Oxford Maths Admissions Interview Question with

Steve from blackpenredpen answers a real Oxford maths admissions interview question set by University of Oxford Mathematician Dr Tom Crawford. This exact question was used by Tom in the 2018 Oxford maths admissions interviews. The question looks at surfaces and volumes of revolution via a famous shape known as Gabriel’s Horn, which has a volume of pi but an infinite surface area.
This is part 1 of the interview - watch the 2nd part on the divergence of the sum of the reciprocals of the prime numbers here: • Prime Reciprocal Serie...
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Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: www.seh.ox.ac.uk/people/tom-c...
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[@TomRocksMaths]

Watch part 2 of the interview on the sum of the reciprocals of the prime numbers here: kzbin.info/www/bejne/pZqXdodmgN-Jn6s

[@sonic5d]

I didn't know there was a part 2! I will be checking it out now.

[@blackpenredpen]

Thanks for all the editing and the opportunity to collab. It was super cool and super fun!

[@dekunut6416]

looking forward to part 2!

[@owen7185]

Steve I just came across Tom's channel now. Your videos are awesome Steve

[@owen7185]

Passion is contagious and Super cool

[@KQJ_Diya007]

Yay

[@OH.Tousif]

After a long day I came to know ur real name is steve😂

[@drpeyam]

Awwww, my two favorite mathematicians 🥰🥰🥰

[@inovexa4039]

Pls dr peyam , u should be there as well. U are my fav

[@martinstent5339]

An oxford professor with a Poké Ball tattoo on his arm. That, if nothing else, makes me feel old.

[@aRskaj]

Not a professor, but I get your point.

[@martinstent5339]

@@aRskaj OK, then just an Oxford Fellow. But he actually says at 00:47 "being an Oxford professor myself". I guess he was jumping the gun a little!

[@jeffreyadams648]

Earrings.

[@TomRocksMaths]

You're only as young as you feel Martin!

[@LETTYONLY1]

I think he's trying to be down with the hood ....it's that annoying look how cool AND. clever I look .....🙄

[@cosmo1413]

Note to self: impress the examiner with your pen-wielding skills during the interview

[@andrewlees494]

Today I broke 1500 on my chess rating. I felt pretty smart. Then I watched these guys and I realise that I’m the TikTok to their Wikipedia.

[@3prismaticpulsarmanuupadhy535]

Don't worry, man. I assure you, I can solve these question easily as well, but you'll probably defeat me in a game of chess! :)

[@spoony01428]

brool

[@TomRocksMaths]

You would no doubt destroy me at chess Andrew...

[@samuelreese5826]

similar thing happened to me except it was when i won my first tournament 1800-1900 10-0 and they broke my mind at the surface area of gabriels horn

[@DieLazergurken]

it's because you don't get smarter by playing chess. You are getting better at playing chess.

[@Mark-sc4bu]

Brilliant. It makes you realise just how good at maths you have to be to get into places like Oxford and Cambridge. It also highlights really well the difference between being able to 'do' a topic in maths and really understanding it, and once you've got the understanding how you go about applying it. It's also great to see how much fun you can have doiing maths.

[@jameszhang9326]

With good education brought up in developed nations, you should be quite geared towards those academics, especially math, which is largely logic. Th rest of math, you can explore on your own when you're interested (dopamine rush?), but do take note that most mathematicians who are too into their games might at times, lost touch with the current world...

[@user-uh9bo2im1h]

Well kinda not I mean those questions surely are quite hard to someone who hasn’t looked into math. However for someone who enjoys doing math (in their spare time)at least the first question should have been easy. I’d also like to note

[@sebgor2319]

Nah this question wasnt hard. Im 15 years Old, and I would solve it, but it is really cool that surface is infinite, but volume is finite

[@Philgob]

this is easy as shit what level would be the people doing these interviews? like good high schoolers?

[@artophile7777]

​@@Philgob The level when people try to get into universities. Now you guess when that is.

[@daykbd]

I feel like I'd have so much fun doing this interview if I knew a whole lot more about mathematics. It`d totally feel like playing a game.

[@TomRocksMaths]

It is meant to be fun yes!

[@william7yifans]

You are cooler than any math prof I've ever had, by quite a wide margin...

