Hello everyone, welcome to this remote interview, mock interview, for Mathematics. I am Dr Tom Crawford I'm one of the Maths tutors at St Edmund Hall and this is my colleague Professor Luc Nguyen who is also a Maths tutor at St Edmund Hall. So we are going to be conducting a mock interview with one of our second year students to give you a feel for what the interview process looks like. Generally, we're just going to be working through an interesting maths problem with both myself and Luc adding some hints and tips and helping guide the candidate in the right direction. We're not in any way trying to catch anybody out, we're generally just trying to work together as a team to try and solve an interesting maths problem. Luc did you have any other thoughts or anything to add? Yep just about the general view of how an interview is going, so we are going...as Tom was saying, we are just going to do some math problems together so you know as a candidate one shouldn't be worried if one gets stuck in an interview, just keep thinking and keep pressing at it, we tutor will input and then help you along the way. We'd like to know more about you, we're not trying to see the bad of you but we want to see how you struggle, how you deal with the math problems so that's the general way how we go about interviewing. Brilliant and hopefully it should be an enjoyable process for everybody, that is ultimately what we are genuinely trying to do here and you will probably get the best experience of seeing what we mean by that by watching the mock interview that we're about to begin.
Hi Bethan, I'm Luc Nguyen, one of the tutors at Teddy Hall and joining us today is Dr Tom Crawford, another tutor here. So we're gonna do a conversation in mathematics so the way it goes is that we're gonna ask you a question and then you're gonna give us what you think about it and hopefully eventually arrive at a solution. Along the way if you get stuck we will give you some input on how to proceed or how to help you think, so don't panic, just keep pressing on that problem, eventually we get there. Ok so before we start can you say a couple of words about yourself. Hiya, yeah so my name is Bethan, I'm yeah from Cambridge and I'm yeah really interested in probability and stats within maths. I really like how, especially probability, it's it...you can come up with some really really pretty kind of solutions to problems but they're always so like useful kind of in applying them in stats like it's really...you can really see kind of where where it's useful in the real world as well. Ok. So unfortunately we won't be doing any probability or stats today, however, we're going to do a problem that I hope you will agree is pretty so I'm glad you did mention that. So the the way it's going to work is we we have this interactive whiteboard which I will be able to draw on, so I'm going to start by just drawing a square which should hopefully appear on the screen for everybody and you yourself Bethan will be able to annotate this and that will also be visible to both myself and Luc, so we're all kind of working on a shared document live so you know you don't need to worry about holding up anything, just get all your thoughts down on here and of course you can speak to us as well. Now the reason I drew a square is that's going to be the starting point of our problem and what we're interested in doing with this square is filling it with rectangle-shaped tiles. Now there are obviously lots of different rectangles that we could use so we're going to slightly limit it and say that the rectangles have to be in this ratio where the long side is twice the length of the short side, so the ratio of all the rectangles can has to be two to one. Now those numbers can be anything, so it could be two to one, it could be four to two, it could be a half to a quarter, just the ratio has to be two to one because the idea is we want to fill the square with a certain number, let's say 'n', rectangles. So for a simple example, I've drawn a square here, I could take a rectangles, two identical rectangles like that, so again the square can be any size, we can shrink the square, we can make the square bigger because of course the only key thing about the square is that all the sides are the same. So here for example this would be an 'n' equals 2 tiling, we have filled our square with two of these two-by-one rectangles. And what we are interested in trying to work out is, for what other values of 'n' can we fill the rectangle? So which 'n' tilings are possible. And a final thing, so before you sort of jump in, is just to say the rectangles do not all need to be the same size, right? So the example I've given you has two rectangles in this particular size, you could use six rectangles with three in one size, two in another, and one in a third different size that's all ok, they just have to have this aspect ratio of two to one. Ok. So as I said, in general we just want to know for which 'n' can we do this, can't we do this, it's a very...hopefully wide-ranging kind of open problem for you to have a think about. Yeah that sounds really interesting actually. Ok so I think straight off the bat I'm kind of...my instinct is that because it's integer values of 'n', I always think of like an induction kind of a proof so I think I'm probably going to start with trying out a few kind of smaller values of 'n' see...and see if there's a way I can kind of...I can kind of put the squares together so, I don't know, yeah like find some sort of trick that I can keep building off of ones I already have, I think. Yes. Ok so...right so for 'n' equals one
Ok so 'n' equals one. Yeah we can see that. That doesn't work obviously because like it's a one by one square you can't...yeah put two by one rectangle on that. Yeah. And then so yeah 'n' equals two is fine. Yep
That's good. Ok so then for 'n' equals three
so I'll try and fit some together. I'm not sure if three would work really, it doesn't seem like it's. Yeah, can you...can you explain why
Yeah so
Anytime you're doing this, there's always going to be one rectangle that's the top left corner, right, so maybe I'll say this corner so this is...I'm not going to put any like this side or this side and then I'm either gonna have to
put one here, which is gonna be like any size,
or maybe it could work, I don't know, and then so if I say this is a one...one-by...or maybe a two-by-one rectangle then this can be x by 2x
so then this length...But before you move on, does it have always have to be, you know, for the top side, does it always have to be two or more than two
I mean two rectangles, right, so just to cover the top side you need at least two rectangle or you need fewer? I think you can just have one but I think in this case I could just rotate it, so kind of rotate it this way and then I can take this like x by 2x rectangle to be the top side anyway, so I think I don't need to consider that case. Ok, sorry my tablets just...
