviour. Now, when it comes to themes, the first is that of apartheid. So do bear in mind that apartheid was a comprehensive system of segregation, racial segregation that divided the people of South Africa along the basis of one single feature, which was the colour of their skin. The coloured people and the African people were the natives of the land. By cunningly fragmenting the ghettos where the coloured people could live, the whites prevented them from getting together. The whites also forbade people from different races into marrying and inside the African and coloured ghettos the conditions were really inhuman. There was not enough space or amenities which were very primitive and these conditions gave rise to a lot of violence. The other important theme is the alienation of African people. So society in South Africa was divided along racial lines. Every facility in the country was segregated with the best reserved for the use of the whites and the black African people had to make do with inferior services and amenities. They lived in specialised ghettos that separated them from their own people and free movement of people between ghettos was also not permitted. The other theme is the denial of freedom and rights. So the African man, the coloured man in this story, referred to derogatoryly as hot knot is taken away for questioning. Accused people did not have to be produced before magistrate and summary executions were the order of the day. Probably that's what also happened to the man in this story also. African people really had no rights that were protected by any rule of law during their apartheid era and most charges against them were just trumped up false charges. So that's all. If you found this video summary useful, do make sure you sign up for a stories of ourselves course which goes into more detail on all the different stories. Also check out our website which is www.fresredeachers.com where you can find plenty of other English revision worksheets, model answers and online courses covering all the major similar verses including edXL, AQA and IGCSE. Thank you so much for watching.


Thank you very much. It's a real pleasure to be here. I've had a lot of fun hanging out with some of you today Especially talking with some of the students. I want to tell you a bit about Some of my favorite math fun facts and to do that. I first have to tell you what a math fun fact is So what are math fun facts? I? started When I started teaching several years ago my lower division course courses such as a calculus course I would often start off with a five-minute math fun fact which could be Just an interesting idea for many different areas of mathematics just to show my students that math is actually pretty cool So what are they? Well, they're their ideas that often Involve a change of perspective so it'll change the way you think about something. It might generate some questions The idea is to arouse your curiosity and fascination with math and they should be fun often There's some element of surprise Or that you get from from hearing the fun fact and as I mentioned I've used them not not just in classes to wet wet people's appetite for mathematics, but also for project ideas So why do I do fun facts? Well, I do them just to show that math is actually more than just Dealing with numbers or dealing with just learning calculus I mean after all calculus is is a very useful subject and if you're a scientist, it's important to know but There isn't you don't get to see in calculus some of the very cool and neat things that have been developed more recently than calculus Calculus is 400 years old, right? So you sort of get a sense that it's all cut and dried and it's just about formulas if you just end your Mathematical career with calculus So I like to do that and I also Want to show my students that math is both a science and an art It's the science of patterns and the art of meaningfully engaging with those patterns And so hopefully you'll get some surprise insight or joy out of seeing some of these and So if you're interested I actually collected all my the fun facts many fun facts on a website you can find if you just Google math fun facts and I started this as a hobby back when the internet was still young If I were to do it today, I probably would be writing a blog But it's it's in the form of a website and it gets about a million hits a year If you don't have internet connection and you have an iPhone well, there's there's an app for that Okay So what I'm going to do is I'm going to show you some of my favorite fun facts I of course I love them all but I'll show you some of the ones that I think you might enjoy and along the way I'm also going to try to trace some of the The the the people we have to thank for some of these fun facts some of the contributions by different groups of people Now why is it hard to tell to give a talk like this? Well, one reason it's hard is that some of you know some of these fun facts already. Okay, and so if that's True for you then don't spoil the fun by shouting out the answer if you know it Let the people who haven't seen it before Enjoy the fun fact, but you can you can obviously play along and think along with me So I hope you won't spoil the fun for others. Okay, so let's let's just get started So I'm going to start with a very simple trick.  Let's say pick a number between one and ten Okay in your head everybody done that. Yes. Okay good now what I'd like you to do is To square the number Which means multiplied by itself. Yes, have you done that great Okay, and now I'd like you to from this new Quantity in your head. I'd like you to subtract your original number from this quantity Yes, good Okay, great now take this quantity that's in your head and divide by your original number Everybody done that yes Okay, great and Let's see You tell me your name Holly what's your favorite number? three I'd like everybody to add three to your To the number that's in your head right now. Yes okay, great and Let's see. What about you with the cap J. What's your favorite number bigger than ten? Well, it's just favorite number whatever Fifty oh, that's actually pretty big. Yeah, go ahead and add fifty to your the number in your head and blame it on him if you Can't do it Okay Okay, as everybody have that everybody should have some number in your head. Yes Okay, now what I want you to do is I want you to subtract your original number Got that Yes, okay great So now I'm gonna make a prediction that the number you're thinking of right now and if I get it right don't hold back your applause Is 52? Okay So this is I'm sure you've all how many people played a game like this before maybe back when you were kids Okay, now to be honest. I actually didn't prepare this trick I just made it up in my head as we went along and I'm gonna show you a trick I'm gonna show you how I did it. It's actually pretty simple It's based on a very simple idea. It's basically based on algebra, which all of you learned in in middle school So you guys start with a number right like Holly. What did you start with? Three okay, because it's your favorite number great. Well, I started with another number. I called it x Yes Oh, that's great.  