[@phnml8440]

i love this video so much you can really see how he feels happy after he gets the right answer

[@Eudaimonian42]

You are seriously one of the coolest guys I have ever seen who is like genuinely into math, keep up the good work!

[@TomRocksMaths]

[@ryaneakins7269]

Turns out that even mock interviews I'm not actually part of can make me feel as nervous as an actual interview.

[@TomRocksMaths]

good practice at least!

[@sohamsankar2490]

man I think u have an impressive personality(happy , cheerful) for a mathematician

[@TomRocksMaths]

But, I have the best job in the world so of course I'm happy :)

[@cocobread2569]

This is how uni professors should be like, casual, approachable, smiling. i feel jealous i hate my professors

[@mikeheyburn9716]

Absolute joy this. It's great on so many levels and especially in helping the kids I teach to see some of the interview process. Thanks guys.

[@Bobbius-il9rf]

This was so awesome to watch! 😁 Sad I've only just seen it now. I had 6 interviews for Chemistry at Oxford but unfortunately didn't get in. I cherish that experience because it really taught me about what interesting questions actually are. I've thought a lot about the answers I gave and the likely answer expectations, but my GCSE grades were meh compared to the average applicant and there were no entrance exams for Chemistry when I applied about 12-13 years ago. It's always been a dream to study at Oxford. I have a Chemistry degree. I'm 2/3 through a maths degree and doing well and loving it. One day maybe I could come back to Oxford and ace the interviews. Well done Steve for making that look easy! In the normal case there would typically be a lot more give and take between interviewer and interviewee I presume... it depends very much on the experience of the interviewee (and interviewer to a degree). Plus the passion in the way Steve communicated, written and verbally, would surely have out-shone anything incorrect he may have written accidentally on the day. Then you have me who blurted out an incorrect answer to the first question I was given in my first interview. 🤣 I'll never forget the bond angle of CO2 ever again though! 🤣 Luckily idiocy is often a very useful quality! 🙃

[@565ChAr]

This is such an awesome collaboration between two of my favourite maths KZbinrs. Unfortunately I was rejected before interview this year which sucks especially considering I was rejected after interview last year haha, but that was so much fun to watch!

[@TomRocksMaths]

I'm glad you enjoyed it, but sorry to hear about your application. Keep your head up, there are SO many other brilliant universities out there that would love to have you :)

[@565ChAr]

@@TomRocksMaths Thanks for the kind words! I wasn't expecting a response haha. At the end of the day, there is still the opportunity for a masters degree at Oxford should I choose to pursue it, and I will be sure to make the most of wherever I go. One things for sure, I will stick with watching your videos for as long as I can!

[@alexwhitfeld]

Same thing just happened to me now , do you reckon its worth it to take a gap year and re apply , or should i go to warwick and apply for masters. Are you doing a masters now or next year?

[@razvanpauliuc5410]

okey this is giving me such wholesome vibes, you two are awesome

[@TomRocksMaths]

[@AnniePrettyFace]

This was so lovely to watch! I like Steve as a student :)

[@mathhack8647]

Great content. I am grateful for both of you for your commitment, modesty and high quality of your video content. 👍

[@tooprock]

Once again we witness that people are cool, when as feel confident about something. Not being able to solve a problem makes us all nervous. Great video and wow, what a knowledge man.

[@musicramblingsbyeben]

Steve is very good! Tom really rocks! I really enjoyed this as an ex Uni maths lecturer decades ago.👏👏👏👏

[@TomRocksMaths]

For more special guest appearances check out the 'Interviews' playlist including videos with 3blue1brown, Hannah Fry and several famous mathematicians: kzbin.info/www/bejne/i6S1l3h5eLWmebs

[@jattprime2927]

wow, this was so amazing! thank you! I kinda always thought all professors at Oxford were really old and closed-minded, thinking only they are right, keeping minimal interaction with "normal" students or others, this has been really eye-opening for me to see how such fun ppl are at Oxford uni. Also, it was great to see these interviews for real and the things that go on in them!

[@TomRocksMaths]

I'm glad you enjoyed it Jatt :)

[@nyunai298]

Wonderful Steve. You're admitted into oxford

[@jonahbresler4317]

I love watching both your channels and I never realised you had a collab! I love how happy red pen black pen looks when he knows the answer😂❤

[@quahntasy]

*Loved the collaboration, this was amazing*

[@TomRocksMaths]

Glad you enjoyed it - and thank you!

