Well, for number one, the first one is going to be True because 300 equals 15 times 20. Okay, so we've shown that 300, is a whole multiple of 20. The second one is False, because if you apply the, division theorem, then you find that 35 equals two times 17 plus one. Okay, the third one is True, because zero is a multiple of five, it's zero, whoops, times five, okay. the next one is False, because zero is excluded. This is explicitly excluded, you can't talk about divisible c, plus zero and the last one is two because negative 21 equals negative one times 21. And obviously, here I'm, I'm giving overkill, I'm actually expressing these, these results in terms of of the Division Theorem. In reality, because we were familiar with divisibility, we could recognize each facts. Okay this is definitely overkill, I am doing this for paregoric reasons just to relate these issues with division theorem. Okay, well, so much for number one. So question two asks us whether this proof of this theorem is correct or not. So, let's just go through and see. we begin by, and I'm going to look at this both in terms of the logical correctness of the argument, and whether it succeeds from a communicative point of view. Okay, so we, let's see what we've got. We, we begin by stating an initial assumption, there's no standard method being used here, so we're not sort of seeing by induction or contradiction. And we're just going to do some reasoning, but we said we're starting from. We're starting with a Division Theorem, and in particular, the consequence of the division theorem, for when we are looking at divisibility by four. the remainder is going to be either zero, one, two, or three. So this is just an immediate consequence, a direct consequence of the Division Theorem. So we've stated the assumption. We've correctly derived an instance of that assumption and we've stated an instance of that assumption. And let me see what we got here. Therefore, any odd number, as one of those two forms, because So now I've particularized these fact, okay? So, for starting fact, and then I've got a particular consequence for particular case. Starting fact, that we specialize in fact, starting factors of Division Theorem, then we specialize it. So this case subject visibility by four. Then we look at the particular case and see that odd numbers of that form. And then when I square, assuming the algebra is correct, so, yes yes, the algebra is correct, and in both cases, the result is one more than multiple of eight. Okay, so that's correct, that's correct. I've made the observation that they're both one more than a multiple of eight, and then I sign off by stating that this is, this in fact proves the theorem. So I've got a beginning, a middle, and an end, so it succeeds communicatively. I've given reasons for each things. Notice I didn't just square these things. I didn't simply square them, I said I was squaring them, so that the reader's not left wondering what I'm doing. Now, to red it might be a little bit [INAUDIBLE] why I started in this way and that's basically experienced mathematicians can fairly regularly moderately come up with this kind of thing. If you're a beginner it can take you longer. Okay, and then when I've dome that calculation I make a specific conclusion, and this is a conclusion. A conclusion to the reasoning, not to the whole proof, because the conclusion of the whole proof is the observation that this proves the theorem. This is the drum roll, this is where the cymbals crash, this is where we take a bow. We've done it, we're out of there. Okay, so the answer is yeah, it's valid. And we just checked that it's valid by going through all of these arguments, okay? Let's move on to number three. Well, this proof is in fact valid, okay? Let's just make sure that it works on all levels. we're arguing by contradiction, okay? So, this is the beginning and this is the beginning step, suppose the conclusion's false. Then there'll be a natural number n for which a is not true, so there'll be a least one. Now about the first condition, in our case, at A one, we know that the m can't be equal to one, because A does at one, so m is bigger than one. So m equals n plus one for some n. Since n is less than m, we know that n holds because n is at least one of which it doesn't. Then by the second condition, A n plus one holds, i.e., A m. That's a contradiction, and that proves the result. the only thing to notice is that here the n has been used in, in a quantified form, so this is a variable that's been quantified, right? Okay, in this case, what we're saying is if the, if the, if it, if this fails, then there'll be one of these n's for which it fails. And at this stage, the n is sort of a variable. But at these points, when we put the m in, m is a specific number. We don't know what its values is. In fact, we're going to show that no such m exists. But on the basis of a false assumption, there will be a, a place where it fails. And m is going to be a specific number, so the m here is specific within the proof. Okay, when I get down to here, the n is also specific within the proof. Because it's equal to m minus one. So in these cases, once I'm down here, I've got a specific m and a specific n. Up here, it's just a general n, okay? So everything here is specific. Now, in fact, there never was such an m, and hence there never was such an n. Because the conclusion as a result is valid. But within the text of this argument, these guys are specific. Here, there's a, there's a, this is a variable that's quantified, alright? In any case this is proof this is true. On number four, we have to use the course rubric again. first of all, let's see what, what