You're all thinking of three You wouldn't have been impressed right But this is what we did and you can follow along here x what did I say? Square it then what did I say? Subtract your original number then what did I say divide by your original number? I mean if you take x squared minus x and you divide by x you get x minus one then let's add Holly's number Which is oh it was three right x plus two and then let's do J's Trick what did I what do you say to add 50? Then get x plus 52 and Then we said subtract your original number and what you ended up with was 52. Yes Nifty right, so this is actually a points out something really useful And that is even if you are learning something that you think is boring in some class You can turn that around into an opportunity to be creative right come up with a magic trick that uses this idea Right this is how a lot of these kinds of mental tricks work now. What did I write down here? I said thanks here. Well What did we just use we just used? Algebra like the beginnings of algebra and one of the people that we have to think of course There's there's been a long history of people Many cultures developing the basic ideas in algebra, but al-hwad is me who is a Persian mathematician From long ago Probably wrote what we would call the foundational book in algebra where where people start taking Objects are taking equations and subtracting the same thing from both sides and manipulating them as we might do today In fact, if you look at the name of the book, what do you see in the name of the book? Algebra algebra Algebra that's that's where actually where we get algebra So this is let's we can all thank al-hwad is me for algebra All right Great, that's the first fun fact. Let's let's give al-hwad is me a big. Thank you actually another cool fact is that This name when when transliterated into Latin actually is is pronounced algorithms me So it's where we get our name algorithm the word algorithm in in computer science Okay, great Here's a second fun fact What I'd like you to do is to look at these four quantities and try to decide which one you think is largest Okay, so a is the number of stars in the universe B is a number of seconds since the Big Bang C is the number of orderings of a deck of 52 cards and D is the number of ways to fill out an nc double a bracket okay, and Don't calculate just give your intuition. Okay, don't calculate. I know some of you can figure this some of these things out Just use your intuition. Okay Which one do you think is largest? So let's see by a show of hands. How many people think the largest quantity here is a B C D Okay, wow pretty evenly split although many there were actually probably fewer D's All right, well, let's see what the right answer is I'm gonna go from smallest to largest the smallest here in place number four is oh Yeah, I didn't even ask you what you think is smallest I saw what you thought was largest. Let's see. Which one do you think is smallest show of hands? How many people think the smallest number on this list is stars? seconds orderings and Number of ways to fill out an nc double a bracket. Okay, that's even more evenly divided. Okay, the smallest quantity on this list is B The number of seconds since the Big Bang Yes, believe it or not number of seconds and so as we I think we estimate basically the the universe is about four three point seven billion years and in seconds. That's about ten to the 17 a little less than ten to the 18 Okay, a number that's 18 digits long It turns out the next largest is the number of ways to fill out a bracket and that's about the same size But a little larger. That's about ten to the 18 just over. Okay, that's two to the 63 it turns out because there's 64 teams in the tournament ah Okay, and the the the second largest is the number of stars in the universe. That's about ten to the 23 Really? Okay, that's a number. That's 23 digits long. This is what scientists estimate and the number of orderings of a deck of 52 cards is of course That's number one But guess what it is a number. That's basically about ten to the 68 a number that's 68 digits long That's so large that if for every star in the universe there was a whole nother universe of stars That would be the square of ten to the 23. That's still smaller Then the number of ways to order a deck of 52 cards What this also means is that if you take a deck of cards and you shuffle it you produce another configuration Right a deck of cards. I suppose somebody was sitting around shuffling a deck once per second since the beginning of time You wouldn't even run through a tiny infinitesimal fraction of the number of orderings of a deck of 52 cards You know what this means This means that every time you shuffle a deck of cards You're making history That's the first time most likely that that configuration's ever been seen in the whole history of the universe Isn't that amazing? You are making history by card shuffling Making history by Card shuffling. Hey, that's that's awesome. And of course, who do we have to thank? Now what did I say this? What did I say this? Thank the Chinese What are we? Who are we thanking for what? Yes, well, I think I just put that because I'm sure lots of cultures have developed playing cards, but Probably the first instance that we we know of our Chinese playing cards, okay and Playing cards have actually motivated a lot of interesting mathematics because after all there's a lot of patterns to study Right and so it mixes in combinatorics probability And just a lot of basic questions you might have fun with understanding patterns Okay, great. That's the second fun fact. You like that fun fact.  