 Like. Oh, so good morning good afternoon or good evening. Whenever you're watching this, my name is Annette Margoliff and I'm an academic English teacher who is on who's also a mathematics teacher and I'm on loan to the Foundation program at Birmingham University. And this isn't something we do on foundation, but I do like number theory. Ac, oedd i mi ddweud, o'r ffyrdd yng Nghymru, ym Mhwylwyr Kenzie, o'r catholiciaeth Cwylwyr, ym Mhwylwyr Byrmeng, a'r ymdweud i'r colli. Felly, mae'n rhai'r hyn o'r hwnnw, o'r Jeiwet Hwdy, o'r cyffredinol, oherwydd mae'n cymdeithasio'n gwybod, un soron hwnnw i mi ddweud arfer mae hynny yn oed yn gwahoddiadau meddlist. Diolch yn fawr, Annett. Felly, mae'n Fhwylwyr Kenzie, mae'n cytholiciaeth Cwylwyr, a fyddai o'r Mightfyrdd, iawn i'n gwahoddiadau cwestiwn, mae'n gofyn sydd ar y rhywun yn gweithwyl i meddlist ac mae... Dwi'n ddweud yn ffas i ffasiasio. Stat o お rhai, dwi'n meddlist cyfwilio'r cyffredinol, Yn eich bod mwynhau, dyna lleddiadau. Sut i drwsion, dwi'n meddwl, rhaid o fel ystafell a'i'n rhan oedd y fwrdd, gyda'r syniadau. Felly oeddwn i... Felly nes oeddwn i'n garadwch, ac mae'n wedi amser fy mod i hyfforddiol. Mae hwnna'n ddod yn edryng hwnnw, gyda'n, nid wneud i ddweud hynny'n gweld, mae'n gweithio'r cwestiynau 12 ymlaen nhw, a nhw, yn ffodus, ymlaen nhw'n gyfwilio'r cyffredinol, yn gyfwilio'r cyffredinol o'r sefydlu sy'n gweithio'r cyffredinol. Yn gyfwilio, ond mae'n ffordd i'r cyfraith bwysig ymlaen nhw. Ond mae'n ffordd i'n bwysig o'n cechwilio'r cyffredinol yn bod gennym ni'n gweithio'n gweithio. So on the first day you get a partridge and then on the second day the poor recipient receives two turtle doves and a partridge so they have then they have three gifts so there's a difference of two there and then on the third day the recipient has three and I don't know it's not got any colours it's French hens isn't it three French hens two turtle doves and a partridge and so you've got six gifts and then if we add another fall to that on the next day they get 10 gifts and so it carries on and the days that are interesting to me are day one and interestingly is day eight because day eight you get 30 if you add up all the numbers here you get 36 gifts so one question could be how many days will we have to extend Christmas to this was the original one after Christmas I suppose this went to epiphany this 12 here went to epiphany but how many days would we have to extend Christmas to get another square number square triangular number and and the answer is quite a lot I would say so I'm going to do a little bit of mathematics but nothing that you won't recognize so we can actually give a term for each of these ones so if we know what day it is we can work out how many gifts and there's a formula for that so I'm going to do a new share and go to the white the white board because I don't want to mess up that beautiful picture so on any it's actually strange enough it's actually a quadratic sequence on any given day a lucky recipient will receive n over two times n plus one gifts so I suppose on day one if you put one into that you get a half and then two times a half is one on day eight so one is a good number because that is a square triangular number I won't put it in because it okay but if we had day eight can I just say that my writing on here is appalling but it has improved so on day eight we'll get eight over two and in fact if people want to work out the days of how many gifts in total you can do it like this I believe there's a fill and add up each day I believe there's also a formula for the whole shebang as it were on day eight twos into eight go four times and you get 36 gifts okay I don't know where you're going to put all these gifts it's a bit like the problem we have at the moment I don't know how much stuff you're ordering online you know where do you put all this stuff still got boxes trying to remember what was in it so if we look and I confess I have got a crib sheet here so if we look on day one we'll have one times two over two and that is equal to so I'm going to put D one so that is one squared times one squared so one condition is that this here has to be two squared numbers times together and on day eight we have eight as I said before eight times nine eight is double a square number which is so that's the sequencing and then you have two squared times three squared now there is a pattern to this do you want me to leave it I can do the next one for you if you like no let me see I think sorry you mean the next there is a pattern you can predict when the next one will be okay but doing maths on the spot is never easy so I don't let me see where it's three or three five would that work I suppose it would be there I know that I think