YOUR ULTIMATE GUIDE
The Definitive Starter Guide To The MAT (2023)
Written by: Matt AmalfitanoStroud
The MAT is an admissions assessment required for applicants who wish to study Maths or a related course at some universities. In this guide we’ll cover the essential test information, the different sections of the MAT and how to prepare for the test, along with some practice questions to get you started.
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MAT BASICS: EVERYTHING YOU NEED TO KNOW
First, let’s cover the essential MAT information, including the different test sections, key dates, scoring, results and more. There’s a lot of information to process so we’ve broken down your most important MAT questions to help you understand everything. Let’s get started!
What is the MAT?
The Mathematics Admissions Test (MAT), is exactly what it sounds like – an admissions test for those applying to Mathematics related university courses. The MAT is used by a variety of universities in the UK and is taken at both Batchelor’s and Masters’ level in some cases.
It is produced and organised by the University of Oxford in partnership with Cambridge Assessments Admissions Testing. Despite their involvement, the University of Cambridge does not require the MAT for any courses.
It is a written test, made up of some multiplechoice questions and some longer written questions. It’s designed to be approachable by all students, including those without Further Mathematics ALevel or the equivalent.
What is the Structure of the MAT?
The MAT lasts 2hour 30minute. Its format is fairly unique compared to other admissions tests such as the TMUA. Essentially, the test paper is split up into seven questions:
Question 1 is a multiplechoice question, with ten parts, each worth 4 marks.
Questions 2 – 7 are longer, written questions, each worth 15 marks and you should attempt four of these depending on which degree you are applying for (learn more about this here). This might seem like a lot, but don’t worry; the questions are broken down into smaller parts for you to work through.
In this format, Question 1 is worth 40 out of a total 100 marks so it’s worth spending a good chunk of your time on.
Here’s some Maths for you: 40/100 × 2.5 = 1 so if you split your time proportionally, question 1 should get a whole hour!
Why is the MAT Used?
The MAT is designed to test the depth of your understanding, and your ability to solve problems. Universities use it to differentiate between candidates who might look similar on paper otherwise – e.g. to decide between two applicants with similar ALevel predictions and GCSE grades. Universities can see your overall mark and how you did on the parts of each question. They use this to identify your strengths and weaknesses and how you think which helps them see if you would thrive in their courses.
The test requires you to think more deeply than your ALevel papers (or equivalent), and so at first, it will seem harder. But don’t worry, this is on purpose! Unlike ALevels, it is very rare that people get 100 on the MAT, as they don’t want to set a paper that people will find easy – this won’t help them to differentiate between candidates.
Universities want to make you think so that they can see how you think.
Who Needs to Sit the MAT?
Since the MAT is a test of mathematics, it is commonly used to test applicants of relevant courses, including Mathematics, Computer Science, or any joint honours courses, such as Mathematics and Philosophy or Mathematics and Computer Science at an undergraduate level.
In the UK, there are six universities that use the MAT for their various mathematics degrees. Out these six, the MAT is compulsory at two universities while the other four accept the results but do not require them for entry:
Universities That Require the MAT
 G100 Mathematics (3year course)
 G103 Mathematics (4year course)
 G125 Mathematics (Pure Mathematics)
 G104 Mathematics with a Year Abroad

G1F3 Mathematics with
Applied Mathematics/Mathematical Physics

G102 Mathematics with Mathematical
Computation
 G1G3 Mathematics with Statistics
 G1GH Mathematics with Statistics for Finance
Universities That Accept the MAT
Bear in mind that your results will be automatically sent to Oxford and Imperial if you are applying to either of these courses. Your result will also be sent to Warwick, Bath and Durham but they will not have access to them unless you grant permission.
Nottingham will not receive your results automatically, they will have their own systems in place to receive them but do not require you to send them if you do not wish for them to have access to them.
Which MAT Questions Should I Answer?
As previously stated, no applicant has to answer every question in the MAT, with certain subjects and unis requiring certain questions to be completed.