And people watch football Yeah, okay, great. That's good enough Good enough. How long is a football field? It's undefined. Did somebody who's undefined? Okay, so let's see so a football field is a hundred yards. Yes, that's true It's a hundred yards and but let's suppose I include the the end zone. So there's two goalposts here. Yes So how long is it from goalpost to goalpost? 120 yards. Yes. Okay now suppose I run a a rope Across the field so that it's tight between the the the two poles. Yes, okay And now let's suppose that I add just one foot of slack That means a little an extra foot. Okay, so now it's a little loose. Yes Suppose then I pull The rope up at the center at the 50-yard line and ask what's going on there So now what's going to happen the the rope gets pulled up a little bit. Yes And I want to know how high does a rope get pulled up at the center of the field? Okay? Don't calculate just use your intuition. Okay, and you'll have four choices How many people think Here your four choices that when you pull up at the center that it'll it'll still be pretty tight And you can barely put your fingers under that's a the next one is you can crawl under that's B Next one is you can walk under and the the last one is you can drive a truck under. Okay, so vote for one of those Everybody vote. Okay, even if you don't just use your gut. Okay How many people think when you pull it up at the center by adding a foot to 120-yard long rope that it's Just enough to put your fingers under How people think you can crawl under Walk under Drive a truck under Okay, that's that that's probably The one that got the least number of votes or maybe the first one, but it turns out in fact Indeed you can drive a truck under What? it's actually gonna be about 13 feet high and How would you calculate it? Well, it's just a it's just a consequence of of what? It's just a consequence of the Pythagorean theorem and you can you can just calculate it at home and see See what happens. So we can thank Pythagoras for this Hmm.  Here's a little bit of intuition Here is my here's my rope at the center. Here's the center of the field. Here's the rope and I'm gonna lift up this end. Yes Suppose I add just an inch to my arm How far will I have to raise this? rope to get an inch Several inches. Yes Right, and of course if you use the Pythagorean theorem, which says that the sum of the squares of the two Short sides of a triangle Equals a set the square of the long side. You can you can check this out. Ah Pretty nifty. How many did you like that trick? How do people like that trick? Yes, awesome. That's that's that's a fun fact Okay, while we're on the top of Pythagorean theorem, this is a very of course very very old theorem dates back to antiquity Pythagoras was perhaps the first Known proof although that's another thing about some of these really old facts is that Many of them were discovered independently by many groups of people and of course we are living here in In the West and we're often influenced I mean we tend to know the things that were the the cultures that we're part of But it's it's quite likely this is discovered in many other contexts But the first thing we know about is Pythagoras Pythagorean proof proof let me show you but there are many proofs and let me show you just a couple of proofs of Pythagorean theorem because they're cool One of them was actually proved by somebody who's not professionally a mathematician And this was one of our former presidents James Garfield came up with this proof Okay Which goes to show you that people can do math for fun, right? And it's a very simple proof and it basically what what what Garfield did is he drew if you have a right triangle and The two side lengths are C. So these are the green triangles. They're the same right triangle They're just oriented so that a the if one side is called a you can't see that but I'll write it here And one side is called B Well, we'll put them down so that one of them has a on the bottom and one on the has B on the bottom. Yes Okay, and then you form the the the yellow triangle Okay, and that's basically got two side lengths C Okay All right, so this is how this is what Garfield's argument was this is kind of nifty Garfield said hey, let's just use the formula for the area of a trapezoid Now of course who knows the formula for the area of a trapezoid Well, our president did right Okay, and it's not hard to actually to to guess what it is Right you how many people here know the formula for the area of a rectangle Yes, hype times width, right? Well, would you agree that if I just Lengthen one of the ends and shorten one of the ends that the thing would have the same area So basically if I take the left height and the right height and average them a Rectangle with that height and the same width would have the same area So that's what we'll do. What's the average of A and B? Well, this whole thing is area of the trapezoid is One half a plus B. That's the average of the two heights times the width and what's the width here? a plus B Right. Oh, okay. Great.  That's the area of the trapezoid. Okay. Now. What's the area of the triangles? Well, the area of the triangles this is one half a B and this is one half a B. Yes Right. Okay. Great. So here's a question then. What is the area of the? Triang of the yellow triangle. Well, it's just the whole area minus the area minus the green area Yes So if I take this expression and I subtract the green area, would you agree? I'm taking this expression and subtracting a B because a half plus a half a B is a B. Yes. Oh Okay, great. So the area of this piece this final piece is simply This is one half a oops a squared plus B squared Because I'm subtracted off the this portion here with This portion here Yes, oh But what is another way to calculate the area of the triangle with two side length C? That's right Well, that's just also equivalently one half C squared so a squared plus B squared equals C squared if you just multiply everything by 2 Okay. Oh Pretty