the theorem says. It says that if we take the Fibonacci numbers. And we square them and we take the first n of them and square them another. Then the result is equal to the nth Fibbionaci number, the last one in this sequence here, multiplied by the one after that. now this is actually, this is true, by the way. This is true, it's a valid theorem. It's one of many identities about Fibonacci numbers that show that they're connected in In what at first, a very surprising ways. they're very typical in, that they're almost always proved by induction. And when you look at the induction proof, you realize that it's really that the identity, which at first, seems surprising. Is actually just a disguised version of the definition of the Fibonacci numbers. What's behind the, all of these interesting identities. Is the fact that the the n plus second Fibonacci number is the sum of the nth Fibonacci number and the n plus first Fibonacci number. So they all go back to the fact that it's defined in this iteratively additive way, okay? well, let's just see if this one, see, see how this one goes, okay? first of all, we'll look at logical correctness. worrying about things like reasons and so forth later. Let's check that the, the first case is true. F1 is equal to one, so F1 squared, and in the case, n equals one, there is no sum it's just F1 squared. So the left hand side is just one, on the right hand side is F1, which is one. And of course the second Fibonacci number is also one. Because the Fibonacci sequence begins with a pair of ones. So we have two ones on the right. So, this is logically correct. then there's the, the induction step. Let's just check the algebra here. here it's just the taking taking a sum up to n plus one, and pulling out the second one, the last one, the last term in the series. So we've got the sum of n plus 1 is the sum of the first n together with the last one. we'll look at the issue of, of reasons in a minute. this will be the induction hypothesis, and nicely stated as, as a reason. that this sum equals Fn plus one, this is just Fn plus one squared I carried through. take out Fn plus one as a common factor okay. let's see, this is definition of Fn plus two, is the sum of that one plus that one. And then we've, we've got the other identity n plus 1. Okay, so, so all of the logical steps are correct. This is a valid induction proof. So I'm going to give four marks for that. is it clear? Yes, I think this is clear. again, there's going to be some issues of reason as to explain things, but everything is clear. It's well laid out, it's easy to follow steps, even when I had to figure out what the author meant. I'm going to, we'll talk in a minute as to whether, whether I should have had to. I, it, it was, it was easy to follow in that sense. So it, so it was clear. there's an opening, it's well opened. It's a proof by induction, there's a standard method. And, and they, they, the good way to start. The correct, well, almost to start to prove using a standard method is to state what the method is. So I want to get full marks for that. conclusion when the conclusion is stated I'll, I'll look at whether it's properly stated in a minute. But it certainly it, it, stated when the proof is complete And it's been laid out that it's going to be an induction proof. So, I think we're going to get four for that reasons, a couple of quibbles. I think the order should have said, separate out fine term. this is good, stating the, the use of the induction hypothesis and an induction proof is, is always, is critical. I, I, I think that's this is such an important step. by algebra, yeah, you could say something like check out a common factor but the, this is, this level of mathematics. When we're doing proofs in number theory as I mentioned with the last problem set. we, we can assume that people can, can spot things like taking out common factors. this I think is important. and this, this this is critical, the fact that we're using the, the definition of the, of the Fibonacci numbers. the, these kinds of identities, as I mentioned a moment ago, these kind of identities actually only hold because of the way the Fibonacci sequence's identity is, is defined. which establishes the identity for n plus one. That's good that's a local conclusion, which is good. the proof is complete, here the person should have said, by induction, or by the principle of induction. let's put out in four principle of induction, because there is a powerful fact about what the, the natural numbers that's been used here. Okay, so, what I'm going to do for reasons, I'm going to give two, I think. Because I can't give four. you know, I, if, if this one arguably I, I, I would tend to lean on having this in especially since it's an opening step in the proof. But if that was the only thing that was missing in this context, I might not have even deducted any marks at all. this one however, I, I have to deduct at least one, and I think, I think I've really deducted, I think really I've deducted the two for this one. It was, it was a bit of a judgement call. you know, I have to allow for the fact that the author of a proof maybe made a slightly different judgement call. you, you have to try, judge how well you think a persons putting down a proof. and, and you can't really say, I always do it this way, therefore you always should do it this way. Because people have different, they come on different sides of these issues. we're really looking to grade this as an overall thing. You know, one of the problems with using a rubric is we're trying to take something that's holistic. And is basically an overall judgement call and reduce it to number of parameters. This