For Mathematics, Mathematics and Statistics, or Mathematics and Philosophy at the University of Oxford, you should answer questions 1,2,3,4,5.
If you are applying to the University of Oxford for Mathematics and Computer Science, you should answer questions 1,2,3,5,6.
If you are applying to the University of Oxford for Computer Science, or Computer Science and Philosophy, you should answer questions 1,2,5,6,7.
For all nonOxford universities, including Imperial College London and the University of Warwick, you should answer questions 1,2,3,4,5.
What are the Key Dates for the MAT?
The dates for the 2023 MAT have not yet been officially announced, but the major deadlines for the assessment usually stay consistent from yeartoyear:
Applicants Deadlines 2022  

Registration Opens  1st September 2022 
Modified Papers Registration /Apply to Become a Testing Centre Deadlines  16th September 2022 
Registration Closes  30th September 2022 
Submit Your UCAS Form (Oxford)  15th October 2022 
MAT Testing Date  2nd November 2022 
MAT Results Released  January 2023 
Where is the MAT Sat?
You need to sit the MAT somewhere which is registered as an exam centre with Cambridge Assessment Admissions Testing. For most people, this is your school or college. It is your responsibility to tell your school that you want to take the MAT and make sure you check with your Exams Officer whether your school is a registered centre or not. If it isn’t, it has until September 30th to register. If you can’t sit the exam in your school/college, you will need to take it at another centre – you can look for one local to you here.
You then need to register as a candidate with your test centre. Registration is open from September 1st to September 30th at 18.00, but some centres will have their own internal deadlines, so don’t leave it to the last minute! Registration is separate from your UCAS form, but you will need to know your UCAS number.
If you have a disability or special requirement and are normally entitled to support for exams, let your centre know when you register as there will be access arrangements available for you. If you are requesting modified question papers (e.g. large print), the deadline for this is September 16th.
How Much Does the MAT Cost to Sit?
There is no entry fee to sit the MAT, but some test centres may charge an admin fee for running the test, so you should check this with them.
How Hard is the MAT?
The MAT is designed to test you by applying the concepts you’re familiar with from ALevels but in unusual ways. You are expected to think more deeply than your ALevel (or equivalent) papers, and so initially the MAT is likely to seem harder. Also, you may find the MAT harder if you are a less creatively inclined mathematician, but with the right preparation you won’t need to worry!
Can you Resit the MAT?
If you aren’t happy with your MAT score, unfortunately, you are not able to resit later in the year, as the test is run only once a year. If you decide to reapply to these universities next year, you will have to sit the MAT again just as you do this year.
How is the MAT Scored?
The MAT is scored out of 100 marks. Question 1 is multiple choice with 10 parts each worth 4 marks. Marks are awarded only for correct answers but you are encouraged to show any working in the space provided. There is no negative marking.
Questions 27 are each worth 15 marks. These are longer questions and part marks are available. Candidates will need to show their working as they will earn you marks.
How do you get your MAT results?
Oxford Applicants are automatically sent an email with their results in January. You do not automatically receive a result for the MAT. If you want to find out how you have done you should email the university’s maths department with your name, UCAS ID, and MAT registration number and request feedback. They will not be able to give you any feedback until the admissions cycle is over, so you should email after March 31 next year.
What is my MAT Registration Number?
Your MAT Registration Number is a unique identifying number that would have been given to you when you were registered for the test. It being with the letter M followed by 5 random numbers (e.g. M19127).
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MAT SYLLABUS: WHAT DO YOU NEED TO REVISE?
The MAT is designed to be accessible to any Maths student at the start of their second year of ALevels or equivalent. All you need is knowledge from the first year of ALevel Maths, and a couple of topics from the beginning of the second year, so let’s run down everything that may be covered in your MAT paper.