nifty. How many people like that proof like Garfield's proof? Yeah. Yeah, good. Okay. Some of you liked it Okay. Now this is not my favorite proof, but it's it's kind of nifty because it shows that that Even politicians can do math which you might not know today, but Okay, no, sorry. I'll just stop with the ragging on Okay, great Let me show you let me show you what what I consider to be my favorite proof my favorite Pythagorean proof And this one's going to go by really quickly So here's the idea Let's so of course we're looking at a triangle like this and I think we agree that that What I'm trying to verify is that the area of this which is a squared plus the area of this which is B squared is The same as the area of this Which is C squared? Right, would everybody agree with that if the side lengths are a b and c Okay, now What did you agree if I'm trying to show that a squared plus B squared equals C squared? That it's equivalent to Multiplying both sides by some constants, right? I mean you can't you can't stop me for multiplying both sides by the same thing Would you agree that if that if one is true the other is true? So it's enough for me to show that a multiple of a squared plus B squared equals a multiple of C squared Yes, you with me on this. Oh, but the multiple K is just a scaling Right, so I could change. I don't have to be looking at squares. I could be I could change the thing. I'm looking at So for instance instead of showing the sum of the areas of the two smaller squares is the area of the bigger square I could instead take some other object like. Oh, I don't know a I could take a smiley face right and I could show that the area of that smiley face is the sum of the areas of These two smiley faces Right if I wanted to right Any I could put any figure there I want Okay, good. So if you buy that well, I'm not gonna put that figure then here's the figure. I'm actually going to use I Will take my original triangle and Instead of using a smiley face or a square I will show you that if I take a copy of the original triangle a similar triangle It's actually the same triangle there and put a copy of the triangle on every side a similar triangle on every side This is not exactly the same Thing but there's a copy of the triangle oops Try again. This has got to be similar. Okay, and then here's another one Okay, those are basically similar copies of the triangle around all three sides. Yes All I have to verify is the sum of the areas of the two small triangles is the same as the area of the Big triangle. Yes, so here's how I'll do it. It's very simple all I'll do is Drop an altitude So that I have Cut the the green triangle to two pieces and I'm just going to fold over This triangle and fold over this triangle and you see they actually completely match when they fold over and they match To the same size as the triangle on the other side because that is that's just the big thing if you fold this over oops If you fold this over it'll cover the whole thing So the sum of those two triangles actually is the bigger triangle area Okay, cool how people thought that was cool Okay, you don't have to say that just to make me happy, but I think it's a little I think it's pretty nifty. Okay, great Let's move on. Oh Did you know that there is in fact a spherical Pythagorean theorem Spherical Pythagorean theorem really what does it mean to what does it mean to be a spherical Pythagorean theorem anybody have any ideas? We should be looking at what? Triangles yes, where on a sphere and one of the angles in the triangle better be a Right triangle so let's draw a picture Now you pick three points on a sphere now this the sides can't be straight straight So they're gonna be a little curved But you're just drawing the shortest the shortest path between two points on the sphere and they're gonna be slightly curved yes and To talk about an angle being right you mean that basically if you look really close at a corner It's gonna have a right angle. Yes everybody with me Now the weird thing about spherical triangles is That the sum of the angles don't necessarily add up to 180 in fact you could have a triangle where all the angles are right Right I go to the take the north pole go down to the equator move over 90 degrees and then come back up to the north pole All the angles there will be right But when we talk about right just pick one of them that happens to be 90 degrees and I will call the two sides next to the right triangle a and b and the third side C and Now the question is what's the relationship between a b and c and so here's a here's a relationship it turns out Turns out that the cosine of C equals the cosine of a Times the cosine of B Oh, I have to be really careful here. This is for a unit sphere. That is a sphere of radius one Or if you don't like that if you don't want to deal with a sphere of unit radius one This is for the case where if I have a sphere of radius R. I divide all these things by R Okay, so that this is radius R Interesting this is the spherical path I think it doesn't look anything like the regulars Spirit but that green theorem does it does it Well, in fact, it actually is so this is kind of nifty this is quite nifty and The Let me show you why this is really cool. So how many people have here had a little calculus just a little bit of calculus awesome Okay, well, so check out what happens here the first thing to notice is Would you agree that if my sphere got really big Radius gets big this thing gets flatter and flatter and flatter. Yes And so that triangle for this formula should change into the regular Pythagorean theorem if R goes to infinity. Yes, oh Okay, but if R goes to infinity, what's cosine? What's C over R doing if R goes to infinity C over R is going to something Really really tiny. Yes, okay, and if you take the cosine of something really really tiny Then what do you get? Very good. You get one but let's let's that's well So if that's true then one equals one times one and we knew that already So let's do a little take a little finer approximation Cosine of something really really tiny equals one plus another term minus another term Taylor series Cosine of X is one minus. What's the next term of the Taylor series? Well, if you calculate it's x squared over two it turns out, okay? Oh really? So