is not how professionals go ,they look overall and say ,this is a good proof and then assign a number ,but now space for years of expertise. Pulling us apart this way is a good for beginners because it allows you to focus on one only individual things, but a professional looks at all these things in, in one. And you have to sort of balance things out. and that's why we're going to all these videos, to try and give you some indication of how a professional, and in this case the professional is me. How we go about it and how we, tacitly, and when we're doing this, this part of our every day work, when we're grading work. whether, whether we're grading student's work or we're evaluating proofs of other mathematicians. This is all part of the tacit process of grading. and, and in writing out a proof, I've just, tried to isolate the things that I implicitly and automatically look at in grading proofs. as indeed does, does any professor, professional mathematician when they're grading a proof, okay? and so, and what I'm really saying is, this is. Not really perfect in terms of giving reasons. arguably this one is more important than that one, for example. You know, it's overall, however, I'm going to get four. because these are sort of niggling. and the reason they're niggling is the person has laid out the fact that this induction, has made it clear that they're assuming it for n, have proved it for n plus 1. they've definitely stated the use of the induction hypothesis. as I said, this one would have been nice, I would have liked to see it, but this author. Presumably decided it really was, was, was patently obvious what's going on. this I think is an important one, because that's critical to the proof. this is not something that's typical. The farther you can pull this in depends on the, where the Fibonacci sequence is defined, and this is the only part in the proof where we make use of that fact. So you really should Should mention this one. So this ones important, and this one's important. And the reason is until we've got to the last line, all we've really done is we've shown that the thing is true for an equals one, two. Actually, further equals one, use the further and F2 was what it was. So if there were few, they were put through for the first case, or observed in the first case. And then we've shown that if it holds stage n, it follows, and it holds stage n plus one. So we've proved two simple facts. One, the fact about the first one, well, it's the first two for that matter. And secondly, we've proved an implication from n to n plus one. The conclusion is that this holds for infinitely many number, for all of the actual numbers. So somehow proving, two statements, one simple observation and one implication, has proved that something is true for the infinitude of all natural numbers. Now admittedly, induction has a natural it's always an obvious thing, this sort of a, self evident truth to the principle of induction, you know. You can think of it in terms of, of knocking rows of dominoes over or something. So But the fact that the, that the, that the principal of introduction, or the part, the method of induction has got a sort of intuitive obviousness to it. Shouldn't obscure the fact that this is actually a deep result. Making a conclusion about an infinite set is non trivial, you know? The, the Hilbert Hotel tells us that infinity's a very, paradoxical domain. We've got to be very careful. So, this actually holds. We can make the conclusion that it's true for all n. Because of the principle of mathematical induction. This, in other words, this is a big deal. This is a big, big deal. And when big deals are involved, you should mention them. You know, if there's a big guy in the room. it's polite, if not, [LAUGH] a matter of self preservation to observe that fact and make it clear. So you really do need to state the principle of induction here to state that it's been used, or at least to say, by induction. I, I, you know, if I was feeling, if it wasn't for the fact that the rest of the proof was laid out so nicely. I might well have just deducted more here. But as it is given everything else was laid out so well and given that the proof is nice and elegant. And this is a slick proof, there's almost no superfluous lines in. And I think overall simply deducting two marks is about right. So I've got 22 out of 24 for this one. and I feel reasonably good about this. you know, the, couple of a small points, [COUGH] well, it's one small point, one moderate big points, one really huge point, I think. I think, this is being generous. But I think this is proof deserves generosity. overall, I'm happy with that. And, and this is really how it, it splits up. Okay, let's go ahead and look at number five. And number five is another of these Fibonacci sequence results. This one says that if we take the first n Fibonacci numbers, add them together, the result is the the next but one. Fibonacci number, we skip over Fn plus one, we go to Fn plus two. Okay, so, okay, so as is typical for these results it is proved by induction. So see how the proof goes. For n equals one, the left-hand side is F1, that's why you get if n equals one, there's no sum, it's just F1 itself. And the right-hand side is uh-oh, oh dear. If n equals one, the right-hand side would be F3 and F3 equals two, and one does not equal two. So this isn't even true, it's not valid for n equals one, which means the theorem's not valid. Oh, good grief, this is such an obvious mistake. It's the kind of mistake that anyone could make. Doesn't really reflect on their ability as a mathematician. It's just a human error. So common this kind of thing, and yet this