Creativity
Part of the Oxford Maths Admissions Test Livestream 2026
Solutions to follow, will be available here
General Advice
We often want to invent an example or counterexample, and this requires some ingenuity.
- The TMUA Content Specification has a section on proof and conjecture and counterexamples, reproduced below.
- Note that for a disproof by counterexample, a single counterexample is enough.
- The more experience you have with examples of things, the better.
- You can invent counterexamples by experimentation. Critically evaluate your creations.
- Try to push the boundary.
TMUA Specification (April 2025, Section 2 § "Mathematical Proof")
- Follow a proof of the following types, and in simple cases know how to construct such a proof:
- Direct deductive proof ("Since A, therefore B, therefore C, ..., therefore Z, which is what we wanted to prove.");
- Proof by cases (for example, by considering even and odd cases separately);
- Proof by contradiction;
- Disproof by counterexample.
- Deduce implications from given statements.
- Make conjectures based on small cases, and then justify these conjectures.
- Rearrange a sequence of statements into the correct order to give a proof for a statement.
- Problems requiring a sophisticated chain of reasoning to solve.
Warm-up
Is there a 3D shape with exactly 13 edges and exactly 8 vertices?
Hint: do you know any 3D shapes that have either 13 edges or 8 vertices?
Extension: for which integers $E$ and $V$ can you find a 3D shape with exactly $E$ edges and exactly $V$ vertices?
Questions
TMUA 2021 Paper 2 Question 4 (modified)
Consider this claim about positive whole numbers $a$ and $b$ and $c$:
If $a$ is a factor of $bc$, then $a$ is a factor of $b$ or $a$ is a factor of $c$.
Find a counterexample where $a$ and $b$ and $c$ are all even numbers.
Find a counterexample where $a$ and $b$ and $c$ are all odd numbers.
Find a counterexample where $a$ and $b$ and $c$ are all square numbers.
Find a counterexample with $a\lt b\lt c$ and a counterexample with $a\gt b\gt c$.
TMUA 2020 Paper 2 Question 4 (modified)
Consider this claim about the positive integer $N$:
If $N$ is greater than 6 then $N$ can be written as the sum of two non-prime integers that are each greater than 1.
Find a counterexample to the claim.
Extension: find all counterexamples. Hence find the smallest integer that could replace "6" in the claim to make it true.
TMUA 2022 Paper 2 Question 3 (modified)
Consider the following claim about the positive integer $n$
if $n$ is prime, then $n^2+2$ is prime
For each integer $n$ from 1 to 10 inclusive, decide whether it is a counterexample to the claim.
For each integer $n$ from 1 to 10 inclusive, decide whether it is a counterexample to the converse of the claim.
TMUA 2020 Paper 2 Question 20 (modified)
For each of the following claims, find a function $\mathrm{f}$ such that the claim is true but the converse of the claim is not true.
- $x \ge 0$ only if $\mathrm{f}(x) \lt 0$
- $x \lt 0$ if $\mathrm{f}(x) \ge 0$
- $x \ge 0$ only if $\mathrm{f}(x) \ge 0$
- $\mathrm{f}(x) \lt 0$ if $x \lt 0$
- $\mathrm{f}(x) \ge 0$ only if $x \ge 0$
TMUA 2020 Paper 2 Question 10
The real numbers $a$, $b$, $c$ and $d$ satisfy both
\[ 0 \lt a + b \lt c + d \]
and
\[ 0 \lt a + c \lt b + d \]
Which of the following inequalities must be true?
I $a \lt d$
II $b \lt c$
III $a + b + c + d \gt 0$
(A) none of them
(B) I only
(C) II only
(D) III only
(E) I and II only
(F) I and III only
(G) II and III only
(H) I, II and III
TMUA 2020 Paper 2 Question 18
In this question, $\mathrm{f}(x) = ax^3 + bx^2 + cx + d$ and $\mathrm{g}(x) = px^3 + qx^2 + rx + s$ are cubic polynomials.
If $\mathrm{f}(x) - \mathrm{g}(x) \gt 0$ for every real $x$, which of the following is/are necessarily true?
I $a \gt p$
II if $b = q$ then $c = r$
III $d \gt s$
(A) none of them
(B) I only
(C) II only
(D) III only
(E) I and II only
(F) I and III only
(G) II and III only
(H) I, II and III
Part of an Interview
Adapted from an interview question used by James Munro for Maths interviews at Oxford. Reproduced here with permission.
We're going to look for functions $y(x)$ that satisfy
\[ x^3 \frac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\frac{\mathrm{d}y}{\mathrm{d}x}-y=0. \]
This is called a second-order differential equation, because it's an equation involving a second derivative. If that first term weren't there, this would be a first-order differential equation, because it would still involve a first derivative.
Let's start by looking for simple functions $y(x)$ that work. Can you think of anything?
In general for differential equations of this form, if you have a solution $y_1(x)$ then you can try $y(x)=v(x)\times y_1(x)$, and you will get a first-order differential equation for $\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}$. Give this a go!
(You should substitute $y(x)=v(x)\times y_1(x)$ into the differential equation, where $y_1(x)$ is the solution you've already found, and $v(x)$ is an unknown function. There will be terms involving $\displaystyle \frac{\mathrm{d}^2v}{\mathrm{d}x^2}$ and $\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}$ and $v$ but after simplifying, one of these disappears.)
(Depending on whether you've seen integrating factors or not, we might solve this first-order differential equation for $\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}$ and then integrate for $v$.)