Right it's coming back
and then yeah so then this distance is 2x minus 1 and this distance is going to be 2 plus x
so yeah that's never going to be
double
why is the bottom one two plus x?
Oh sorry yeah...oh that's just two isn't it, just two, so then oh yeah I guess if I could put two...two x minus one equals two then equals. Oh equals oh four, sorry. Equals five and then x equals five over two, so actually yeah if I had it being. Oh but wait that doesn't work
four oh so this would have to be four anyway
that doesn't work
Well so you've worked out a value, a value...potential value for x and as you said it looks like this is going to have to be two and four and you're right in that that one would have to be four, so then what can you say about the other side of the square. So you know the left-hand side, you know the total length now of that left-hand side. Oh yeah so so then it will be a five by
Oh yeah so so this one's gonna be...wait what's that one gonna be...that one's going to be five, this one's going to be over two
plus two. Oh well maybe it's not gonna be two. Wait sorry I'm confusing myself I'm not quite sure what I've done.
Yeah no rush just keep...
Ok so that's 1 by 2 and then that's x and 2x, so five...no that means it's five by three
So is it a square? Yeah.
Yeah sorry what was that? Is it a square. You just worked out. Yeah wouldn't be a square. So yes you know x, don't you, because you've said...you said the right hand side has to be five. I mean you kind of got there but didn't perhaps put the pieces together, right? You worked it out, you sort of figured out that x had to be two and a half so then the top length is four and a half and the right hand side is five, it's not a square. That's right, there's no way I can. Yeah. Ok great so I think, at least I'm happy enough, let's see, so three isn't possible and I think you know that was I...you know a reasonable argument to say why, so right so if we go back to your list, we've said that one we can't do, two we can do, three we can't do. Can't do . And then four I think
let's have a think. So again I'll just start with one in the corner
and then
if I were to kind of
put one like that, just kind of...it might there might be a one which actually does work . Hmm oh wait I don't know how to get rid of that
Ok.
Three, four, no that's not going to work I don't think. Exactly one and a half and then that's one. Yeah so that wouldn't work that would be like that
or I could do
one, two, three. Oh no they don't have to be the same size do they
Sorry I'm just kind of having a mess around at the moment. No, this is good, this is what you should be doing, yeah. Ok sure
So...I think...I think if I did that it's probably not useful now but I guess I could use that as the...That's useful for something else, yeah, so that that pattern is useful for some things but not not for the k is then equal to four but it's useful like..Yeah that one can be for later not for now. Why why is it potentially...why do you think it's useful for later? Well because I've left a little square so I think that's what I can use to maybe go by induction and if I w-...if I were to find another. So what statement would that give you? So suppose that we look at that configuration so what does it say about the possible value of 'n's. So if I find any n that does work then I can add four to that n as many times as I want to so...That could be useful so maybe you want to write it down to return to it later. So four...so it's going to be n zero plus four, pretty much. And then actually if I could find something that's co-prime to the four, so without any common factors, then I think past a certain point then all integers would work. Yeah we return to that later, we just wanted to keep this in mind. Let's press on with the case. Sorry I'm getting a bit ahead of myself.
Is there one that should work or am I looking for a counter proof to it? Maybe you want to think back about the case and you do the three, just try to have a more robust way of arguing why n equal to three doesn't work and maybe that can say something more n equal to four. Think about the corner. Ok right so
along...so do you mean like along the top side I guess I have any number up to w-...technically up to four of rectangles at the top. How many corners can, you know, can you fit into a rectangle. Imagine that you have four...three or four rectangles sitting there then you look at the corner, you have four corners yeah, so what's the relation between the number of corners and the number of rectangles?