[@user-en5vj6vr2u]

fellow andrew dotson commenter

[@fisherman_02archibong7]

I appreciate you Dr. Tom

[@TomRocksMaths]

And I appreciate that!

[@jameszhang9326]

Got to catch them all (all the equations).

[@jayringo77]

I'm forwarding so many of these to my youngest. He's a budding STEM major in his freshman year of high school. Hopefully will give him the confidence to consider some of the top schools seeing what the interview questions involve.

[@TomRocksMaths]

awesome - best of luck to him!

[@cariogenic]

Thank you very much for this video. Before I always treated dx like part of notation and just did integration as though it was algebra but now this video actually explained what dx is.

[@OscarMorales-uj7nz]

Best collaboration ever

[@TomRocksMaths]

Thanks Oscar - glad you enjoyed it!

[@Siance1992]

@Tom Rocks Maths i love the fact that you break the traditional visual presentation of teachers, while being good at the subject.

[@TomRocksMaths]

[@safwanislam111]

This was soo much fun!!

[@AbhishekSachans]

This was fun to watch!

[@TomRocksMaths]

Thanks Abhishek!

[@SanneBerkhuizen]

This was fun to watch, can't wait for the next one

[@TomRocksMaths]

Glad you enjoyed it Sanne. Part 2 should be up in the next few weeks.

[@dusscode]

@@TomRocksMaths I know that this is quite late, but any updates on part 2?

[@TomRocksMaths]

@@dusscode as luck would have it I started editing last weekend so hopefully will be online in the next few weeks :)

[@sampresman5128]

@@TomRocksMaths can't wait

[@technodris2780]

These question looked really fun

[@Thrawnio]

The Horn of Gabriel, Fascinating!

[@athysw.e.9562]

Wow great to see Steve as a student for once !

[@TomRocksMaths]

He did well don't you think?

[@athysw.e.9562]

@@TomRocksMaths Of course, as a good teacher, he was certainly a good student as well.

[@wesleyrichardson3427]

blackpenredpens so humble and a quality maths proffesor and all round person I wish nothing but the best for him...

[@TomRocksMaths]

Amen

[@VimokshaBandara]

I respect him ❤️

[@jasonfuller7073]

Yes, the volume of the pi of the formula and axis. Definitely.

[@dabossbabie3605]

Thanks for the insightful problem - I've heard of Gabriel's horn and did these calculations a couple years ago, and now looking at problems to hopefully have some questions for an interesting school math competition, and just to practice my own maths, this is pretty amazing to watch. Right now my high school is new [only around 2 years old right now], and I really don't have much "history" to go off of, so these are the types of problems I hope to add... problems that require no more than a basic intuition of derivatives, integration, and a bit of cleverness since at the end of the day, cleverness is what separates the great from the best ya know. If I'm smart enough to apply - it would be pretty fun to have you or a similar interviewer, since games are a lot more fun than a 80 year old staring you down lmao. tldr; tysm. i needed this, and it was pretty fun to follow along

[@owen7185]

This stuff is gold

[@Almirante1741]

6:02 Cavalieri's principle 😍 I have been told about that in my uni

[@ShirleyBekker]

I have only just discovered this channel. I absolutely love that you are an Oxford math professor with personality and character. You probably (without knowing) are making a lot of students feel like they could envisage themselves at an institution with professors like yourself thereby encouraging them to apply! I can’t put into words how happy it makes me to see this.

[@TomRocksMaths]

amazing - thank you!!

[@Yash42189]

Take this guy to Oxford!

[@jennishaagrawal1622]

It’s amazing you can fill the paint but you can’t pain🥶🥶

[@israelvicentemartinez5558]

Thank you for sharing with us....

[@TomRocksMaths]

My pleasure Israel!

[@ycm8888]

This video is awesome! Will there be a second video on these interview questions? I hope they will help me preparing for my Cambridge interview in December.

[@TomRocksMaths]

Yes, part 2 will be out soon (and hopefully before December)

[@guyguy1811]

Such a humble guy man.

[@TomRocksMaths]

Steve's fab isn't he?