is mathematics. Ultimately in mathematics things are right or wrong. I mean, you know, if, if this mathematics being used by an engineer to build a bridge, and the bridge falls down and people get killed. You know, that, that engineer could be held liable. i am just defining or so you know at the end of the day we call that thing goes to false ,i am have to 0 for logical correctness before, we are go further ,i mean it is just a plan to evolve false result you could make the result correct ,you could make first result correct by subtracting one And it fact it turns out as, and I'll come back to this, that if you put a minus 1 in here, then the identity is true for all n. So there is a theorem here, and the get, we get at the theorem by noticing what went wrong with this proof. Incidentally, this is very typical in mathematics. often in mathematics, the statement of the theorem when it's proved isn't the one that the author originally tried to, to do. Very often in mathematics we, we make a conjecture. We try to prove it. the proof has gone wrong and so by analyzing the proof we've thought of go back and change the statement. So it's often the case that statements of theorems actually come after the proof. Not many proofs in mathematics began as proofs of something else that failed, but then the, the statement has changed to what's been proved. So it's not always the case that the mathematicians sort of formulated theorems and then proved them. They often formulate a theorem, develop a proof, find out the proof is wrong, go back and restate the theorem so the proof works for that restatement. Okay, that's just the way mathematics advances, it's, it's part of the process of, of getting you knowledge. Okay, [COUGH], well we're going to have to come back and, and, and sort of look at how the thing works as a proof but, but it, it isn't a proof as it stands. Okay, well let's go through the mechanics of it and see if, if all of the other steps are okay. So, assume the identity holds for n, then well what's going on here? This is a case of separate, I'll, I'll, I'll give grades for, for these ones later, but let's just mention that what's going on here is separate out the last term. Okay, let's see, that's the reason here. Sum up to n plus one is the sum up to n together with the final term. this is induction hypothesis. That guy, equals that. Incidentally and I'm jumping ahead to when I corrected this, at this point there should be a negative one in here, and if we do that it carries through. So this, this is the proof as we're going through it will work validly. Providing that we stare at the theorem correctly. Okay, so I'm not going to give double jeopardy deduction for this. We've already knocked a ton off for that. okay, so the rest of the thing is, is actually, the logic is correct, in of itself. Expect for this glaring mistake at step one. But that, that topples the whole interphase. So I'm going to give four for clarity because it's absolutely clear, well laid out. it's proof by induction, it states it, stated the method at the beginning. So four for the opening, the conclusion is certainly stated, I'm going to give four marks for that. What's the problem I have with this is that, it's not mentioned that this is an induction proof. And as I elaborated on a great length with the previous question, was question four. the fact that something is, you know the fact of an infinite three/g, and infinite three statement. The statement about something which we've all end. The fact that that follows from a couple of little facts like an observation and a simple implication that's a big deal. and that takes us into the realm of the infinite. And what takes us into the realm of the infinite is the fact that we have this thing called The Principle of Mathematical Induction. Which is difficult to prove if you try to prove it, or you have to assume it is, it is an axiom or some kind of principle. So this is not a trivial thing. It's intuitively clear, I know. But it's not it's, by no means easy to prove it. So you know, their, we're pulling on something powerful here. We should state that we're pulling on something powerful. Okay, no reasons, I mean I was addressing those as I went through. this ones missing, that's in, that's good. You know, the, the, the, the fact that the, the definition of the, I mean it, thi, this res-, this result holds, I mean, when I modify it to make it true. It holds by virtue of the way that Fibonacci sequence is defined. So you should stipulate the fact that you're using that. this is good, using the induction hypothesis. I've got something good here, I've got something good here. that's a bit of a problem. That should've been in. this one, I, you know, again, this is this is a judgement call, I would like to see that there, but that's just me. All of the things being equal, I would of ignored that, but all of the things I'm seeing, of course, this person's already missed this out. So, got a missing reason here, missing reason here, but a couple of good ones here. so I'm going to give two, okay. I think it's about right, overall valuation? The thing is false, I mean, I can't possibly give four overall for a theorem that's plain false, you know. Much as I, I'm sympathetic for the fact that it was a simple slip, the most I can give is two. And I think I'm being generous there, quite frankly because it is a false result. On the other hand the, the, the course is focusing on mathematical thinking and mathematical communication, the ability to formulate and present a proof, and there's lots of aspects to that. It's not just about whether things are right or wrong. You know high school mathematics is, is