Ok so in each rectangle there's four corners and then
four...You're looking at the corners of the original square. You have only four corners of the squares and then you have now four rectangles. Oh so there's gonna have to be...wait sorry I'm not quite sure what you're getting at with that. But where should those corner or reasonable corner of the square belong to? You know, how many of them, well I don't know, try to find a relation between the the original corners and the little squares that you have, sort of you know, where each of the corner belongs to. So if you...if you think Bethan, if you say we have a square and it has four corners and now you're limiting yourself to saying I have a maximum of four rectangles, so obviously there is a sort of finite number of solutions in terms of how many corners each rectangle can touch in your square. Right ok so, for example, could each rectangle touch one? Could you have one rectangle that touched two corners and what would that mean for what was left? Right that's interesting, yeah okay so each rectangle's either gonna be in one or two corners and it can't be any more...more than that. Why not? Because if you had all three corners then there's like not really any space to put another rectangle and it's going to be a two by one shape. It doesn't fit in the square, right? You've turned...your rectangle is hanging out of your square, basically . Yeah, basically, yeah. Oh yeah so if I'm thinking of it like a...like that and if I wanted to touch all three corners then it's got to be outside of the rect-...It's going to hangout. Yeah so you can have only either one or two inside, or maybe not. Ok so if I'm...right, if I'm gonna say that this one maybe has...is touching two of the corners then I have kind of a little...another two by one to fill with three rectangles. Yep. Is that possible?
I think, well you can't just stack them together, that wouldn't work because the dimensions would be off. Yep. You couldn't...you can't just have one that's. Oh yeah, again, you can't just have one that's touching two of the corners because then that will be the whole. Yeah, so each of them has to touch one corner but there's only three rectangles so that doesn't work. Yeah what about the case, you know, each rectangle contain exactly one corner. Ok and then
so that would be the case of
one. So yeah I can't have...really have two like that because
because then if I were to put any this side then this wouldn't be a corner and if I was putting any this side then that wouldn't be in the corner either, this one. So
I would have to stack them like this in order to have two in a corner
and then that applies to all the corners as well because like they're not really unique corners so yeah that leaves with this kind of arrangement and that leaves the square in the middle so that's yeah, so four doesn't work either. Ok. Ok good so we've got...I'm happy enough with that so we've got: one obviously we can't do, two we can do, three is a no, four's a no. It might be helpful to have something else that we can do at this point. So yeah I guess we could just put put some rectangles together and count them. Yeah put them together until they make something good. Ok so I've got one...I'll...maybe I'll do four together and then like that, which is
where...I think that's a square so that'll be two, one
one and then. Oh my thing's gone off again. And then a half times four so yeah that's the square. Yeah yeah, so five does work. Ok good. And that automatically means we have...from the thing with the four back here that means you automatically have 9, 13, 17 and any number added on
Should we start keeping a track of these lists? Oh yeah. So what kind of integer have you see. This is a prime number.
You said five and then five plus. Oh yeah so that's any five plus 4m for any integer m. Ok.
Yeah and then yeah so. Don't forget...don't forget that you had another number that you could do previously. Oh...oh yeah and then also two plus 4m. Yeah and then that's like half of the num-...kind of the numbers past...past five anyway because two and five both different values when you kind of...like modulo four in like. Is it half the cases? Sorry? Is it half the cases or not quite?
Yeah because it's every...because it's a multiple of four plus two and a multiple of four plus one at this point, so yeah we just need to find a multiple of four plus three and then a multiple of four plus zip plus nothing. Yeah. So if I find a multiple of four then it'll be any multiple of four...four past a certain point so...yeah so that...I can just put four... four of the blocks of two together, so that's the same block
these are all the same block of two, so there's two, four, six, eight. So this is a block of eight. Does that make sense what I've drawn? Yes. I think so, you're just taking the original one we had of two and then just doing four copies of it, I think. Yeah yeah so then this is an eight plus 4m. Yeah. And then I just need to find one which is a...That leaves what. Multiple four plus three, so maybe I could try seven or something, but I'm not sure...seven...Yeah if you manage seven then you manage all, right? Is that...? Yeah so if I manage seven then I can manage all of them past...past
six, I think. Past six? Well you already did six, you already did five. Yeah I mean so if I can find seven then that means I've all got all integers past six. Five. You have five as well. You drew one earlier for five. Oh yeah of course, so all integers past five...So for seven
hmm
Yeah so again I guess I could try with having
this one in each corner
maybe I could try that and then
one, two, so then that's two by three and then I need another one
here maybe there's one here
one, two, three, four, five, six and then does that make seven? I don't know if this will work. One, two, three, four, five. This is five by four. Oh no that's not a square, I don't think.