[@sechi7239]

If the paint is infinitely thin it would actually be possible to cover the whole surface with LESS paint than pi units. The only reason Gabriel’s horn seems paradoxical is because we instinctivily apply physical properties, eg. like paint having thickness, to a mathematical construct which is impossible to build in the real world. But it is actually possible to imagine paint with finite thickness being used to cover the surface. Create two horns, one bigger than the other, fill up the big one with paint and then insert the smaller horn inside it. Remove the bigger horn and you are left with a horn that is both filled AND covered by the same amount of finite paint :)

[@sonic5d]

Steve, you have been cordially accepted to the University of Oxford!

[@blackpenredpen]

I feel honored!

[@koshin6585]

wow, my two favourite mathematicians together 😀😀

[@marius4363]

this man is so smart

[@andrewcorrie8936]

Steve (bprp) more of a Cambridge guy I reckon... but this video was a delight. Thanks both!!

[@TomRocksMaths]

Glad you enjoyed it Andrew :)

[@Synchrowize]

Intuitively i'd say the volume is heavily influenced by Y in the formula Pi.y2.dx. whereas the surface is heavingly influenced by dL which clearly will diverge.

[@Roarshark12]

Wow what a superb video!!

[@TomRocksMaths]

Glad you enjoyed it!

[@sssilky3317]

pretty proud of myself for managing to solve them along side backpenredpen, just took my calculus 1 final yesterday.

[@TomRocksMaths]

I hope it went well!

[@sssilky3317]

@@TomRocksMaths it did, I'm taking calc 3 now

[@jamesl8640]

Wow no wonder I never thought about applying to Oxbridge I can do everything at home but I was nervous just watching this

[@IBM1111]

Oxford interview made easy

[@gabrielbarrera3447]

Great video

[@discostu483]

These 2 makes a guy feel about as smart as a rock

[@julialovisa4239]

I didn't understand much but I really like your voice. My boyfriend thinks I'm crazy and that you don't watch advanced maths-videos you don't understand simply because you like the voice. But here I am

[@vivekgusain7311]

Now that's some amazing stuff im your subscriber since you were having 3k subs But it's always awesome to watch these kind of videos with 3b1b also 😁 I wish you could have a collaboration with the veritesium also

[@TomRocksMaths]

yes I love veritasium!

[@TheAyhan76]

Best Teachers the next generation ✌️✌️👍👍🤯😂🤣

[@shk439]

Is it blackpenredpenbluepen now?

[@mobizoid2571]

That's what I've been thinking all along dude😅

[@firemonkey1015]

This man single handedly got me my degree

[@Qermaq]

To me, the real paradox is why the integral from 1 to infinity of x^-1 dx diverges but the rotated one converges to pi.

[@markmcpeake715]

This was great. I loved the real world interview question. I guess it's removed from the interview question list now! Best of luck. Be safe all.

[@TomRocksMaths]

Glad you enjoyed it Mark. And yes, I can confirm I will no longer be asking this question!

[@markmcpeake715]

@@TomRocksMaths Thks. One thing that springs to mind about this problem is that we know the outside surface cannot be painted, but the inside volume can be filled, so my question is, can the inner surface be painted?

[@Jooolse]

​@@markmcpeake715 Filling the inside (finite) volume with paint is the same as covering the (inner or outer) surface with a coat of paint decreasing in 1/x. So, both can be done with a volume Pi of paint. But painting the (inner or outer) surface with a coat of constant thickness would necessitate an infinite amount of paint!

[@markmcpeake715]

@@Jooolse As I thought, but wanted to hear it definitively. Many thanks.

[@RoyceDima]

Alternative solution for the final integral is substituting u=x^4+1, giving the integral of sqrt(u)/4 which clearly diverges

[@mehdikarimivagargahi5178]

Awesome!

[@TomRocksMaths]

Thanks!

[@aviralsood8141]

I SPENT 30 MINUTES TRYING TO FIND OUT WHAT HE MEANT BY SURFACE AREA BECAUSE THE ANSWER I HAD WAS ALWAYS INFINITY. Man I should really just sit back and watch these videos.

[@zanerobison2956]

I’m excited to apply in 2021! Hopefully I can get in lol. p.s. great channel keep up the great work

[@TomRocksMaths]

Thanks Zane - and best of luck!