if I take the next term in the Taylor series this thing just becomes one minus Approximately C squared over two R squared and we're saying that's equivalent to approximately one minus a squared over two R squared times one minus b squared over two R squared and When all is said and done if you start Canceling things and ignoring things which is what we often do when you work with approximations You'll find that C squared is approximately a squared plus B squared and viola the regular Pythagorean theorem 50 Who do we have to thank for this? Well, it turns out that we have a lot of astronomers who love mathematics a lot of astronomers Both in the Greek and Islamic world Astronomers to thank for for the origins of the spherical Pythagorean theorem you can find evidence of this in their writings Cool, how many people like that? Is that cool like that? Awesome. Great Here's a fun fact. I really love I love this fun fact. This is so awesome. So here's a question for you If I take a number like the square root of two and I square it, what do I get? Two Okay, no interesting So it's possible to take an irrational number and raise it to a rational power and get something rational Yes rational means written as a fraction Okay, so is here's a question is it possible to take something irrational and Raise it to an irrational and get something rational Is that possible? How many people think? No How many people think yes? Okay, hmm well Let me prove a theorem for you Which just means let me show you why it's true theorem. Yes, it can be done That's a great theorem just say yes Okay Here's an argument. This is one thing that's really clever. This is what's fun about this fact is not the result necessarily But how you prove this result? Because what's gonna happen is We're gonna show that this is true without showing you a specific example of why it's true What it's completely non-constructive and this is this is how it goes. This is really great. This is awesome So here we go. So here's a question. Let's look at How about let's just do the square root of two the square root of two Is it rational? Well, I don't know But if it is if yes We're done We found an irrational to an irrational that's rational correct. Yes So what's the other possibility? If no, then the square root of two to the square root of two is irrational. Would you agree with that? Okay, so you can't stop me From raising it to an irrational power and what I get is a square root of two to the Square root of two times the square root of two otherwise known as two and that's two and we're done How many people like that? That's awesome Nifty so either way we know it's true But we don't we we don't know whether root to the root to actually is rational Actually, some people do and what do you think the answer is? It turns out to be irrational Turns out to be irrational who do we have to thank for this? I have no idea This is one of the things that's passed down folklore and people just go ooh and ah and they share it It's kind of like sharing a joke on Twitter that you didn't write Okay. Oh, I love this one. This one's great This one's awesome if you have a piece of paper you can do this take a Take draw a triangle any triangle just make sure it's not special like make sure it's not a right triangle for instance Oh, that looks right. Yeah It's this actually really hard to draw something very general, but I'm gonna try okay. Here we go Here's a triangle and I'm gonna ask you to mark nine special points on on this triangle related to this triangle, okay? If you want to do this on your own sheet of paper you can I What I'll do is I'll ask you to mark the midpoints of each of each side. Yes That's one set of points Midpoints of every side. Okay, great Now the next thing I'm gonna ask you to do is to drop an altitude From each corner to the opposite side But what that means is pretend that that one side is the floor Drop something from the corner and see where it lands. Are you with me? So that's one that's that it's gonna be a right angle in this case Now if I imagine this side being the floor and I drop something from it It'll actually go to there approximately. Yes And if I drop something to that side from the other side, it'll actually Go here and it's actually a fact that all these altitudes across somewhere. They will cross Across somewhere which is not a point. I'm interested in right now But they happen to cross the three points. I want you to mark though are these points the points where they They land on the other side. So here's three more points Are you with me? That's six points Okay, now look at where the altitudes all cross That's called the orthocenter by the way We're at all crop where those cross and look at the point That's halfway between the orthocenter and the three corners. So in this case This point the point where they all cross halfway to each of the three sides is Is this point here's where there's one here There's one here and There's one here Those are nine points And if you've done this right if your things to scale by the time you're done those nine points all will lie on a perfect circle. Oh really, this is called the nine-point circle and For that we have to thank a German mathematician fairbac fairbac nine-point circle and This is early 1800s 1822 or so Nifty now, I know some of you drew an acute triangle or a obtuse triangle. It still works You just have to extend the edges the altitudes may not fall inside the triangle to fall outside the triangle Okay, but it still works. It's nifty Alright, how many people like this one like this one awesome great Great, we're on our way. Oh, this one's a great one. I love this one because it's all about friendship This is awesome. This is called the friendship theorem Friendship theorem says this suppose you go to a party and At the party suppose it's the case that every two people Have exactly one friend in common Okay, every two people have exactly one friend in common then something amazing is true if this is true then You read that. Oh, yeah, then it turns out that there will be There'll be one person who is friends with Everybody. Oh, yeah, really Yeah, so if you go to party where where any two people have exactly one friend in common There's one person who actually is friends with everybody and in fact what what what we know a little bit more It turns out that the graph of connections between people basically looks like a bunch of triangles that are connected at a Look that looks like this.  