largely focused on, K through 12, mathematics is largely focused on things being right or wrong. we've, we've moved beyond that now. Right and wrong is still an important factor which is why I give zero to this part. But there's other things we're looking at. So I will be giving 16 for this one, okay? Well, that's said eight max lost out of 24, so this pair lost a third of the max for this. So one could say that this is generous for false proof for false result. On the other hand there's a lot of good stuff here. It was simply a silly mistake right here at the beginning. Okay, that was very unfortunate. Okay, life's like that at times, let's go on and look at number six. Well, this is another one about the Fibonacci sequence. what does it say? It says that the n Fibonacci number is at least equal to the number three over two, to the power n minus two. Okay, let's see how this person does this. we have, oh, okay. Interesting way to start, making a statement about F1. which is equal to one, true, I mean, I, I wouldn't look at logical correctness. I'll just follow it through, and see what this person's doing. and two over three is equal to three over two to the negative one. Okay, well I guess that's showing that it's true for F equals F1. so maybe this is an induction proof, although there's no way of knowing just by looking at it, right? well this is interesting. The person now goes on and, you know, you usual with an induction proof you prove it for n equals 1 and then you stop. This person now proving it for F 2. well, let's see what they're doing. So they're saying F2 equals one, which is does. And one is indeed three over two to the power zero, which is true, because anything to the power zero equals one. So the inequality is valid for n equals one, two, alright, absolutely correct. Now assume the inequality holds for n, where n greater or equal to two, so, we assume, but we're not told, that this is going to be an induction proof. This person just jumped in, so this is, this is already not a, not a proof. This isn't telling a story. It's, it's, it's, it's presenting us with a, with a who done it or a what done it. Or what are they doing? so this is going to be a mystery, where n's greater than or equal to two. Okay, then, let's see. Fn plus one equals Fn plus Fn minus one. True, that's a definition of the Fibonacci sequence. Although there's no explanation of that fact. that's greater than or equal to, well, ha. Where's this coming from? well looking ahead to the fact that this is the, the person's is almost certainly doing an induction proof, but hasn't written it, written it down. What should of been said here, was for wholes the in, inequality wholes up to and including it that, I think, is what's is what's meant here. Because we're assuming, we're using it for two cases here. We're using it for the two previous cases, and we're saying that is greater than or equal to that one, that's greater than or equal to that one. and then then we're taking out a common factor. this is interesting, there's almost no reasons given for anything else. And certainly the person makes the obvious statement that this is bialgebra. Which indeed it is, taking out 3 over 2 to the nc is a common factor. And one of them now is a digit of one there. then 3 over 2 plus one equals 5 over 2. Now we're spelling everything out in goal we have mathematical detail. 5 over 2 is 10 over 4, why are you doing that? Well, because because then you can put a 9 over 4 here and make it smaller. And then nine over four is just 3 over 2 squared. Which gets you everything back to 3 over 2. Which establishes the inequality for n plus one. Okay well, this person can certainly manipulate fractions and also really impressive. Because you know a large percentage of the world's population has trouble dealing with fractions and inverting fractions and things. So, so this person has a lot of procedural skill with fractions, but is it a proof? Heavens no, I mean this, this so much missing here. I'm going to give four for logical correctness, because the manipulations and the logic and everything was, was, was fine. as a professional mathematician, able to figure out what's going on. I could recon I am, and I could recognize fairly early, fairly early on that this was an induction proof. But no thanks to the person writing it down, and just because I've got a lot of experience. never the less I'm going to give four for the, for the logical correctness. I'm just going to get four for the clarity, because it was sufficiently well lit out. That one side realized that there's an induction proof going on here. I was able to follow the steps. so I didn't have to solve, bury myself in a, in a marage of details. To find out what was going on it was clear. Opening, well there wasn't one, there absolutely wasn't one. This person just straight in, and did so very obscurely, because even if looking at the first case indicates that we're doing induction. Why look at the second case when we almost never do that in induction. And then we had to make a, a modification to, to even make sense of what was going on. So, no opening, what about the conclusion? Well no, there isn't a conclusion. what the person presumably should have said, or maybe meant to but didn't. Hence, by induction, we could simply say the theorem is proved, that's fine. That's absolutely fine. It bears fruit, this is alright. but it's missing. There's no conclusion stated, and there'll be a zero here. Reasons, there aren't any. There aren't any reasons. simply no reasons. I mean, this one doesn't count. I'm not going to give credit for that. That's just a, here, we should have said something like well, by, by