Challenging the eternal truth of mathematics

I learned that 0,999... = 1. I believe it was in M381 that I learned to prove it. The proof is quite simple actually.

Let
x = 0,999..., multiply both sides with 10
10x = 9,999..., now subtract x from 10x
9x = 9, and we have our result
x = 1.

What I like about mathematics is that it is timeless, i.e. we can still read math books of hundreds of years old and still learn things from them. That's not an advisable strategy hfor any other science than mathematics. Of course, new branches of mathematics appear, new discoveries are made, but they don't invalidate the truths of the past.

Today however I found someone who actually is challenging some of the established truths in mathematics, like for example that 0,999... = 1. I don't think that he is a crackpot, although I have no doubt that the mathematical establishment, professors who are 'safe' by all means, will call him like that.

The man is arrogant though, he calls the proof above, 'juvenile' for example.

Who is he, what are his ideas and how did he disproof that 1 = 0,999? The links below will help you abshereing these questions.
- The New Calculus - The first rigorous formulation of calculus in history.
- Proof that 0.999 not equal 1.pdf

Computers are simple

Computers are simple. They must be. Consider how fast the industry developed and the only conclusion you can draw is that it must be simple. Once you understand how easy it is to implement the basic operations add, move, compare and jump to a machine, and that you can build -any- program from these operations you'll know. Alan Turing started it all.

The smartphones of today are so much more powerful than the mainframes banks had in the sixties and seventies in terms of computing power. Over a billion people on Planet Earth live with a computer. These computers connect us with other people and are turning into our best friends. And that is something what was beyond what Alan Turing foresaw about computers.

Computer Science and Mathematics students need a deep understanding of the Turing Machine. OU M381 did not teach the Turing Machine but a slightly simpler model because the Turing Machine was considered too complex. Not anymore with today's simulators if you ask me.

Turing Machine simulator

M381 'Challenge Exercise' - Revisited

Sorry, Paddy and Jobidaker for the late reply. ( Auto-accepting comments has its drawbacks too. )

My solution for the 1/2, 1/3, 1/5 and 1/7 case is the following. Follow the pattern for the solution to the general case. Click to enlarge pic.

Please let me know if you think a smaller number qualifies.

M381 - Challenge Exercise

In the Open University course Number Theory and Mathematical Logic the additional exercises sections of the workbooks are complemented with several 'challenge exercises'. This is one of them.

Let $n$ be an odd positive integer.
Prove that there are $\tau(n)$ ways of writing $n$ as a sum of consecutive positive integers.
For example, if $n=9$, $\tau(9)=3$ because $9$ has three divisors $1,3,9$ and the three sums are: $9$, $4+5$ and $2+3+4$.

To be continued.

Shanks-Tonelli algorithm for solving quadratic modular equations

M381 unit 6 is about the Law of Quadratic Reciprocity. An application of the LQR is solving quadratic modular equations like: $$x^2 \equiv 499 \ \text{mod(617)}.$$ M381 contains a method that can fast determine if that equation is solvable. It does not contain however a fast algorithm for finding the actual solutions. One such algorithm is the Shanks-Tonelli algorithm and can be found on Planet Math which as often gives a much clearer presentation than the messy Wikipedia-entry.

See also: Tutorial for Quadratic Equations

Quadratic reciprocity in a finite group.

Law of Quadratic Reciprocity

Let $p$ and $q$ be distinct odd primes. Then $$\displaystyle \left({\frac p q}\right) \left({\frac q p}\right) = \left({-1}\right)^{\frac {\left({p-1}\right) \left({q-1}\right)} 4}$$ where $\displaystyle \left({\frac p q}\right)$ and $\displaystyle \left({\frac q p}\right)$ are defined as the Legendre Symbol $\displaystyle \left({\frac{a}{p}}\right) := a^{\frac{(p-1)}{2}} \pmod p$.

Gauss considered his work on the Quadratic Reciprocity Law among his major achievements. I don't 'get that', not now anyway, that's a call for more study on the topic.

Now and then, when I browse through papers, or otherwise, I find an interesting mathematical paper... ( that I can actually read ). Actually, I was browsing through a book called Reciprocity Laws, from Euler to Eisenstein by Franz Lemmermeyer, it contains more than 100 proofs of the Quadratic Reciprocity Law. I hoped to find a proof I could appreciate by it's beauty. Although most proofs are based on Gauss's Lemma ( as the proof in M381 ) but there are proofs in other realms of mathematics like Group Theory. Group Theory -as we know it today- did not exist in Gauss's time. That's why I am going to spend some time studying the following paper 'Quadratic reciprocity in a finite group.'