Yeah that's not a square. So seven is quite a difficult one to think about. Seven is possibly the hardest integer to look at, so I wonder whether we can...so I really like your approach of using modular arithmetic and saying if we get certain remainders, modulo 4, then as you said yourself, if you can get 7 then you're done, or maybe if you could get 11 or 15 then, you know, you would have all the numbers past that point. Now I wonder whether there's another possible way of constructing these rectangles so you came up to earlier the thing that you starred when you said, oh I can put four around the outside of a square and that's why you were doing this discussion about plus 4m. Now there is actually another way of generating higher values which isn't modulo 4, it's modulo a different integer, and that can be very helpful to rule out a lot of these remaining cases without having to worry about seven. Sure, ok. If I can, so if I can find something that's odd then that...automatically gives...gives me the answer because then four and this odd number, if I can find one, wouldn't have any common factors so then I can always kind of construct them in a way that
any integer. Yes, that's a certain point, like you said, yes. Yeah so if I can find the three...oh I think the four...
that maybe made a score
Ok, maybe if I can have
one here
another one
here and
oh that's not going to make it square
So remember here, rather than trying to find a specific solution, you're trying to find a method, like you said, sort of like...of induction being able to generate more rectangles. Yeah. So I might suggest if you go back to the very first one that I drew of the two rectangles. Now, from it...from that configuration, is it possible to, let's say, divide it up into...starting from that configuration but kind of divide it up to get a different solution, a possible number you already have but using a different method?
That kind of makes sense. I'm not sure, wait, so
so using a number I already have. Wait could you explain that again, sorry, I didn't quite...Right so if you look. Sorry, go ahead. I was just going to say so the...the example...so if we go back to my drawing of two and let's see if I can use a different colour to highlight this for you, so let's do this in blue. So you sort of started with these two and then you had this really good idea about saying, well I can just always add these ones around the outside? Yeah. Right and then that was kind of like a...I think of it as like a building up method, you can take a solution you have and you can always add four and that's how you were generating all of these other ones that you had. Now I'm saying, is there another way of instead of like adding four by doing something could you add another number ? Right ok. To create that kind of sequence. Sure yeah. In a similar manner and starting with two is hopefully a good...a helpful place to begin. Oh ok.
So maybe if I can put one here that kind of overlaps
there
and one there
one, two
Yeah that could lead to something...
You're basically back where you were with the last one. I keep on getting back to the same configuration. oh dear. Instead of adding them on, can...or like...can you like divide up the ones that we already have?
So instead of adding...so like you...you've added up or, I call it the building up method, of adding these four around the outside, can you take the configuration you have maybe focus on one of the rectangles can you divide that up into smaller rectangles such that you get a configuration that works? Oh right ok that's
Hmm. Yeah so if I can divide one of the rectangles into
a different number of rectangles. You've actually already done it in your solution for n equals five. So if you compare your n equals five solution with your n equals two solution. Maybe we can put them next to each other, if we draw them next to each other
Oh ok. Oh of course, I can just...oh yeah ok that's...I can't believe I missed that, ok, so it's just...I don't think I explained it very well either so that's...my apologies too. So what's happening...what's happening going from the two case to the five case, what is it that you're doing there? You're adding three to go on to. Right. So how does that help us? So that means you can also have any...so anything you already have so say two plus 3m
Yeah and then you can also...so if we say that m is three because it's multiplying by three and then to have...well I'll put this as n as well and then n plus...m four because three and four are co-prime then that means you can just. I think, let's go back, we already said that we already have the three plus 4k that remained so now you just learn another way of settling it can you find something...does that give you something or not fine? Right so if you manage any multiple of four plus three then you would get all the bigger numbers, right?
So what have you managed? So any integer past five I think...if we assume that seven work then that means...No we didn't know that seven works. Oh ok. Oh right so...but now you just learned a new trick so does it say anything about this multiple four plus three...or not quite?