[@zamamkhize6579]

Amazing video! My first time hearing about Gabriel's Horn. Really fascinating.

[@TomRocksMaths]

It's one of my favourites for sure

[@s.y.4968]

If all Oxford mathematics interviews are like this, I'll definitely encourage my son to apply. Maths should be fun! Do you ever teach computer science students?

[@TomRocksMaths]

Afraid not, just maths. I do try to make my interviews fun though for sure :)

[@exopolitikgermany1767]

When you start to study it is not than entertaining most of the time because you have to learn the basics of higher mathematics. It is great when you start to see the patterns and when you can apply the rules easily to solve different problems.

[@mikejackson19828]

I never knew he was called Steve!

[@facr]

I Like the way you present your videos and explain things. Kind reminds me of a math teacher I had. Very cool.

[@TomRocksMaths]

Awesome thanks :)

[@darcash1738]

Epic problems. Glad to see i havent gone rusty in my comparison tests 😆

[@gledsonjuniormonteiro2995]

Isn't anyone gonna talk about how beatiful the answer to the volume question is?

[@TomRocksMaths]

'unexpected pi'

[@jameszhang9326]

Magic of Infinities. (Infinity might have been a strange function rather than an exceedingly large number...)

[@user-tk2tt7do4y]

I think Oxford university is very important . Especially , because of it, more students are interested in science thanks you. Mr Sir I love mathematics . Even I found some comforts from mathematics can you help me? I will wait a day

[@dancroitoru364]

It's analogue to a model of the universe, geodesics on the surface (light traces) travel for ever without ever coming back to the same point so we say the Universe is infinite but the circumscribed volume is finite so therefore there's finite mass in the Universe.

[@losis5092]

I don't understand much, but love tgeir genius.

[@TomRocksMaths]

Don't worry, you'll get there if you keep working hard :)

[@MarkusDarkess]

If you set Gabriel's horn in space with the horn's hole or orpheus facing down and it rains what water would fall and the floor of Gabriel's horn is gravity. The liquid will flow down. And collect in its opening and will fill to compasity and any excess water will fall into space. I learnt of Gabriel's horn from red pen blue pen. Several days ago. When he told me the filled paint couldn't cover the surface.

[@HenrikMyrhaug]

The paradox isn't really a paradox. If you paint a surface in the real world, you would need to apply a layer of paint with some thickness, and below a certain thickness you wouldn't consider it to be properly painted. But the horn gets narrower and narrower, so no matter what thickness you pick for the paint layer, the horn will be thinner than that at some point, meaning a finite ammount of paint wouldn't "properly" paint the surface of the horn in the real world. Any volume can be split into infinitely many 2d surfaces however, giving a volume an infinite ammount of surface area. So mathematically, a volume of paint can cover an infinite surface area. It is kind of like how you can travel a finite distance by halving the distance to your destination an infinite number of times. So long as the time steps also become infinitely small, you can do this in a finite time, but if you needed a set time per halving, you could never finish.

[@ektasaikia7153]

The whole time I was mostly looking at the pokeball tattoo. On a serious note, what a refreshing way to look at interviews! This was so interesting! And I'm from physics background 😅

[@TomRocksMaths]

Glad you enjoyed it!

[@johnroberts6876]

I remember in primary school when i learned my timetables and believed i had cracked maths. Halcyon days.

[@user-lh2nd2cp3p]

Классный канал, от подачи не устаёшь

[@MrCyanGaming]

I don't think it's a paradox, it just shows that there is an infinite amount of 2d space in finite 3d space. The outside surface area of the horn is the same as the inside surface area since the horn has no thickness, so we already know we can paint the surface area of the horn with an upper bound of pi units of paint. Then if you imagine removing a volume from the inside of the horn using another horn that is infinitesimally smaller in volume we can see that you can paint the surface area of the horn with practically 0 units of paint.

[@shk439]

Sample interview that would get one into Oxford, noice

[@joshmcdouglas1720]

This Gabriel’s Horn reminds me of something else I saw! Say you have a cake, cut it in half, and then cut one of the pieces in half again and stack one of the small pieces on top of the big piece. with the other small piece, cut it in half and stack one half on top of the others, and keep doing this forever. Here, you’ve created something with infinite surface area but finite volume!