The MAT syllabus covers the following ten areas, so it’s a good idea to make sure you are familiar with them early on in your preparation:
 The quadratic formula
 Completing the square
 Discriminant
 Factorisation
 The Factor Theorem
 Simple simultaneous equations in one or two variables
 Solution of simple inequalities
 Binomial Theorem with positive whole exponent
 Combinations and binomial probabilities
 Derivative of x^{a}, including for fractional exponents
 Derivative of e^{kx}
 Derivative of a sum of functions
 Tangents and normals to graphs
 Turning points
 Second order derivatives
 Maxima and minima
 Increasing and decreasing functions
 Differentiation from first principles
 Indefinite integration as the reverse of differentiation
 Definite integrals and the signed areas they represent
 Integration of x^{a }(where a ≠ 1) and sums of these
 The graphs of quadratics and cubics
 Graphs of sin x, cos x, tan x, √x, a^{x}, log_{a}x
 Solving equations and inequalities with graphs
 Laws of logarithms and exponentials
 Solution of the equation a^{x }= b
The relations between the following graphs:
 y = f (ax)
 y = af (x)
 y = f (x – a)
 y = f (x) + a
 y = f (x)
 Coordinate geometry and vectors in the plane
 The equations of straight lines and circles
 Basic properties of circles
 Lengths of arcs of circles
Solution of simple trigonometric equations. The identities:
 tan x = sin x / cos x
 sin^{2} x + cos^{2} x = 1
 sin (90° – x) = cos x
Periodicity of sine, cosine and tangent. Sine and cosine rules for triangles.
 Sequences defined iteratively and by formulae
 Arithmetic and geometric progressions
 Their sums
 Convergence condition for infinite geometric progressions
Exams.Ninja Tip
The MAT syllabus was changed in 2018, adding and adjusting required knowledge throughout most subjects. Some of these changes were minor, but it is still important to make sure you are reading the 2018 version of the MAT syllabus when beginning your preparation.
This will also affect your use of past papers. When using past papers, be aware that the papers dated 2017 or earlier will be following the old syllabus and thus may have questions which are no longer relevant in the MAT. It is still worth taking on these papers though as they provide tonnes of practice questions from the official source.
You can download the official MAT syllabus document here.
Sequences and series are the only topics which you may not come across until the second year of ALevel Maths. Ask your teacher if you can cover these before October halfterm, or if you can borrow a textbook to teach yourself these bits.
You won’t get a formula booklet or be allowed your own when you take the MAT so you should make sure you have all the formulae and identities here memorised for when you sit it.
Exams.Ninja Tip
Sequence and Series is typically covered on Question 7 of the MAT, so you won’t need to dedicate much time to revising it unless you’re applying for Computer Science at Oxford.
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MAT REVISION TIPS
Once you are comfortable with the material on the syllabus, it’s time to get your practice and preparation started! The tips will help you make the most of your time before the test, so let’s do this.
1. Create a Plan
Before you think about jumping into past papers and practice questions, you should first think about how you’re actually going to approach this. In other words, you’re going to need to make a plan.
We wouldn’t recommend leaving any less than four weeks to prepare for a typical admissions test, but mathematics can generally be more difficult to get so we would suggest six weeks as a minimum. We’ve even outlined how you can structure your preparation over six months!
The essential three elements to include in this plan are revision, practice questions and mock exams. Start off by using your notes and resources to gain a better understanding of the theory behind the maths. Then take some time to get to know the question structure and techniques with some practice questions (preferably with worked solutions). Lastly, as you get closer to the test date, begin to sit more and more past papers in realistic exam conditions to ensure you get to grips with your exam technique.
Exams.Ninja’s MAT Preparation Platform contains everything you need to work through your preparation, so be sure. to create an account today to try it for free!
2. Pay Attention in Class
As we’ve already said, all of the required knowledge for the MAT will be covered in your first and early second years of A Levels (or equivalent). Your class is where your MAT preparation actually starts, so be sure to make the most of the learning time you have.
Be sure to make detailed notes that you can refer to later, this will be a lifesaver when you’re in the early stages of practice questions for the test. Some teachers may also provide their own teaching resources for you to take home, such as worksheets and presentations. Anything you can get may come in handy when you’re fully stuck into your MAT prep, so gather your resources early and keep them organised.