M381 exam

Did M381 exam. All the questions were doable, easy as a matter of fact. Honestly. I am sure I have full marks for the first question I made. But when I was done and looked at the clock it was already past three o'clock. "Should I have been able to do that question in 10 minutes?", I thought. Maybe. I have a painful ear-infection at the moment, and I was drugged of course, prescription painkillers, doctor's order. And antibiotics, of course. I felt like walking on the moon, in an astronaut's suit. Whatever the outcome may be, I can set it right. More later on the exam, when I feel better.

Primitive recursive function

Normally you calculate n factorial with Factorial[n] or short n!. Mathematica handles the details of the function for you and prints the result.

In the course M381 you have to prove that functions like factorial are a primitive recursive function. This basically means that the function can be defined only in terms of itself, add one, or set to zero. A primitive recursive definition of factorial would look as follows in Mathematica.

suc[n1_] := n1 + 1
add[n1_, 0] := n1
add[n1_, n2_] := suc[add[n1, n2 - 1]]
mul[n1_, 0] := 0
mul[n1_, n2_] := add[mul[n1, n2 - 1], n1]
fac[0] := suc[0]
fac[n_] := mul[n, fac[n - 1]]

As you can see no other Mathematica functions than "+ 1" and "= 0" are used. The functions suc, add, mul, fac are defined for the first time.

For example:
In[67]:= Factorial[6]
fac[6]

Out[67]= 720

Out[68]= 720
.

Goedel, Escher, Bach - Lecture 6

In the first six minutes or so Curry gives a fairly good summary of Goedel's Theorem. Unfortunately this is the summary of the previous lecture which was not recorded. It seems nothing is free, not even free video lectures because it turns out the best ( not implying the rest is good ) is missing.

After rushing through formal stuff he wastes five minutes about a three-layer stupid joke about a book he had not read.

I quick-scanned through the rest of the video. Not worth watching, really. Too bad. I looked forward to this.

The take home message of the course. All provable things are true but not necessarily al true things are provable.
Justin Curry

Now that I am mostly through all M381 stuff I am glad it included mathematical logic. I would -not- have done it as a stand-alone course. Logic is hard in the beginning, like most new subjects. It needs time to work on you. I will get back to this in the next M381 post.

Previous posts on the series:
- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4
- Lecture 5

About M381 (1)

Regular readers of this blog know that I have a sort-of rage-button like the Hulk: it is called MathCad. Thank God, I have the anti-dote almost always open and ready: Mathematica. I am not going to repeat why MathCad is a danger to your mental health, but I have to press the MathCad button at least once.

In almost all mathematics courses you can do at the Open University there is software involved. They either deliver a standard package, or ship custom software especially developed for the course ( i.e. MT365 ). The house-package of the Mathematics Department of the Open University is MathCad, version 2001. I have argued that MathCad alone is a reason -not- to choose for the Open University. What a disgrace...

( Calming down. )

They do however recognize that software, computers, tools are relevant in mathematics. Especially in Number Theory computers are used in active research. Another area where they use software in active research is: mathematical logic. Stronger: research in Number Theory is impossible without computers.

These facts are not even mentioned in M381. There are many open source tools available for Number Theory, even more for Mathematical Logic. Not a word about it in M381. One, if not -the- reason is the fact that course development in the Open University is done in a project organization. A project is created with the objective to create course X which will then be used for the next 10 or so years. It is exactly the opposite of what one would expect of a university education. It is not reasonable to expect the Open University to be at the forefront of mathematical research. Simply because other universities in the UK have that role. But it is reasonable to expect more than a static expose of 19th century Gauss number theory and early 20th century logic from Church, Turing and Goedel. In fact, the field is presented as abstract and of theoretical importance only. But Number Theory and Mathematical Logic are extremely relevant and applicable in many industries! But I did not learn that from the course and that is sad.

It took me a lot of work but I found some relevant learning tools in the fields of number theory and mathematical logic. More about those later in this blog.