[@TomRocksMaths]

Nice idea - can you come up with a formula for the surface area after n steps? That would be how I would go about showing it tends to infinity as the number of steps does...

[@sssilky3317]

The interesting thing about this is that I would argue that it should be possible to paint the outside using a finite amount of paint, it would just take an infinite amount of time. lets say that the horn is infinitely thin, it has a volume of pi units, so that means we could fill the horn completely using just pi units of paint. When the horn is filled that implies that every part of the interior surface area is coated in paint. If the horn is infinitely thin then the outside surface area should be almost identical to the inside surface are (I think), so that implies it should take fewer than pi units of paint to entirely coat (given that the paint coating the interior is also filling the vacant space between the walls of the horn. I'm obviously wrong given the result but I'm genuinely curious about why this isn't true, or if it's just one of those quirks of infinity.

[@master4755]

As a year 12 student doing AS maths I'm suprised how I actually understood nearly all of this

[@manswind3417]

To be honest you're actually supposed to, since the calculus you learn in A Maths is basically a more meaningful and extensive discourse in understanding and computing limits, derivatives and integrals - the key concepts remain the same, nothing new. Besides, don't forget that these questions (Oxford entry) are meant for Y13 students so... :)

[@andrewfleet2010]

I don't understand any of this, but it's absolutely fascinating.

[@5gjmlch9]

My Further Maths teacher told me that only kids would say 'oval' , but 'ellipse' for mathematicians.

[@marcstewart9169]

Is ‘Gabriel’s Horn’ what happens when watching Equations Stripped?

[@TomRocksMaths]

Nice Marc, nice.

[@pr_some148]

🥴

[@emilioyared]

I understood everything besides the dL part

[@zyrphath]

Great video, and an interesting result. But I don't think it's that weird that one is finite and the other infinite, if we give it a little bit of thought - it rather seems that we can think of this as any ordinary "infinite amount of steps gives finite result" case, except in three dimensions rather than two, where you can think of the steps, the volume and the area as the three dimensions we're examining simultaneously. The brain's comparison of the two isolated cases (steps+volume vs steps+area) makes it seem weird since they both describe the same figure, but the weirdness is an illusion: because the two geometric quantities are inherently differently sized, it's a somewhat natural result. Imagining a physical representation of a point-like geometric figure, we can then also imagine an infinitesimal area. But the circumference of that point will be, though very small, inherently and necessarily much larger (by comparison) than the volume, because it has to go "all the way around". Looking at the rate of change gives the mathematical confirmation of this intuition: for 1 > r > 0, reducing r by one decimal place reduces the circumference by one order of magnitudes, while the area is reduced by two orders of magnitude. Meaning that as the area approaches zero, the circumference is (and will always be) twice as large, and that also means that they can never both be "as close to zero as possible" at the same time, because that posits 2x = x, which is only ever true if x = 0. But x approaching zero is not the same as saying that x *is* zero, as that is rather the case limits are solving to begin with. The result is that 2x can never be equal to x. So there exists no radius in the real numbers where both the area and the circumference can be regarded as "as close to zero as possible" at the same time, and the intuitive result is that we can think of this as if the infinite side of the horn being sliced into disks with infinitesimal height, each individual disk past some approaching-infinity number would have area equal zero and circumference equal non-zero such that circumference > area = 0. This of course means that the volume doesn't grow, but the area does. Once this intuition is grasped, a natural observation to make is that this can also to some careful extent be intuited directly from the formulae (and from the intro of the problem in this video!) - circumference takes radius, while area takes radius^2, and we compare these ratios with the infinite series 1/n which diverges to infinity and 1/n^2 which converges to a finite real number. Remember that Tom started the video by having Steve graph 1/x versus 1/x^2.

[@sardarbekomurbekov1030]

So interesting

[@SteleexLS]

Best crossoveeeer

[@TomRocksMaths]

Glad you enjoyed it!

[@alejrandom6592]

Epic crossover ♡

[@TomRocksMaths]

Glad you enjoyed it Alej!

[@ismailshtewi8560]

People: "avengers endgame is the most ambitious crossover event in history" me, an intellectual: let me introduce you to tomrcocksmaths and blackpenredpen