Lastly, if you’re not getting something in class, don’t be afraid to seek extra help from your teacher, even if it’s after class. It’s vital that you understand as much as possible from the syllabus, so it would be a waste to not take the chance to learn more from an expert.
3. Practice Questions are Key
Mathematics is a pretty practical subject in reality. There isn’t a lot of factual information to memorise other than rules and formulas. These are incredibly important to remember, but we feel that the best way to cement them into your brain is with practice questions.
It’s common knowledge that in many cases, the best way to learn is by doing, and implementing your maths knowledge into practice questions is very easy and effective to do. We wouldn’t suggest you take all your questions from past papers, as you’ll need those for mock exams (although there 18 to currently pick from). Instead, look into using an MAT question bank – the best ones should have 100s of unique questions to answer.
Alongside the number of questions, you should also seek out worked solutions to go with them. You’re not going to get every question right when you’re starting out, or even when you’re sitting the final exam! Therefore, worked solutions will allow you to see where you went wrong and understand what the correct solution is in a clear manner. You can see how good worked solutions should look down below.
4. Test Your Skills Regularly
Sitting mock papers shouldn’t just be saved for the last few days of your preparation, you should be taking them on regularly. With papers available for free dating as far back as 2007, you aren’t likely to run out if you use them sensibly! (Remember that pre2018 papers are based on a slightly different syllabus).
We would actually say that looking at a mock paper should be one of the first things you do in your preparation plan. At this stage, you shouldn’t be too concerned about completing it within the 150minute time limit or getting a good score. Instead, just use the time to get an understanding of the exam format and the types of questions that will be asked of you.
As you move forward in your prep, you can gradually become more and more strict with yourself until you’re sitting papers in exam conditions and achieving scores that would get you an offer!
As well as learning from the papers, it’s also important to make sure you’re monitoring your progress. Mark your papers thoroughly and pay attention to the types of questions you’re getting right and wrong as this will help you influence your revision to strengthen your weaknesses.
Exams.Ninja Tip
It doesn’t matter too much what order you take the past papers in, but we would suggest saving the most recent past paper for the last days before your test date. This paper will be the most similar to the one you will be sitting as it was developed most recently. Of course, this may be untrue if any syllabus or format changes have been made on the year you take the test.
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MAT PRACTICE QUESTIONS
With all the basics of the MAT now out of the way, the final thing to do here is to check out some questions. This collection of practice questions will provide you with a good entry point to your MAT preparation.
MAT Question 1 Example 1
The equation
4^{x –}^{ 1} – 2^{ x }^{+ 1} = k
where k is a real number, obtains its minimum value when
A) x = 0
B) x = 1
C) x = 2
D) x = 1
E) x = ∞
The correct answer is C.
4^{x –}^{ 1} – 2^{ x }^{+ 1} = k = (2^{2x}/4) – 2(2^{x}) = k
Let 2^{x} = y, then y²/4 – 2y = k.
We want to find the minimum of y² – 8y = 4k, so we can initiate and set to zero to see that y = 4 is a stationary point. You can take the second derivative to verify that it is a minimum (as 2 > 0). So we must have 2^{x} = 4 ⇒ x = 2. Alternatively, you can check each option quickly to see this must be the case.
MAT Question 1 Example 2
Let n be an integer and define the functions f(x) and g(x) by
f(x) = x²
g(x) = (x + 4)^{n} – (x + 1)^{n}(x+1)^{n}²
Then x² + 3 is a factor of g(f(x)) for
A) Even n
B) n = 0 only
C) Odd n
D) n = 0
E) n = 1 only
The correct answer is D.
We know that (x³ + 3) is a factor of g(f(x)) which implies that x + 3 is a factor of g(x). So in particular, we have that g(3) = 0 ⇒ (1)^{n} – (2)^{n} (2)^{n} = 0.
We see general even and odd values of n do not work. Plugging in values for n = 0, 1, we see that both are valid solutions.
MAT Question 1 Example 3
A square is inscribed in a circle of radius √2. An isosceles triangle is inscribed in the square. What is the area of the shaded region?