So at the moment we don't have...we don't have seven which means we don't have 11 or 15 or 19 or 23 from your original method of adding fours. However, we now have a new method of adding threes so can we get any of those numbers? Oh yeah, yeah so you can add four and add three to things so, oh yeah, so for every...yeah. So what would be the lowest one that we can get now because it's not seven yet. Oh. Yeah. So we've got
the smallest would be...so wait, wait what was the number...we've got two, fi-...Yep you had five, yep. Did we have...we had six as well didn't we. Yeah six. Then eight. We've also...we've got...why do we have eight? Oh yeah we have eight as well. Yeah I think the only thing that we care at this point are 7, 11, 15, 19 and so on and so forth so which one of those can you solve. So for 11 what we could do
is eight plus three. Yeah. Perfect. So eight with this kind of yeah...
and then...So that means that...15...Yeah. We can do for...how would you do that, so you could put
eight plus, no six plus. No, no you had...you had two method of generating either you add three or you add. Yes, sorry I'm just trying to like work out which one you add together to get 15. What are the two methods that we generate new numbers? You add three or you add four, like any number of times. I don't know why I can't quite see it though. Wait so the...You...you just showed us...You just showed us you can do 11. Yeah, yeah, yeah, right? Good, there we go. So you just showed us, so that was the new thing, right? 11 was new because that plus three method allowed you to take your drawing of eight. Of course, yeah. Add on three and then like there's a lot going on, don't worry, and then you said oh but we can just keep adding four so now you've got 11, 15, 19, you've got them all, so it all comes down to, rather than being left with like a quarter of the numbers you now literally just have seven as the only remaining...as you don't really one. Now there was a reason I...I sort of stopped you when you were thinking about seven because seven is really hard. Seven is possible but we are pretty much out of time so I didn't want you to spend all of the remaining time trying to figure it out because it took me a very long time to do it but I will show you the solution just before we go because I'm sure you want to see it, so seven is possible which, and I'm gonna attempt to draw this now, is gonna look like...so you have a big one at the top... now I have to remember this, you have these sort of smaller ones along here
and then...so that's now four...you need three different sized rectangles and then I think you have two more of these small ones and then that big one like that, so I think that will be true for seven, so you can...let's check my numbers so what that's going to...Take a six by six square. Yes, that's a better way to do it, thank you Luc. I should have drawn this using the actual squares on the app but that one should work if you think of it as a six by six...six by six square you can figure it out and you've got three different sized rectangles there to get the seven, so...Yes so the answer to the question...what was the answer to the question because we said for what values of n can we tile the square with two by one rectangles...what's the final answer? I think we got it was any integer greater than five or two as well. Yes, perfect, so it's...I find it surprising that that you can do it but almost every single number apart from just three of them. Awesome, so just...just before we go, do you have any final questions for us or anything that you want to ask at this stage, there's no...you don't have to just your opportunity in case there's anything that's unclear? I think that's all clear for now but yeah thank you very much. Thank you, I hope it was fun. Yeah, thanks a lot and good luck. Thank you, Bethan, bye-bye.
So now that the interview has concluded we of course will have a very brief chat amongst ourselves as tutors to to check that we agree on the candidate's performance, so Luc how did you think that that in general went and what were you looking for from a candidate there? Yes I quite like how the interview went. It's clear that at the beginning the candidate struggled quite a bit and made some silly mistakes which we probably don't care about. Nope. Yeah, what what has clearly shown us is that the candidate didn't have a prior knowledge of the problem, was trying hard to solve it and she keep crushing at it, eventually arrive as a solution. We can clearly see the thought process at which the candidate arrived at the problem. I think that she has shown that her enthusiasm towards mathematics. Yeah. Yeah, I think she definitely enjoyed the process, that was something that came across to me which I think as interviewers is something that's always really nice when you have a candidate that is actually having fun because as we mentioned at the beginning we...we want...you know you are meant to enjoy this process as much as is possible in the circumstances it is just, you know, three of us as mathematicians kind of working together to try and solve a problem, so yeah I think you made a very good point about the beginning. Obviously she was a little nervous, a little flustered about trying to figure out the sides of the square but we don't really care about that, like you said, like you know that's not what we're looking for in these interviews, it's that thought process and that ability to be creative and to spot patterns so I think being able to see the relationship between the drawings and realising that you can continue to add four so, you know, that was spotted very early on and it was really nice to see the candidate then see the use of that, so not just realising that we can add four but immediately see, oh that allows us to generate a sequence, so I think...I think that was a really...you know really good thing to see and yeah the kind of thinking definitely that we're looking for in a candidate. So yeah I think, overall, I think that was a good interview.