(*) - I may have misunderstood the concept of 'University' in the UK. I think many universities in the UK are what we call in the Netherlands 'schools'. They deliver professionals with a degree in all fields through excellent education but they don't do research and so on. They don't add to the body of knowledge. They process and transfer knowledge. That description fits the Open University as well. - A marketing issue is that students like to have a 'university' education. And marketing people love empty heads boxes, they have a fancy word for it too: the 'packaging'. Does that make sense?

Goedel, Escher, Bach - Lecture 5

A few months ago I started to watch the MIT video lecture series on Goedel, Escher, Bach. Due to time constraints I wasn't able to complete watching the entire series. Today I continued with watching lecture 5. I have learned quite a lot on the subject through M381 and I am about to really 'get it' as far as the Goedel Incompleteness Theorems are concerned. My first reading of GEB took months and now parts of the book begin to look simple. If you don't know what I mean browse through a mathematics book you thought was hard, a few years ago. It often seems if there is 'nothing in the book'. The odd thing with Goedel ( and with all mathematics, I suppose ) is that in your mind you think you can explain it to a laymen in one or two sentences. ( It is -that- simple, I am afraid. ) The power of mathematics is that it can capture an entire knowledge tree in a single word. That word remains meaningless without understanding of all the words in the knowledge tree.

A bit about lecture 5.

Dress shows arrogance

I wonder if Justin Curry would go to a job interview in that Club Med outfit. Students are paying customers ( and a pool of cheap labor for lucrative research deals the university makes ) deserving respect from teaching staff.

Formal number theory ( as in M381 )

Justin talks about Typographic Number Theory, ( formal number theory in M381 ). For example $$\forall x ( \neg x = \mathbf{0} ( \exists y x = y') )$$ can be interpreted as

"Every x that is not equal to 0 is the successor of some y."

Leibniz was the first to propose a formal language for number theory. He asked whether it was true that an algorithm could decide if a statement in number theory was true. - Although in M381 this question is answered negatively that does not mean computers can not play a role in proving mathematical propositions. There is an abundance of ( open source ) software for proving theorems.

( Not in video: ) Isabelle a formal proof theory assistant has been used in testing an operating system kernel written in C and assembler. It not only verified that the spec was implemented correctly but it also discovered hundreds (...) of programming and design (...) errors which were not found by traditional testing methods.

Previous posts on the series:
- Lecture 1
- Lecture 2
- Lecture 3
- Lecture 4

Exercise ( logic ).

Knowledge of mathematics is not required to solve the following exercise, but it will sure help ;-)

Dr. Who asked you for a ride in the Tardis. Naturally, you couldn't decline, it might be a matter of national, if not global, importance. The Tardis landed on the Planet of Truth which is is inhabited by people who always tell the truth. A minority however decided to lie, always. It is Doctor Who's mission to seek and destroy all liars. First you must get to the Capital of the Planet. Two roads fork out. Should you go left, or right? An inhabitant approaches, greets you and gives you the privilege of asking him one Yes/No question.

Ask him where the capital is, left or right on the fork, but beware he might be a liar!

Credit follows with the answer.

Take the challenge, test your ability to think logically.

Is M381 True or False ?

In facebook I have a photo album with mathematics related pictures. Sometime ago I added the picture below. I thought it was a joke...

... but I could not have been further from the truth. The picture could have been taken at any Open University M381 tutorial. This student is not clueless, on the contrary: he contemplates about some deep mathematics.

Anyway, if you have always wanted to give an answer to the layman's question: "But can you -prove- that 1+1 = 2?" then M381 is your course.

Goedel, Escher, Bach - Lecture 4(1)

My brain ran those neural network algorithms.

Justin Curry

The reading assignment was chapter 6 'The location of meaning'. It is basically about coding and decoding. Of course Hofstadter mentioned the Rosetta Stone in this context, the key to ancient Egypt. It contained a parallel text in three languages and was deciphered in 1821 by Champollion.

A recent example in the context of chapter 6 is space archeology. Archeologists and Egyptologists were able to interpret satellite pictures of Egypt which lead to the sensational discovery of new pyramids.