Well, I don't know I mean go to have go to a party and start asking people who they know it might be kind of creepy Okay, but who do we have to thank for this These are several famous Combinatorialists Paul Airdish some of you may have heard of one of the most prolific Mathematicians and people like to count how many paper connections they have with Paul Airdish How many people have heard of an Airdish number? Okay, that's Paul Airdish An Alfred Renye is Hungarian and Vera Sosh is Hungarian And if any of you have gone to Budapest you have an opportunity I had a student once studied with Vera Sosh and They've got a great Combinatorialist there who proved lots of really cool theorems the friendship theorem Awesome, great. Okay Let's do a few more here So here's a here's a fun fact if you take a bunch of cubes The cube of one plus two plus three plus all the cubes one cube plus two cubes Let's let's count that actually can we do that? What is four cubed? 64 what's three cubed and what's 64 plus 27? mmm 91 right plus eight 99 plus one 100 really? Oh, that's interesting if I just add one cube plus two cubed. What do I get? nine oh 100 nine what if I add one two cube cube three cubed 27 plus nine is 36 Something funny going on nine thirty six a hundred those seem really special In fact, what are they? They're they're all squares and in fact the sum of cubes in this case turns out to be the sum of the the numbers squared Ten squared is a hundred. Yeah, right Six squared is thirty six and two three squared is nine So this is a this is a cool fact. It's it's true in general okay, and If you want you can prove it by induction those of you who know what induction is and Of course, we have a couple of people to thank here and the reason I said we can thank them is big is because Well first of all first of all our Aria Bada is one of the first mentions of this of noting this but al-karaji is probably the first person to prove it and What al-karaji is also known for is for Developing the method of induction This is another Persian mathematician Okay, but I'm going to show you a completely different proof which is an induction It's just a it's a proof by picture and this is this is cool enough for me to do okay, so here we go I am going to Would you agree that if I? want to To do this let's see Would you agree that this one cubed is just the volume of a little cube of side length one? Okay, and two cubed is just the volume of a cube of side length two. Yes Yeah, and three cubes three cubes. Okay, you get the picture All right now. Oh just wait. This is so awesome. Look Okay, I feel compelled to do one more. Is that okay? Okay Perfect Okay now check this out I'm gonna show you why the if you if you stack all these cubes in a different way You'll see the right-hand side is the same why I'll just put everything in one layer And I'll just show you the top so there's the first cube, but I've shown you just the top Well, if I want I can break up the blue two by two into a Two by one a two by one. That's the first layer of four squares Yes, and each layer and the second layer goes right here and Would you agree this is one plus two squared? And if I want to do one plus two cubed I can just well I got to show you why one plus two plus three squared is the same. Yes So you can't stop me from adding Three more squares on each side and then filling out the rest with a two by one A three by three by two and a two by three and a three by three This my friends is layer one, which I'll label a This is layer two, which I'll label b. I'll have to I'll have to to use one of these Together with one of these and then c will be the third layer. Yeah That's one plus two plus three squared on that side, right? And now you can see what's going to happen in the last case. I'll just I'll just draw this out Um Just one more thing and if you like it you can you can clap at the end There we go two by four three by four four by four four by three four by two four by one And if you want to match these up a b c d However, how am I going to do it? I'll call that a I'll call that b and I'll match b up with this one I'll match c up with this one And I'll match d up with This one one plus two plus three plus four Squared same number of squares how many people like that awesome Yeah, because you get insight from uh from doing this uh doing a picture proof all right I think we just have a couple more a couple more here. So here this how do you know what a magic square is? Yeah magic square. It's a array a square. Will you fill in numbers? Such that every row column and two diagonals add up to what? The same number whatever it is. Okay, so here's an example of a magic square Here's one uh one two three four five six seven eight nine Oh, cool. How did you do that? That's so awesome. Well, I used a special trick I won't tell you what that trick is though But this is a magic square size three in every row column diagonal adds up to 15 okay now Magic squares have been around a long time celebrated studied in many cultures china persia some of the the in the year in europe and even in africa and the reason we know this is that We have the the writings of muhammad ibn muhammad from an area that's now nigeria It wasn't at the time and he came up with a construction a very interesting construction For doing a magic square and i'll show you what his construction does in the case of a five by five There are other methods. I just want to show you this one because it's kind of nifty So here's what he said Basically The rule is the following You start in it actually doesn't matter. We'll start in the upper right square And then we're going to take we'll do night's moves All in the same direction. So if I go one and then I go down to two That's a night's moves on it on a board chest board, right? And I'll then I'll go down to three. Yes And now what I have to imagine is this board wraps around So if I leave the bottom, I enter at the top and leave the side under the other side. Yes And so I'll keep going as far as I can and let's see if I go down Make sure I'm doing this right. This is four. I believe