**Skipping core modules restricts your potential.**

We have heard about a recent Oxbridge graduate in maths and philosophy who wanted to apply for a more advanced course at another university with an entrance exam. The entrance exam had 4 problems in basic topics such as calculus and probability. To our surprise, the student did not study any probability and decided not to apply. Yes, the probability theory is a key tool of modern mathematics and should be studied by all undergraduates who want to use their mathematical ability in the future.

**Why are there so many optional modules?**

The same student eagerly argued that their university offered the widest possible choice of maths modules in the UK including ‘Axiomatic Set Theory’, ‘Analytic Topology’, and ‘Banach Spaces’. To avoid an inevitable disappointment, the student wasn’t told that all these topics are actually compulsory for any maths undergraduates at many overseas universities. The main reason for creating optional modules is the tuition fees and the consumer culture at UK universities. Compulsory modules can make some students unhappy, which is considered unacceptable when the top priority is the student satisfaction, not the learning outcomes.

**How a wide choice creates wider problems.**

Many graduates with the same 1st class degree from the same maths programme at the same top university often have completely different knowledge and skills. Hence employers can not quickly and fairly select best candidates and need to further analyze all examined modules (and their content) or simply design their own entrance exams or job interview tests.

Similarly, all top UK universities require proper entrance exams such as STEP and MAT, which seem better for choosing brightest mathematics candidates than an inevitable random selection from a large pool of school students with the highest grades A*A*A* at A-level.

**How can you benefit from flexibility?**

One of our students couldn’t get required STEP grades for a maths programme at Cambridge. However he was flexible enough, talked to admissions tutors about other options and got in Cambridge on a joint programme with computer science. Indeed, many universities allow students to change their programme later, so a purely mathematical course at Cambridge was still a option. Moreover, a solid foundation in two closely related subjects such as mathematics and computer science is simply wider, hence more robust for a future career.

Returning to our first story about the Oxbridge graduate, yes, we think that any computer science student should learn the probability theory, at least because randomized algorithms are now a large and active research area in computer science.

**Is it really possible to learn more? Yes!**

Another story is about a student from Strasbourg, who was on an exchange programme for a year at a top UK university. The student enrolled on 7 modules instead of required 6 despite clashes in the timetable between lectures. The student gained 90+ marks in all 7 exams and came on top of all her classmates.

**What to learn: broad range vs narrow focus.**

In the past all great inventors were universal specialists in many subjects. For example, was Leonardo da Vinci a painter, sculptor, architect, musician, mathematician, engineer, writer, anatomist or “all in one”? Actually, the word “university” hints at *universal knowledge*.

Distant research areas converge to each other and become inter-disciplinary. For instance, *cancer genetics* uses fast algorithms from *computational geometry*, which are rigorously justified by methods from (applied?) *statistics* and (pure?) *topology*. Analogous interactions of distant subjects in the *computer vision* have led to robotic cars that you will be able to buy soon.

It seems that the world of the (near) future will need only *super-specialists*. If you play chess, you hopefully know standard positions when a king can keep 2 perpendicular aims in mind by making diagonal moves, which are simply longer. Similarly, by learning topics on the interface of different areas, you can keep your options open and make a right choice later when you have more information. We finish by the following riddle with our usual reward.

**Riddle 4**: find the product of the roots of of the equation \(0.5x^2+52=99x\).**How to submit**: to write your full answer, simply submit a comment.**Hint**: there is no need to find the roots since only their product is needed.**Warning**: please avoid using a calculator, because you can be smarter.**Restriction**: only the first fully correct public answer will be rewarded.**Prize**: free 1-year access to one of our interactive web tutorials.**Update**: after attempts 1 and 2 Ariella has won the prize.

If you wish to receive automatic e-mails about our new posts (for a quicker chance to answer a riddle and win a prize) or distance courses, please contact us and tick the box “keep me updated”. You can easily unsubscribe at any time by e-mailing “unsubscribe” in the subject line.

The roots are

positive root : 394.95 to 2dp

negative root : 1.05 to 2dp

The exact product when not shortening the answer is exactly 416.

The equation is ( 198-sqr38788 )(198+sqr38788)

coming to the answer of 416.

Dear Ariella, thank you for your attempt to solve the equation 0.5x^2+52=99x. You have probably used the quadratic formula to produce these rather complicated numbers. First, we without trying to repeat your computations, we simply substitute your second root -1.05 to 2dp (roughly -1) into 0.5x^2+52=99x and get 0.5+52=-99, which seems very far from a correct identity. Second, we have actually advised not to use a calculator, simply because you have no chance against the computer power. While you are solving one quadratic equation, a modern computer will solve a billion and always correctly! Third, a correct answer to our question requires computations without any square roots and are accessible for a primary school student, though a proper understanding is more involved, of course. We hope that you will make another more successful attempt. Thank you in advance!

The product of roots formula is c/a.

0.5x^2+52=99x

0.5x^2-99x+52=0

then a=0.5 b=-99 c=52

therefore the answer is 52/0.5

so 104 .

Dear Ariella, yes! Now all computations are much easier and correct. We shall probably explain the general approach in one of our future posts. Please e-mail blogger@master-maths.co.uk the title of the web tutorial you would like to access at our Master Maths tutorials website.