## Some good puzzles here

A link on another blog, I think provided by Chris at http://chrisfmathsphysicsmusic.blogspot.com/, took me to here: http://www.maths.cam.ac.uk/undergrad/nst-pastpapers/2011/index.html whereupon, of course, I had to have a look at the papers on offer.

Within Part IA Paper 1 lies this funny 2 mark question, which can be solved with GCSE higher level maths: A circle is inscribed within an equilateral triangle. Find the ratio of the circle to the ratio of the triangle (Q5 here: http://www.maths.cam.ac.uk/undergrad/nst-pastpapers/2011/PaperNST_IA_1.pdf)

By the mark scheme, it should take three minutes. How quickly can you do it?

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### 12 responses

1. Chris Finlay

Glad you found the links helpful as for the problem once you realise the centre of the circle must be the same as the centroid of the triangles then you can work out the radius of the circle. Difficult to do that question under exam pressure though.

August 27, 2011 at 6:57 PM

2. Variants of it appear on GMAT tests. Pretty sure I’ve seen them before in that guise.
I’ve been back to the papers you linked to and done quite a bit more – feels good to know *some* of Cambridge’s stuff is within the grasp of non-Oxbridge students.

August 29, 2011 at 11:43 PM

3. Nick

Warning – spoiler below for anyone who hasn’t worked it out!

Love this question! Keep coming back and playing around with it instead of studying unit 26. Here’s quite an interesting result from it…

Let a = length of triangle side
and r = radius of contained circle.

Area of the triangle = 1/2 * a * 3r = 3ar/2
Area of circle is = pi r^2

So the ratio of area of triangle to area of circle… 3ar/2 : pi r^2

Which simplifies to… 3a : 2 pi r

i.e. the ratio of the perimeter of the triangle to the circumference of the circle.

Nice result, not sure what it means! And of course you’d need r in terms of a to simplify the the result for the examiner.

September 1, 2011 at 4:05 PM

4. Hi Nick, nice to see another MST209er posting.
How’s unit 26 for you? I’ve lost the will to read it after chapter 1.

Not sure I quite follow what you did to get your answer but I got pi : 3sqrt3 for the ratio.

September 1, 2011 at 7:35 PM

5. Nick

It’s good to post too. Been reading your blog for months and loving it. I should comment more really. I’ve had the same debate as you as to whether to do MS324 or not. I’ve decided to go for it and then do MST326 the following year. Thinking I want to do an applied maths Masters full time after this degree.

Unit 26 I’m finding very difficult to concentrate on. But that’s probably got a bit to do with having just quit smoking. I need to learn to love numerical methods.

Regarding the Cambridge question – sorry, I didn’t explain myself especially well but we get the same result. Bear with me…

(Forgive the inconsistent use of asterisks for multiplication)

a: the length of each of the sides of the triangle
r: the radius of the circle

Your ratio:
pi : 3sqrt3

Multiplying both sides by 2r:
2 pi r : 6 sqrt3 * r

Substituting r = a / (2sqrt3) into the right hand side of the ratio:
2 pi r : 6sqrt3 * a / (2sqrt3)

The right hand side of the ratio then simplifies to leave:
2 pi r : 3a

i.e. the ratio of the circumference of the circle to the perimeter of the triangle.

September 1, 2011 at 10:05 PM

6. I wonder how many ways there are to answer this question. 🙂
Nice one.
All I can say so far about the MS324 course material is that block 0 is pretty easy. I’m told block 1 is a bit of a pain, block 2 less so and block 3 very enjoyable.
MST326 is the main course I’m looking at, too – far too much juicy stuff in that to ignore.
You’re braver than me quitting smoking during MST209 – I’ve got that penciled in for after the exam.
Glad you’re posting – I’m getting somewhere around 900 unique viewers a month, I think, but only a handful of viewers post (though I’m very glad they do, or I’d assume my posts are only being read by myself and a bunch of search engine bots). 🙂
All the best – four weeks to the TMA and CMA finish lines – then BEER.

September 2, 2011 at 4:36 PM

7. Nick

Got a question for you. Did you consider doing SM358 ‘The Quantum World’? I am massively tempted.

I haven’t studied S207 (or any other physics since GCSE in ’96) but from the ‘Are you ready for SM358?’ page, it seems that MST209 just about covers the prerequisites by itself.