the induction, well actually, there's two things. By the definition, of the Fibonacci sequence is the first equality, okay? And the second one, is the induction hypothesis. This is by the definition of the Fibonacci sequence. you know, it should be said. I mean, this is so self evident. and either way it's written, because this is literally the definition. You know, you could maybe, let this person have have some, some leeway, if everything else was good. But nothing else is good, so I'm not going to, I'm, I'm not inclined to give credit generously. When the so lack of, there's so much lack of reasons give. This is the induction hypothesis applied twice, okay? Once for that one and once for that one, and then the rest is algebra. Okay overall, the only you know I do, you know, if, if I simply leave it at that And say that this is just terrible. Because it is terrible as a proof. The person gets eight out 24, and yet, this is some, this is quite sophisticated by most people's standards. I'm going to give two for the overall. And I'm going to give some sort of compensatory credits in here. For the fact that this person's done some really intricate, manipulations. there's some intricacy in here, this is not trivial but it still just gives 10 out of 24, I mean this is a, a low mark. Now in terms of a grade for mathematical dexterity that would be low, this person clearly has considerable mathematical dexterity. But proving things and communicating things in mathematics is much more than dexterity. it's about telling a story, it's giving reasons. It's establishing why something is true, you know? I'm prepared to believe that this person convinced themselves that this is true, and they understood why it was true. But what they haven't done is express that fact properly. this doesn't even come close to being a proof, as a proof, this one sucks. I mean, this, this is not a proof at all, okay? no mention of reasons. we've still got this mystery of why did the person. Why did this person do the first two cases? Well that was actually an obser, a good observation. mostly with induction proofs, you have to prove. Just, you, you still have to prove the first case. But this one, you have to prove two of them. Because if you look at the induction step, it uses it twice. You have to use it as the two previous cases. So in this case, the induction step depends on having it twice. You've got Fn greater than or equal to that. And you've got Fn minus one greater than or equal to that. So we've got two instances of the induction hy-, the induction hypothesis here. Well, it would, the induction hypothesis has to be pulled in twice. We use two inequalities. So you have to prove the first two cases. And you also have to, when you, when you're making the assumption. You're not just assuming it for n you're assuming it n and n minus one. Okay, you're assuming it for n and n minus one. and, and the simple way to say that is you assuming it up to and including n. So there's a lot in terms of missing reasons here. the fact that the proof, the fact that this person does what he or she does Indicates to me that they almost certainly understand these issues. You know, simply observing you have to do it for two cases, is significant. So I've got a lot of sympathy with this person. They've obviously thought deeply. I'm convinced, actually, that they know what they're doing. And they know why they're doing it, otherwise, they wouldn't have done this. So the low mark here doesn't reflect that this person isn't good in terms of doing mathematics themselves. It reflects the fact that we're talking about the articulation and the expression of, of mathematical thinking and communication. So, you know, this would be a salutary lesson for the person to say look, you're good, make yourself like a professional. You know, you've got the ability, do the right thing. Learn how to express things. Learn how to, to communicate things to others. you know, there's room in the world for a small number of brilliant mathematicians who are really bad at communicating. they often win prizes, we, we're glad they're there, but, but this is a course to sort of get, get everybody up to a nice level, a nice usable level of mathematics and mathematical thinking. And so I'm focusing on different things in this course. and in terms of what we're looking for in this course, you know, ten is actually about right. Even though, in this, I mean, what I would do with, you know? A real, you know, if I'd seen this in, in real, in a real situation. I would have sat down and talked with the person, and said, look, you know? You've got a lot of ability, you really have got some talent. let's work with that talent. And, and, and let's develop the, the technique you need. And you know that to become a, a, function professional and if your truly brilliant, we'll give you huge amounts of slack. And we'll overlook the fact that you, you, you, you can't write in good proof. if your life depended on that. But you know that's the small percentage. Most people have to make their living by collaborating with others, by waking with others, and by, by being good communicators. And and this course isn't really focused on the brilliant person. The brilliant person, we take aside and we deal with them separately. This is focused on, on getting everybody up to, up to an acceptable level. So, low grade overall. But, just, this is not a good proof. It's simply not a good proof. It's not really a proof at all, okay? And we've, we've taken account appropriately. So a sad way to end the problem set, on the other hand, you know? This person looks as though they can probably take care of themselves, okay?