day 49 fits oh gosh is there something before that no I'm just you know was that divine revelation okay so I've changed colour actually perhaps I shouldn't I'll go yeah so I'll do the colour thing here right so do you want to write that shall I write day 40 so day 49 fits did you do that in your head well I was just thinking about other numbers and sort of 49 I was just looking at the squares various squares and okay so would you like to give your reasoning for that and so day 49 is whatever 49 times 50 over two, which is so which will end up being seven squared times five squared. Okay I'm going to put it in the other. Yeah. Order. Yeah. It is an interesting thing though isn't it or was it just me. No it's a yeah it actually does answer one well. So how did you get to day 49 did had you looked at the paper or. I mean I looked at a couple of pages but also I was just thinking of square numbers generally and because one of the two has to be a square. One of the two has to be a square and the other one has to be. If the tool is going to cancel out then it has to be double of a square I guess. Yes. That's that is exact. Yeah that has to be exactly it so I put those two like this. Okay and I'm going to. If I do, then I'll do this to the other one. And then this to the. Okay. Don't know if that makes it. So, could you use. Looking at though it's sort of like do you like Fibonacci. Yeah, yeah. Yeah, yeah, yeah. We don't have to like him, you know the city. I like the series. Yeah. So, do you think you could see where the next one is coming from. So day 49 do you think that we're going to, we're going to extend Christmas to, I don't, I can't see it happening to have 49 days of Christmas just to get a square now triangular number in. That's like, you know, the feast that are for 50 days, you know, seven weeks. Oh yes of course. Yes we can have the seven weeks of Christmas. Okay, so now this is. One, one, two, three, five. So the next 12. I mean that's one, one, two, three is five is the first five numbers of the Fibonacci. Yeah, so, and Fibonacci, it's the previous numbers, the two previous ones in the Fibonacci sequence, give the consecutive one, don't they. So it goes one, one, two, three. So if you were adding up previous squares or something. Yeah, one, one, one plus one, then two, one plus two, three, two plus three, five, three plus five, eight. Okay, but looking at this one. Yeah, there's something here. Well, I suppose the. Well, one of the small, maybe the smaller the two numbers is the sum of the two previous ones. Yeah. Okay. Okay, so what would that take us to. The next one should be 12. Okay, so 12 squared. And then the problem is what is the number next to it. Yeah. If you look at this, the odd num. Can I just does work. Okay. If you look at this, the odd numbers tend to feed will feed into the, the, can I just get. Yeah, the odd numbers will feed into this one here. The odd square feeds into the second number of the sequence, whereas when it's an even square, if it, it maps on to the first number. In, in the formula, let's say. So 12 times. I mean, I think that's food for thought for people. So, is to recap one of the numbers is a square. And the other one has to be. Twice a square. Yeah. It is not. So 144 would be one. Okay. So it's got to be about 142 is 145. Well, the only one that can be twice a square. No. Is is actually 144 itself. Yeah, well. So, so 144 double that. Oh yeah. Okay. Oh, that's the smaller one. All right. 288, which gives you 289, which is a square, isn't it? You can use it. You can use a calculator. You can phone a friend or use a calculator. Let me see 289. That's 17 square. Used to be able to do this in my head. 17 squared. Yes. This 289. Okay. So 288 and 28. Okay. So day 288. Right. Yeah. Okay. I hope we haven't spoiled it for anybody. It's a bit like an Agatha Christie, isn't it? You don't. How much do you say? And so equals 12 squared. 17 squared. Actually, I do like numbers like 17 squared because they look like their prime numbers. You know that. They're not. That's a two there. Okay. Yeah. Okay. I think that's some. I think we'll leave people to discover the others. Do you think there is something godly in this? Well, there's, I think numbers. What is it? What is it? Who said that mathematics is the language with which God creates the universe? I think it might have been Galileo actually. Well, you've said it now. Okay. Okay. I'll stop the show now. Thank you ever so much for that. No, thank you. That was, that was fascinating. I have to go and figure out why it is that that sequence works. Well, maybe that's what the paper. Yes. I think it's, I have looked for other papers, but that one is done particularly nicely. Thank you ever so much. Thank you. I'll.