Let's go to the lecture. ( This is a 1h46m lecture and will be discussed in two posts. )

Curry talked about Goedel numbering again, a method Goedel used to code strings in formal number theory to numbers. What Curry said about coding a string in formal number theory, playing with it and then code it back is only true in theory. Simply because Goedel numbers become -extremely large-. Think of numbers built from pages full of digits. ( Would Curry ever have calculated a Goedel number? This reminds me of a DBA course I attended once. The trainer talked about all the beautiful properties of the then new RMAN from Oracle as if backups could be recovered in an instant. It turned out that he never worked in the trenches of 7 x 24 administration of large databases. )

Dialog "Contracrostipunctus" on page 75 of GEB is discussed. How this dialog has meaning on several levels. The dialog refers to itself that it contains a hidden message. The concept of 'Self' is introduced here.

Starting with what does "Snow is white" mean? he builds an argument that there is an isomorphism between electrical activity in the brain and the interpretation of symbols. ( Thought reading might be possible after all, one day. Isn't it true that man can create everything he is able to envision? )

Adam and Eve

There are at least two phases in the proces of assigning meaning two a string. The first is parsing the string, the second is the interpretation of the parsed words. Interpretation depends on the context of the interpreter.

Message in a bottle

He introduces the concept of information. For example how physicists reduce complex physical behaviour to a small sequence of symbols. Like for example how a pendulum works.

Pendulum ?

Goedel, Escher, Bach - Lecture 3

A guy named Euclid.

Justin Curry

Curry briefly explaines the concepts:
- consistency
- completeness
- and geometry.

A consistent system leads to conclusions that are not contradictory in any sense. A statement is either true or false, and never both true and false.
A system is complete if everything that is true in the context of that system can be derived from the axioms.
Regarding geometry he mentioned that there are Euclidean non-Euclidean geometries.

Then he attempts to explain Goedel's Incompleteness Theorems.
1. Any system as powerful as number theory which can prove its own consistency is necessarily inconsistent.
2. Any system as powerful as number theory is necessarily incomplete.
He explains that Goedel managed to transform the idea of provability to a property of numbers by introducing his Goedel numbers.
He says that students should now have a notion of the Goedel theorems and promises that this is just a first glance at Goedel's theorem. ( Not sure if he meant he would come back at Goedel in this lecture series. )

Trying to explain Goedel

He then talks about Euclid and his postulates.
(1) Any straight line segment can be drawn joining any two points.
(2) Any straight line segment can be extended indefinitely in a straight line
(3) Given any straight line, a circle can be drawn having the segment as radius and the
(4) All right angles are congruent.
===
(5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles

He explains that the 5th postulate is consistent in Euclidean Geometry but not in spherical and hyperbolic geometry.

Hofstadter Dialog - Little Harmonic Labyrinth is removed from the video due to copyright concerns. It is part of what makes GEB such a difficult book. Here is part of it.

The Tortoise and Achilles are spending a day at Coney Island. After buying a couple of cotton candies, they decide to take a ride on the Ferris wheel.
Tortoise: This is my favorite ride. One seems to move so far, and yet in
reality one gets nowhere.
Achilles: I can see why it would appeal to you. Are you all strapped in?
Tortoise: Yes, I think I've got this buckle done. Well, here we go. Whee!
Achilles: You certainly are exuberant today.
Tortoise: I have good reason to be. My aunt, who is a fortune-teller, told me that a stroke of Good Fortune would befall me today. So I am tingling with anticipation.
Achilles: Don't tell me you believe in fortune-telling!
Tortoise: No . . . but they say it works even if you don't believe in it.

About 1% of the Little Harmonic Labyrinth dialog.

He explains the cardinal arithmetic, the arithmetic of infinities.
An interesting definition of infinity is that a set can be mapped to a subset of itself. I.e. the natural numbers can be bijectively mapped to the even numbers. The points on the real line can be bijectively mapped to the points on the line between 0 and 1.

Non-Euclidean geometries

Goedel, Escher, Bach - Lecture 2

It will get a little bit mathy, but that's ok.

Curran Kelleher

Lecture 2 is given by Curran Kelleher. This lecture is all about recursion and ends with a nice explanation of the Mandelbrot set.

He starts out with the traditional examples factorial:

factorial[0]:=1;
factorial[n_]:=factorial[n-1]*n; 

and fibonacci sequence:

fib[1]:=1;
fib[2]:=1;
fib[n_]:=fib[n-1]+fib[n-2];

Kelleher's factorial program

Kelleher's hand-out ( pdf ) contains examples of Java code for drawings of the Koch curve and Sierpinski triangle. Although I fast-forwarded through this part of the lecture, it may be very interesting for non-programmers.