A) 5√2 – 7
B) 1/3
C) 4√2
D) 1/10
E) √2 – 1
The correct answer is A.
By Pythagoras on the bold right angle triangle, the side length of the square is 2. The height of the shaded triangle is therefore √2 – 1, and the height of the large triangle is 1 + √2.
Considering their respective angles, note that the small and large triangles are similar with a scale factor of (√2 – 1)/(√2 + 1). Half the base of the small triangle is therefore equal to (√2 – 1)/(√2 + 1) so the small triangle has area (√2 – 1) * (√2 – 1)/(√2 + 1).
MAT Question 1 Example 4
To be awarded a £100n prize, students must score at least 50% on n different exams, where n is a positive integer. Suppose that 6 students are taking these exams, and for every exam, every result is equally likely. Furthermore, individual exam results are independent. What is the probability that 2 of these 6 students received a £100n prize?
A) 2^{2n}
B) (15(2^{n} – 1)^{4}) / 2^{6n}
C) (2^{n} – 1)^{4} / 2^{2n}
D) ((2^{n} – 1)²) / 2^{2n}
E) (6(2^{n} – 1)^{4}) / 2^{6n}
The correct answer is B.
For these 6 students, the probability of the award is p = 1/2^{n}.
Each student can either get the award with probability p or not get the award with probability 1 – p, independent of all the other student’s performances. This is a binomial probability and so the answer is
^{6}C_{2} x (1/2^{n})² x (1 – 1/2^{n})^{4} = 15 x ((2^{n} – 1)^{4}) / 2^{6n}
MAT Question 2 Example
A function f(x) is called ‘even’ if f(x) = f(x) for all x and it is called ‘odd’ if f(x) = –f(x) for all x.
(i) Show that f(x) = x² is an even function and g(x) = x³ is an odd function.
(ii) Must a polynomial be either an odd or even function? Explain your answer.
(iii) What is the only function that is both odd and even? Show that this is the only such function.
Suppose that f(x) is an even function and g(x) is an odd function.
(iv) Show that fg, gf and f² are all even functions.
Now let h(x) be some function. Define
f(x) = (h(x) + h(x))/2 g(x) = (h(x) + h(x))/2
(v) Hence show that any function can be written as the sum of an odd and an even function.
(vi) Show that every function can be written as a unique sum of an even and odd function.
(i) [2 Marks] f(x) = (x)² = x² = f(x) and f(x) = (x)³ = –x³ = – 9(x).
(ii) [2 Marks] No. A correct counterexample is, for example, f(x) = 1 + x with f(x) = 1 – x which is neither equal to f(x) nor –f(x) for all x.
(iii) [2 Marks] If f is both even and odd, then for all x, f(x) = f(x) and –f(x) = f(x). Adding these together gives f(x) = 0 for all x. Thus the only such function is the function that is zero everywhere.
(iv) [3 Marks] g(x) = f(g(x))(g(x)) = f(9(x)) = fg(x) for all x.
gf(x) = g(f(x)) = g(f(x)) = gf(x) for all x.
f²(x) – f(f(x)) = f(f(x)) = f²(x) for all x.
(v) [3 Marks] f(x) + g(x) = (h(x) + h(x)) / 2 = –g(x) so g is odd. Given any function h we can always construct this f and g. Therefore every function can be written as the sum of an odd function and an even function.
(vi) [3 Marks] Assume that h can be written in two ways, as the sum of an even and odd function i.e. h(x) = f_{1}(x) + g_{1}(x) = f_{2}(x) + g_{2}(x) where f_{i}(x) are even functions and g_{i} are odd functions. This means that f_{1}(x) – f_{2}(x) = g_{2}(x) – g_{1}(x). Note that the left side of the equation is an even function, and the right side is an odd function. So these must be equal to 0, from part (iii). This implies that f_{1 }= f_{2} = f and g_{1} = g_{2} = g. So h(x) = f(x) + g(x) is the unique decomposition in this way.