yes Down and over down over left one down two. Yes left one down two Now what happens here with five? Uh-oh I can't go down to the next one because it one's already filled. Yes So what what he says is then you do a special rule you move over two units And put six there Yes And then you just continue six goes down to seven, which is here oops wrong Seven seven goes down to eight, etc. And just keep going And every time you just remember the rule if you can't keep going move over to The nice thing about this rule is it produces a magic square Not only with all the rows and columns and both diagonals That add up to the same number, but all the diagonals with wrap around add up to the same number Kind of nifty Really neat contribution from an african mathematician And if that wasn't enough magic squares are so awesome Here's another thing that was only recently noted Uh about magic squares Three by three magic squares. So if I look at three by three magic squares check this out Something that's really nifty Is if you take this magic square and you just pretend that the first row is a number Like 816 Uh and square it. Why would you do that? Well because something amazing happens if you read the rows as numbers and square them Guess what when you what happens when you sum up these squares You get a million 35,369. Yay Is that fun how people think that's fun Okay, maybe that's not fun, but if you now take the rows and read them backwards As a number 618 squared plus 753 squared plus 294 squared And you sum them guess what you get Whoa say whoa if you didn't expect that whoa, okay great, but that's not all Guess what if you read the columns as numbers and square them and sum them one way And read them backwards up the columns sum them square them one way You actually get the same number in but not the same number as this one In other words the same property holds uh reading the columns it holds for the diagonals It's kind of amazing fact probably uh the first place that I know if it was noted was in a martin gardener book But uh an undergraduate actually asked the question is this true for all three by threes And uh in this uh undergraduate uh at harvey mud actually proved that it actually holds for all all three by three magic squares You just have to interpret if you have more than one digit you have to carry So if that middle number were an 11 instead of a one the first row would be read as 916 And in reverse it would be read as uh 6 718 Okay, um before I do my final fun fact I just want to point out another fun fact Which is this year what's sort of the noble prize of mathematics uh is uh went to um four mathematicians uh and It was the first time we actually awarded a uh noble prize to a woman First time to a mathematician from latin america and the first time to an indian mathematician And the reason this is significant is if you think a little bit about uh the history of of uh scholarship not just mathematics but scholarship in general There have been many groups for which it's been hard Uh for those groups to be able to study or to have an education right some of the early women mathematicians actually had to do Take extraordinary lengths to actually get educated right so women weren't admitted into into universities at one point And some of the famous women mathematicians actually Started off by using pseudonyms in their correspondence. So sophie germane did that with um gauss Just because because of the fear that they would not be received And so of course we've come a long way But it's taken a long time for us to actually start recognizing the the work of Of diverse groups of mathematicians and of course we we hope this trend continues All right last trick Last fun fact is a trick. It's a card trick And to do that i'm going to Switch to the document camera This is this is really cool. It's one of my favorite card tricks So here's a deck. Yes Looks like an ordinary deck Mixed but of course you don't believe me. So i'm going to have somebody shuffle the deck Who here can shuffle the deck a riffle shuffle? So i'm gonna do a riffle shuffle. Okay, you're a little hard to reach but Um, can one of you riffle shuffle? No, okay You can riffle shuffle. Okay. Good. So i'm going to give him the deck. Just give it one shuffle Yeah, you don't need to give it a flourish. Just shuffle with it. Yeah, very good. Thank you very much. All right. It's a new combination Yeah, that's right. He just made history Okay so Is there any possible way after you shuffle the deck that I could know the the ordering of this deck? No, I wasn't in cahoots with him. I'll pay you later So would you believe then that I was born with an amazing ability I was no Amazing ability to feel the colors of the cards with my hands Now it's actually not a very refined ability. I can't tell you what color this card is but I can pull them off the deck in In pairs Red black opposite colors. Okay, and so just to uh, so you can see that red black. Yes Red black red black Red black now, of course if I'm just doing this by random Randomly then the chances that I'm doing this is actually about one and two for each pair And as soon as I do more than eight of them, it's one in 500 512 to be exact Yes One in the 1024 chance that I got all those to be pairs of red black. Yes And if you like I could keep going Or you could just believe me. I have an amazing ability Yeah, and we could all go home No No Okay, well a magician never reveals his secrets But I'm not a magician so I will This is so awesome because this is a trick it's a mathematical card trick So you don't actually need you don't have to worry about it's not sleight of hand You're not going to mess it up unless you if you just remember the rules it'll work This is so awesome. Let me show you Okay, so clearly I stacked the deck beforehand. Yes Yes Okay, but I had to do so in such a way that no matter how he shuffled I would still know something about the deck and this brings up another lesson from mathematics because math is the science of patterns after all And what we're trying to do is understand patterns that remain after a riffle shuffle. Yes It turns out we know that a single shuffle a riffle shuffle is not enough to mix the deck In fact, it was proved in the early 90s that it takes seven shuffles to mix the deck