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Mathematics interview
Develop a derivation, test small cases and break unfamiliar problems into known steps.
One thinking framework
Use the same five steps for the exercises below. Familiarise yourself once, then focus on explaining your reasoning.
Five steps for your reasoning
1
Define variables, assumptions and a model
State what you are studying and what you will temporarily ignore.
2
Explain your choice of method
Connect your formula, theorem or comparison to the problem.
3
Explain each step as you calculate
Explain relationships, substitutions and intermediate conclusions.
4
Check units, scale and plausibility
Check signs, ranges, dimensions and possible sources of error.
5
Adapt to hints and changes
Identify which assumption changes and revise your approach.
1
There is an 8×8 chessboard, with two corner squares on a diagonal removed. Can you tile the remaining 62 squares exactly with 31 dominoes of size 1×2?
2
Why is the product of any three consecutive integers always divisible by 6?
3
Walk from coordinate to , where each step may only go right or up. The path is required not to pass through . How many paths are there in total? First establish a method that explains the answer rather than merely computing it.
Whiteboard combinatorial counting and generalisation12–15 minutesModellingComplementary countingProve uniquenessGeneralisation
4
Let be a polynomial with integer coefficients, and , . Is it possible that ? Do not try to guess a specific polynomial; find a constraint that is satisfied by all polynomials with integer coefficients.
Discovery and proof of unfamiliar theorems10–14 minutesExploring special casesFactorisationCongruence reasoningNecessary and sufficient
5
The parabola encloses a region above the coordinate axes. A rectangle symmetric about the axis with its base on the axis has its two upper vertices on the parabola. Find the maximum area, and explain why what you have found is a maximum and not a minimum.
Modelling, differentiation and boundary checking12–15 minutesChoice of variablesOptimisationBoundary conditionsScale generalisation
6
Starting from , define . First compute several terms and guess the limit, then prove what the limit can only be if it converges; finally explain why this iteration becomes accurate so quickly.
Numerical exploration, limits and error analysis15–18 minutesPattern recognitionLimit equationError analysisRigour review