 I have a super difficult riddle. Not sure if we will be able to solve it here in time, but I would be interested to see You to tackle a sure drop it in Fit ratios. I'm bad with shapes This one that the ratios I have is not shapes. You can you can drop the riddle for us Dr. Hang We can think about it If I don't know it off the bat We might do the ratios and let that question You know bounce around the head a little bit and see if an answer will come up Why not? Why not? Always good to challenge Is it a math riddle? I'm usually pretty bad at riddles to tell you the truth You know if I was a victim of the riddler I'd die He'd kill me Solve this riddle to free yourself So here's the question I believe At least it took me like forever. Okay, maybe it's easier for you Three points a b and c are placed at random on a circle of radius r What is the probability? Oh that the triangle ABC is acute Acute is smaller than 90 degrees right the angle of twos. I always forget the names Acute is less than 90 right And this is probability. Oh, yeah, this would take me forever as soon as you introduce probability. It's like dang Because I'm not they took probability they used to teach probability in high school In my part of world they took probability out of high school about 12 years ago 10 years ago again, right? They revamped the curriculum and dropped 30% of the content. So basically the question is this the riddle You got a circle of radius one radius of one radius of one Point a b c a b c and the Our place around them on a circle. What is it? Probably that the triangle ABC is acute So basically they're saying they want this But all the angles have to be less than 90 degrees Right all the angles have to be less than a degree. So the way I drew it is not going to be the way you draw it That's assuming would be a Oh, wait a second. What does it mean for a triangle? Yeah less There's no angles greater than 90 degrees. So all these angles are less than 90. So the way I drew it here Let me draw it better That way visually it's more appealing right, so if you have this so It would be like this and the radius is one Okay, so that means the distance from here to any point is one To there is one and to there is one. I'm pretty sure this this has a lot to do with it You would have to start off Something like that and then Ultra all triangles by definition of at least two acute angles Okay, all acute Yeah, all acute. Yeah by definition they have they have to have at least two acute angles, right? So they're all acute they all have to be less than 90 degrees So if they're all less than 90 degrees if the radius is one right you could do this and This plus this plus this has to equal 360 right, there's no other stipulation of See it goes into probability. I can't do the probability aspect of it Have a non-probability on this Right, but I think this is the way you would have to approach it. I believe is this the way you approached it Dr. Heyn to get an idea to get a Because once you do this Once you do this thing you would have to figure out a Point a limit where this angle becomes less than Less than what? Could you use the fact angles? Yes. Yeah It's a good start. It's a good start. I think this is where you would have to start Could you use the fact that they're all they all need to be on the same? Semi-circle to have an angle greater than 90 Could you use the fact that they all need to be on the same? Semi-circle, but they don't all need to be on the same semi-circle and if they are All in the same semi-circle as long as they're not exactly equal distance They must have an up to yeah obtuse Yeah, if they're both if all three on the same semi-circle then the angle would be obtuse so that doesn't work right Elder God an acute angle. Okay. This is definition an acute angle or an acute triangle or acute angle Triangle is a triangle with three acute angles less than 90 degrees an obtuse angle or obtuse angle triangle Is a triangle with one obtuse angle greater than 90 degrees and two acute angles? Yeah to be non-acute They do I think yeah to be non-acute they do they would have to be on the same semi-circle right Yeah I wouldn't know how to go about it. Nice question though. Difficult Difficult, but this is the way I would start it. This is the way I would start it And then If you find the answer if you know well, you know the answer go to our discord page and post the post the solution Please dr. Hank. Here's our discord page We have a math folder or maths folder in heavy topics Not the way I would have Wrote it. Dr. Hank says yeah, very difficult. Feel free to pivot away from it. Yeah, I took it down I'm like I would have to spend all day trying to figure it out and look things up, right? So can that be you somehow you get a free choice on The first two points Yeah, but then the third is Very Constraint for where it can't be and still be acute. Yeah, and here's the thing. Oh You're looking for the probability not possible angles So this is like rolling dice. Let's check this out So that's very good by the way SS SST a I did that problem of school, but I can't remember. So basically your first choice anywhere Doesn't matter your second choice anywhere doesn't matter Your third choice cannot be in the same semicircle as the other two Right, so the the problem would be the semicircle could be If this is the center could be from this point all the way to the other point So if you drew That's a semicircle That's a semicircle. So if you put any the point anywhere along here You satisfy the The question right so the probability would just be that Right, I did that. Is that correct? So if you do this, let's take it to the extreme Let's you say you put one there and you can't put it exactly on the opposite side. Let's go you put it Here just off that thing then the other point could be