MAT Question 4 Example
The diagram shows two lines in the plane, intersecting at a point Q. The line joining O and Q creates an angle θ with the x axis, while the line joining P, Q and R creates an angle α with the x axis, as shown in the diagram. Point P is located at (1,0).
(i) Find, in terms of α, the equation of the line joining points P, Q and R.
(ii) Find, in terms of θ, the equation of the line joining O and R. Hence write, in terms of α and θ, the coordinates of point Q.
Triangle OPQ has area A(α, θ) and triangle OQR has area B(α, θ).
(iii) Give an example of an α and θ such that A(α, θ) = B(α, θ). In addition, show that when A(α, θ) = B(α, θ), A(α, θ) is independent of θ.
(iv) Find an expression for A(α, θ) in terms of α and θ.
(v) Now let α = π/4. Find the value of tan(θ) such that A(π/4, θ) = 1/2 B(π/4, θ).
(i) [2 Marks] Using trigonometric relations on triangle POR, we can see that R is located at (0, tan α). The gradient of the line is –tan α/1 – tan α. The equation of the line is y – 0 = – tan α(x 1) which simplifies to y = – tan α(x 1).
(ii) [3 Marks] We know that the gradient of this line will be tan θ. Since the line goes through the origin, the equation of the line is y = xtanθ.
To find the coordinates of Q, we calculate the point of intersection between the two lines. xtanθ = xtanθ(x – 1) and solving for x gives x = tanα / (tanθ + tanα). The y coordinates is y = tanθtanα / (tanθ + tanα).
(iii) [3 Marks] The simplest example of when A(α, θ) = B(α, θ) = B is when triangle POQ is isosceles and the line OR bisects PQ. This occurs when α = π/4 and θ = π/4.
The area of triangle POQ can be calculated using A = 1/2 bh where b is the base and h the height of the triangle. In this case b = 1 and h = tanα so A(POQ) = 1/2 tanα. We want A(α, θ) to be equal to exactly half this area, so A(α, θ) – 1/4 tanα which is independent of θ.
(iv) [4 Marks] We first calculate the length OQ Pythagoras.
OQ² = ((tanθtanα / (tanθ + tanα))² + (tanα / (tanθ + tanα))² = (tan²α(1 + tan²θ) / (tanθ + tanα)² = (tan²αsec²θ) / (tanθ + tanα)².
Then we can determine the area of the triangle using A = 1/2absinC. In this case…
(v) [3 Marks] When α = π/4, A(π/4, θ). The area of triangle POR is 1/2 when α = π/4. For A(π/4, θ) = 1/2B(π/4,θ), we want A(π/4, θ) = 1/6. Then we must have 1/6 = 1/ (2tanθ(1 + tanθ). This simplifies to the quadratic tan²θ + tanθ – 3 = 0 and solving for tanθ gives tanθ= (1 ± √14(3)) / 2 = (1 ± √13) / 2. For 0 < θ < π/2, we must have tanθ = (1 + √73) / 2.
MAT Question 6 Example
Some people are dealt cards which tell them exactly how to act. They say either: “You must always tell the truth” or “You must lie exactly once”. Each person can see everyone else’s cards.
(i) Suppose that there are two people and one of each type of card. First, person A says: “person B has the liar card”. And person B says: “person A has the liar card”. Then, A says: “Today is Monday”, and B says: “I have the truth card”. Who has the liar card?
For the rest of the question assume that there are more than one of each card (so A and B can have the same type of card).
(ii) Does the above conversation contain enough information to know which cards A and B have? Explain your answer and give any conclusions that can be made.
(iii) Suppose a new person C says first: “B has a liar card” and then “I have a truth card”. Does this change the conclusions you can make?
(iv) A new person D enters the room. And players AD are redealt cards.
Person A says: “I have a truth card” and “It is sunny”.
Person B says “It is after 2 pm” and “Exactly one of us has a truth card”. Person C says “It is not sunny” and “Person A has a truth card”.
Person D says “5 is less than 3” and “It is before 2 pm”.
Who has the truth card?