Explaining the Fern algorithm

Complex number implemented as a class in Groovy 

At around 1:00 he starts with the topic of the Mandelbrot set. Starting with f(z) = z^2 + c he manages to give a nice explanation of the Mandelbrot set. The color of a point in the Mandelbrot set is based on the number of iterations it took f to 'escape' a circle. Since this is done on a pixel by pixel basis and one pixel may generate not one but several iterations this explains the long time it takes to generate a Mandelbrot set.

P.S.
GEB does not seem 'outdated' at all although it was written in the late seventies. A time when there were no mobile phones, no PCs, let alone laptops and the internet was still in its toddler phase.

Goedel, Escher, Bach - Lecture 1

'Understanding Goedel' is one of the major goals I set for myself.

This final unit brings together all the ideas introduced in the course. These ideas constitute the technical machinery that enables us to prove some very important theorems which answer what we have called Leibniz's and Hilbert's Questions. These theorems, Goedel's Incompleteness Theorems, are among the most profound intellectual discoveries of the the twentieth century. Thus you should not be surprised if you find this unit hard going in places.

M381 - Unit 8.

In Goedel, Escher, Bach (GEB) Hofstadter asks the question: what happens when 'things' start referencing themselves? ( Like people do who are in essence not more than a set of linked molecules. )

At last I took the time to watch video 1 of the GEB series.

Justin Curry

The teacher is Justin Curry. He started by telling that most undergraduates don't get through GEB in less than 13 weeks and that it took him seven years to get through the book. I am not sure but I think my first attempt in reading GEB was in 2007 or 2008. It took me almost six months to get through it. Which I thought was really bad. When I finished the book and still didn't understand what he was talking about I started to seriously doubt my learning abilities. I have to admit that I still don't get it but I made progress. And I am getting closer, thanks to M381 Mathematical Logic ( read: Nigel Cutland ).

Anyway, to the point: the lecture.

He starts with the concept of isomorphism. In GEB, Hofstadter explains isomorphism as a map between structures that maps parts with similar purpose to similar purpose ( my words ). This is different than the mathematical definition which states that an isomorphic map is both surjective and injective. Hofstadters definition can be understood immediately, whereas the mathematical definition needs understanding of layer upon layer upon layer. Since what is a map in mathematical sense? What does surjective mean? What does injective mean? Analyzing a mathematical sentence always creates a ( large ) tree structure.

Recursion. The concept of recursive definition. A fascinating concept which I use a lot, since I am a programmer by profession. Curry uses the example of the Fibonacci sequence 1,1,2,3,5,8,13,... and translates it to f(n) = f(n-1) + f(n-2) and the Sierpinski triangle ( fractal ).

Drawing the Sierpinski triangle

( To be continued in the next post. ) Edit: Nope. I'll make a last note about lecture 1 here and continue with lecture 2 next time.

Some remarks, tips for if you want to give it a try ( like myself ). I was not in continuous awe while watching this lecture. You know when like you are watching the latest BBC Horizon or similar. It's not like that. I don't have the feeling as if I have wasted my time, not at all. I am going to watch lecture 2 soon.

- You definitely need the 720+ pages ( 20 chapters ) book. ( Details on the course site. )
- You need to be ( somewhat ) familiar with Bach's music, or at least -know- someone who is. ( What are forums for anyway? ) To fully grasp the genius of Hofstadter's work.
- If you are a religuous person than GEB might not be for you.

There is an audio set in the lecture room. Near the end of the lecture a piece of Bach is played. Students familiar with that music could elaborate on it. Since I am ignorant to most classical music I must have missed a lot of what Hofstadter said. It might be an opportunity to start listening to some Bach, who knows what happens.,

So far for lecture 1,

M381 mathematical logic - or Escher, Goedel, Bach - Revisited

Although I believe the academic world is at least as closed as it always has been, the access to ( mathematical knowledge ) is ( almost ) a level playing field for those in and those outside the thick walls of academia. All thanks to the development of Internet. Some people say we live in a short period of a truly free Internet that once the technology stabilizes ( whenever that will be ) more and more sites will become accessible to the elite only. A bit like today's access to scientific journals for example, which give professors a six month to a year lead.