anywhere along here So are you close to 50%? I don't know that doesn't make sense hmm Yes, that's the picture that was in my head. I don't know how to translate that to probability Well, the probability would be you could put the point anywhere along here so you could get as close as you want to this Point to the diameter the semicircle and As long as you don't touch it because as soon as you touch it you're gonna get Triangle that's obtuse right then any point along here Would make a triangle that's all angles are acute so are we talking 50% Dr. Hang 50% no You got a 50% probability of creating a triangle this Acute that seems way too high something. We're not considering here No, no, it's way too high, but the allowable arc you can pick the third point From Changes depending on where the first two points are I think yeah, it would really depend on first two points Also, I think in the picture below it would be close to 100 Would it be let's assume here We'd be close to 100% Well, no because all of the points on here would not not be included The probability of third point producing acute triangles Continuous function of the choice of second point hence the integral isn't to grow 66% is it beat because check this out like any point here Would make all these angles acute Right as long as this point is not on it Does 90 degree count as a cute or not? No, it doesn't 90 degrees 90 degree triangle. It's not acute. Oh You do not go through the center In the problem below. No, you don't go through the center. You're not through the center Right, you can't be on the center. Otherwise this becomes 90 degrees Right, like that's the That's the circle if you go through that if it's a diameter then any Triangle here makes this 90 degrees and you can't you can't do that if you pick the first two points as Diamet Diametric then all choices of the third point are night. Yeah are 90 degrees as long as the points aren't Superimpose. Yeah. Yes, that is what I meant. Yeah. Yeah So we just have to be off that right that's why I said it is a close to 50% because As soon as you hit it is 50% but okay, so 50% is too high 50% is too high 50% is too high So there's something else at play here And it is a relationship between this Because let's say the two points go the other extreme. Let's say this point and this point Then what do you have to play with go through the diameters? You can only have a point in here for it to be to work, right? So how do you? How do you incorporate that with that? How do you incorporate the two? Exactly, yeah Interesting interesting. I'm not really sure how to go beyond that. I think it's I saw Variable answer and I remember sorry It was 1987 Yeah, so point five is a way too high upper bound. It's a way too high upper bound because The two points the first two points could be very close together if they're very close together on the extreme end here On the extreme end if we draw a circle, let's assume the two points are extremely close together If they're both going through the center that you're limited to this So if you did it this way then that would be almost zero percent If you take the average fifty and twenty five zero, so twenty five percent Was it the answer doctor hang twenty five percent probability that you can make an acute angle. No, it can't be that simple right Knowing math the answer probably involves pot. Yeah Yeah Pi somehow and maybe e as well. Yeah, maybe some of those some of the magic numbers, right? And zero and one right fun Yeah, yeah, yeah Crazy cool though. If you do have the answer post it in our discord page Doctor hang they'd be cool to see it especially if there's visuals with it just to get a better visual of this Funny enough twenty five percent is correct, but I think the argument is not yeah Twenty five percent is correct So the math one one one bound would be fifty percent the other bound would be zero take the average twenty five Now if you and if you did it that way on a test Would you get marks for it? If I was a teacher? I'd give the mark. I Would because it would be the person went through the logic trying to figure it out, right? And because we did it this way So it is twenty five percent really I wonder if this is a legitimate way of doing it by the way I Wonder if this is legitimate with doing it also seven five thousand I Can visualize the geometry, but I have no clue how to make it into probability very interesting problem though very interesting problem right a more of a of an Engineer I write code To pick a lot of random points and count to see if that gave me an insight. Yeah Maybe I'm confused. I Don't know how to God seventy five percent will be the opposite of the twenty five, right? I did it very similar to the way All the God Lined it out really let's check it out How did all the God line it out? It missed it. Oh I didn't know there is out of God the probability of the third point producing a cute triangle is a continuous function of the choice of the second point Hence the integral so you need to go to into in integral That's something place. I'm not gonna go. It's a variable of second point. It's a variable of the second point Which is what we're doing visually, right? But I don't know how you would go about it with the integral Yeah, but maybe it's way too complicated now that you came up with this rationale This rationale seemed it logical. I Don't know if it's like how do you prove it? That's the kicker, right? I think try and stuff out to see if it leads to something that looks sane is Is a valid approach to finding an answer then you can try to find a way to actually solve it Usually that involves induction just to mess with me. Yeah Yeah, I'm proof some horrendous out. So I like a lot of logical reasoning through things but Integral over the angles at which point a non-ecute triangle comes out