(i) [2 Marks] First let us condition on the card that A receives. Let’s assume it is the truth card. This implies that B has the liar card and it is a Monday. But this means that both of person B’s statements are lies, which is a contradiction – as they can only tell exactly one lie. So, A must have the liar card.
(ii) [5 Marks] Exactly the same line of reasoning holds and shows that A must have the liar card. If person B has a truth card then all of B’s statements are true and, provided it isn’t Monday, exactly one of A’s statements is a lie – so this situation is consistent. If person B has a liar card then B’s second statement is a lie, so B’s first statement is true which gives, as desired, that A has a liar card. Person A is then telling exactly one lie, as desired, provided it isn’t Monday. Therefore, B can have either card.
(iii) [3 Marks] If C has a truth card then B must have a liar card. If C has a liar card then C’s second statement is the lie so “B has a liar card” is true. Therefore, B has a liar card.
(iv) [5 Marks] D’s first statement is an obvious lie so D has a liar card and it is before 2pm. This makes B a liar, so it is true that exactly one person has a truth card. Suppose that C has a truth card. Then A has a truth card. This contradicts B’s true statement, so C has a liar card. Thus, by elimination, A must have the only truth card.
MAT Question 7 Example
Sentences in Martian are made up from three Martian words: Zoop, Zap and Zeep, according to the rule that Zoop and Zeep can never be next to each other in a sentence.
(i) List all the sentences of length 2 and all the sentences of length 3 that start with Zap.
Let z(n) denote the number of sentences of length n. Further, let o(n), a(n) and e(n), denote the number of sentences of length n that start with a Zoop, a Zap and a Zeep respectively.
(ii) Explain why o(n) = e(n), o(n) – 0(n – 1) = a(n – 1) and a(n) – a(n – 1) = 2o(n – 1) for n > 1.
(iii) Write down a formula for z(n) in terms of o(n) and a(n). Hence compute z(3) and z(4).
A Martian sentence communicates friendship if reversing the sentence and swapping all Zoops for Zeeps (and vice versa) does not change the sentence. Let f(n) be the number of friendly sentences.
(iv) Show that f(2k – 1) for all positive integers k.
(i) [2 Marks] There are 3 sentences of length 2 starting with Zap: ZapZoop, ZapZap, ZapZeep. The 7 sentences of length 3 start with Zap and are given by ZapZapZap, ZapZapZoop,ZapZapZeep, ZapZoopZoop, ZapZoopZap, ZapZeepZeep,ZapZeepZap.
(ii) [4 Marks] Every sentence starting with a ‘Zoop‘ can be turned into a sequence starting with Zeep by changing all the Zoops to Zeeps and vice versa. Likewise, all Zeepsentences can be turned into Zoopsentences. So there are equal numbers of each type, with length n. Starting with a Zoop, then the next letter must be a Zoop or a Zap. Therefore any sequence of length n which starts with a Zoop is a Zoop followed by a valid sequence of length n – 1 starting with a Zoop or a Zap. That is, o(n) = o(n – 1) + a(n – 1). Starting with Zap, then the next letter can be a Zoop, Zap or Zeep. Thus a(n) = o(n – 1) + a(n – 1) + e(n – 1) = 2o(n – 1) + a(n – 1) since e(k) = o(k) for all k. Rearranging gives the required result.
(iii) [4 Marks] A sequence of length n must start with one of Zoop, Zap or Zeep. Therefore z(n) = o(n) + a(n) + e(n) = 2o(n) + a(n). Using this and the above relations gives:
z(3) = 2o(3) + a(3) = 2(o(2) + a(2)) + a(3) = 2(2 +3) + 7 = 17
and
z(4) = 2o(4) + a(4) = 2(o(3) + (3)) + 2o(3) + a(3) = 4o(3) + 21= 4 x 5 +21 = 41
(iv) [5 Marks] Consider first the case when the sentence is odd. If a sentence is friendly, then when it is reversed, the middle word stays in the same position and so must remain unchanged by the Zoop – Zap swap. Therefore it must be a Zap.
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