I wish we didn’t have to do MST121 and MS221 to get a maths degree but that’s a whole separate rant. My choices seem to come down to choosing any two of MS324, SM358 and the new M347 Mathematical Statistics. I know that M347 could be pretty useful, but the chance to study Quantum Mechanics…

September 2, 2011 at 6:50 PM

8. Nick,
I keep looking at SM358. I’ve initially discounted it because I haven’t done S207 and my physics is non-existent. I’m not worried about the maths element of it.

I’ve come up with a plan for two degrees concurrently, subject to the rules about keeping the transitional fee arrangement plan alive.

I was going to go on to do an MSc in Financial Risk Management after a maths degree but have changed plans (again) and now want two degrees before applying to Oxford for its MSc in Mathematical Modelling and Scientific Computing (probably out of my reach but it’s a goal which will motivate me).

In getting the two degrees, I can squeeze in MS324 or MS327, MST326, M337, M303, SM358, M343, M346, M347 and M373.

I agree about MST121 and MS221. To be fair, I absolutely needed to do both to get my maths off to some sort of start, and, at the time, thought: “cool, 60 points of a maths degree in the bag” though now, as I develop an interest far in excess of “just” completing 360 points, I find myself wishing that MST121 and MS221 could be “set aside” and considered to be preparatory courses and not affect my ongoing ability to accumulate points.

I have a very all-or-nothing personality and once I find an interest, it tends to absorb me. Maths has done exactly that. It’s fair to say if I won the lottery, far from going off around the world, I’d hire a one-to-one maths tutor and simply disappear down the maths rabbit hole. Sadly, at present, an income of four buttons and a half-sucked Polo mint per month precludes anything more than buying a couple of books from time-to-time.

What’s your honest opinion on us handling SM358 without S207? (It would be good if Chris Finley could chime in here, too).

All the best,

Keith

September 2, 2011 at 10:41 PM

9. Chris Finlay

Hi Guys thanks for the compliments here is my two pennorth for what it’s worth
As far as S207 goes avoid it like the plague it has virtually no mathematical content at all MST209 is a much better preparation. If you want a good book on classicial physics then as I recommended on my long post on my blog about preparatory reading for High energy physicists then the book to get is Longair ‘Concepts of theoretical Physics’ would be a natural to read during down time. It covers, albeit concisely, all the main topics of classical physics gives an introduction to chaos theory and general relativity. So I think you should be well prepared for SM358. There is also an introduction to physics and astronomy available from the physics website which is aimed at those who want to do the third level courses in astronomy.

As for your dilemma vis a vis MST221 and MS121 why not ditch the ambitions for a named degree and settle for an open one instead. I’ve done that and have stopped counting MST121 for the degree which meant I could avoid Linear Statistical modelling and has enabled me to squeeze in MS324 before tackling MS326 in October of next year. I feel much freer as a result. I really think the OU should consider the equivalent of a foundation year compulsory for those who don’t have a background in maths or whatever subject but not for those who already have A level’s or equivalent. I don’t regret doing those courses they were warm up for my current courses and I quite enjoyed going back to basics and I even learnt one or two new things discrete maths and so forth.
But there is no way that they should be considered degree level qualifications.

On a final note glad you enjoyed the Cambridge Exams Shall I set up a shared activities forum to see if we can do the exams together or will this informal thing here do. I intend to try and do the 4 Natural science tripos exam papers during ‘down time’

On a final final note Neil (and good to hear from you lets have a post or two from you on your blog, please I would love to know your thoughts about the last part of M337) if you want to learn to love numerical analysis come and join us on the Cambridge Computing projects.

All the best Chris

September 4, 2011 at 5:14 PM

10. Chris Finlay

Sorry Nick scrub that last comment I think I’ve confused you with another guy, The offer to join the Cambridge computing project still stands though it’s a great way to understand the techniques.

Best wishes Chris

September 4, 2011 at 5:19 PM

11. Chris, great input as ever. Thanks. 🙂
So, you in essence think SM358 could be a goer with some downtime reading/the prep notes for third level courses?
That really livens things up.
I’m going to have to have a really serious think about my 2 x degree plan and its timescale – what I go for next depends on how well I think I’ve done after the MST209 exam.
If I stuff it up, I’m going to have to go down the route of M248 and M249 next year, and primarily chase the mathematics and statistics degree first. If I feel the exam went well, I’ll be almost obliged to do MS324 and SM358 next year, which would be perfect, if a little demanding on intellect.

September 4, 2011 at 6:24 PM

12. PS – a shared activities forum for the tripos exams would be a great idea. I’d be able to contribute to that much more than the computing forum you set up.

September 4, 2011 at 6:25 PM