well of 52 cards That was proved by persidae conus and uh But so yeah, so there is something that happens after one shuffle There's still order and so here's what I did before the trick started. I actually stacked the deck in a special way I stacked them Okay, so I stacked them so that they're alternating red black red black red black And this one actually still is quite Ordered like that. Okay. Now it's not but I stacked them red black. I'll just do a few here. Yes Okay, so that's what I did Now of course when you show people at the beginning They won't notice it because the the the cards clump, right? You should show them that it looks mixed then you invite somebody to to shuffle. Yes Now the this is the only part you have to remember if you forget this the trick won't work or may not work. Okay When you hand the deck to your friend To shuffle you have to cut the deck You have to remember to cut the deck so the colors on the bottom are different colors. Are you with me? If you remember that then the trick will work no matter how we riffle shuffles And I advise you when you ask him to riffle shuffle it To make sure you tell him to shuffle it once because if they start shuffling more than once you're in trouble. Okay Or you should have another deck prepared as I did Okay, so Remind me your name again John so John is going to drop cards. Yes, but how is he going to drop them? I don't know If they'll fall off either one side or the other right? Yes So let's watch underneath what's happening when he drops cards from either side of the deck So which of these two sides is going to fall first? I don't know. I'm not sure. Maybe it falls from this side, but if it falls from this side What color must the next card be no matter which side it falls from? Red So the next one will be red even if it falls from the same side Now what happens when that falls? What's the two colors? Black red and we're back to where we started which sides are going to fall from now I don't know. Maybe red But what's the next card have to be? Black And so no matter how he shuffles even if he starts falling a lot from the same side Or if it alternates it will be It'll come off now won't be red black anymore. Do you see here?  Yes And then the only thing you have to do to make sure people are confused about it is you don't show Good save You don't show them what you're doing You you you fish around to make it sound like you're you're picking cards randomly, but you're just picking them off the top Two by two by two Anyways, I hope you enjoyed that and uh, have a good night


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So do you guys want the full Quidditch gear for your character within Hogwarts legacy? Well today guys I have you covered. This Quidditch gear is tied to 20 landing platforms hidden across the entire map and today guys I'll showcase to you the location of each. Alright guys if you do enjoy this video or even if our crew helps out and if you like what you see and want to see more Hogwarts be sure to subscribe. Okay so there are 20 of these landing platforms scattered across the entire map. We're going to go from top to bottom from north to south showcasing to you everyone on the way. Also if there's a specific one you're looking for or a one in a certain area they're all timestamped down below. There's 20 in total like I said and these will give you the Quidditch captain's gear which is pretty cool looking if you're into that kind of thing. Now these you will have to claim from your challenges so don't forget to do that but hey guys if you enjoyed the video leaving a like who helps out if you like what you see and want to see more be sure to subscribe.  Lumos Rebellion!


Namaste. I am Dr. A. Sharda here. Namaste. I am Dr. Priya Chinappa. We are both from the Department of Endocrinology and Diabetes from Manipal Hospital, Millers Road. So, you know, that's how the liver patients are managed. Now, frequently, you know, we hear about kidney disease in diabetes. Can you tell us if kidney disease is diagnosed only with a high creatinine or can they have kidney disease with normal creatinine? That's a very interesting question because diabetic patients tend to have kidney problems, especially if they have had uncontrolled diabetes and long-term diabetes and there is a genetic that is the family history of diabetes with kidney disease in the family. That's called diabetic kidney disease in the family. Like Dr. Priya just asked me, we can have kidney problem in patients with diabetes with normal creatinine. So, and this can be easily diagnosed by doing a routine urine examination and looking at the protein leak. So, if this protein leak is, in fact, the first indication of kidney problem in a diabetic. And fortunately, we are now having proven benefits of glucose-lovering medications like SGLT2 inhibitors, which can reverse the protein leak and also stop the progression to further development of diabetic kidney disease and kidney failure leading to increase in creatinine. Many of my patients are actually having good proteinuria has been reversed or minimized and they are able to postpone the increase in the creatinine for many, many years. That's very good. So, early detection and monitoring for the proteinuria is important. Now, how do we manage the diabetes once the creatinine goes up? Okay. In the beginning, like I said, with only proteinuria, SGLT2 inhibitors are extremely good and the drug called metformin can also be safely used. But when the creatinine is a little up, they develop what is called mild to moderate liver failure. In that case, we can still use the SGLT2 inhibitors till the creatinine is manageable, 1.4, 1.5 with a GFR of around 45 for some and 30 for some of the molecules. And the GLP1 analogs are also indicated up to a mild to moderate kidney problem only if the creatinine has increased to a stage of dialysis or the GFR is less than 30 to 15, then we need to use only insulin. Otherwise, like in the liver disease initially, we can use certain medications in people with diabetic kidney disease. In fact, it is beneficial for the kidney outcomes. So, like in end-stage liver disease, end-stage renal disease, you need to go on insulin but till such time, there are a lot of medications and they can actually benefit, isn't it?