To the point: I still haven't watched the MIT video-series 'Gödel, Escher, Bach: A Mental Space Odyssey' because it is stored in Real format and I don't want to install Real with it's bogus Adware. But these files play well in VLC Media Player which is my favorite media player. Now is the time to watch these videos, I suppose. In the middle of doing M381 mathematical logic. Now that I have two ( real ) books on the subject the Open University booklets seem so much more accessible. In retrospect I think that M208 is an extremely difficult course IF you go by the booklets alone. Enough said over that subject.

I noticed that understanding a mathematical topic is something continuous it's not a discrete 'I get it' versus 'I don't get it'. The Jigsaw pieces are slowly getting on their right place and the image becomes clearer every day, Goedel's Incompleteness Theorems. I remember that when I read Hofstadter's book that I thought that he was an absolute genius ( he might be ) and that it was impossible for me to =ever= understand Goedel at a mathematical level. And now, years later, I am getting closer to that every day. I still have this idea about unreachable highs of mathematical knowledge represented by mountains disappearing in the clouds. I am far from the mountains on low ground. As long as I keep walking I must get there one day.

MIT video, six lectures on GEB.

About M381 mathematical logic or how to study an Open University booklet

How to study an Open University booklet? Well, don't if you have an alternative classic source. Study the source first, then go back to the booklets and do the exercises. Just because the TMA questions are modeled after them and the model answers are written in that style. Remember that your tutor wants to get through your work asap. The more they look like the model answer the better. The rule is 'when in doubt it is wrong.' Tutors are NOT professors, so don't expect them to be.

Studying the source is essential. Because that's what the authors of the booklets used when they wrote them. You should be able to write these booklets yourself and sooner than you think. Besides, the vast body of mathematics is written in source classics, not in the form of OU booklets or lecture notes. - ( An Open University booklet is roughly comparable to a set of lecture notes brick university students get, I suppose. )

Take for example the M381 mathematical logic booklets. What a Layman would understand as Mathematical Logic is called Propositional Logic and is covered early in the track in the MS221 course. The title of the course 'Mathematical Logic' is thus somewhat confusing. And it is not just the title that is confusing. I noticed on the forums that students considered the M381 logic part rather 'difficult'. The principal author of the booklet is Alan Slomson, now a professor in Mathematics at Leeds University. ( A host of other names are listed in the credits including Jeremy Gray: Wise Beard Man, the face of the OU ). Anyway, the source book is: "Computability, an introduction to recursive function theory" by Nigel Cutland.". ( I checked all available books on Mathematical Logic, I have no doubt that this is the book that was used. ) The thing is that the material in the course booklet looks a bit fuzzy, somwehat unclear, rather difficult to be honest. Well, it isn't. Not if you read Cutland's original first. Then you recognize what they ( Slomson ) are -trying- to explain. It's just the presentation in the booklet that makes it foggy. It would have been better if they simply have used Cutland's book ( like Apostol's in the Analytical Number Theory course ) and write a Reader's Guide. In a Reader's Guide you can use informal English to explain things while for the formal math definitions you can refer to the book. - ( Under Slomsons wings another book on Computability has been published by Brian Cooper : Computability Theory. Here too you'll find Cutland's URM and a lot of his original examples. )

Despite my occasional criticism I feel quite at home at the Open University.

M381 Exam setup

Today I received pack 2 of M381 materials and read some interesting news about the exam setup. ( Open University ) course M381 has two fairly independent tracks: Number Theory and Mathematical Logic. The TMA's are thus mostly 50/50. The exam however has a different setup.

The exam consists of 16 questions.
Questions 1-8 cover Number Theory. ( My educated guess is 1 questions for each book / topic.
- divisibility
- prime numbers
- congruence
- Fermat / Wilson theorems
- multiplicative functions
- quadratic reciprocity
- continued fractions
- linear Diophantine equations.
Questions 9-16 cover Mathematical Logic. ( My guess is as follows )
- computability
- primitive recursive functions
- Church's thesis
- formal systems
- formal proofs
- formal number theory ( 3 questions )

All questions are marked and ranked. The nine best results are used to count your grade for the exam. There is one restriction however you can't use more than six questions from one subject. So if all number theory results are better than the logic ones than they take the six best nt questions and then the best three from logic.
eaxc
If, and I suppose it is, ( isn't it always? ) an extremely time-constrained exam then there are quite a number of strategies to follow for sequencing the work at the exam. I will have to think about that when the exam date comes closer, I suppose. A lot depends on my self-