Tips, Tricks & Exam Strategy

General exam strategy

How marks are awarded in subjective exams

• Statement of definitions and theorems you invoke — partial credit even if your final answer is wrong.
• Clear reasoning — write the logical steps in full sentences, not just symbols.
• Correct final answer — the prize, but rarely the bulk of the marks.
• Verification / sanity check — proving the answer is consistent (e.g., plugging back in).

Time management

• If a question is worth $m$ marks in a $T$-minute paper with $M$ total marks, allocate $\sim m\cdot T/M$ minutes.
• Leave the last 10% of time for review. Re-check arithmetic, re-read your proof structure.
• If you're stuck for more than 1.5× the allocated time, skip and return. Cost-benefit is real.

Writing proofs that earn full marks

Examiners want to see structure. Cultivate the following habits:

1. State what you're proving. "Claim: …" or "We show that…". One line.
2. State the method. "Proof (by contradiction)." or "Proof (by induction on $n$)." Don't make the reader guess.
3. Carefully introduce variables. "Let $a, b$ be even integers." Make their type and quantifier crystal clear.
4. Cite each step. "By the inductive hypothesis…", "By Lagrange's theorem…", "Applying De Morgan…". This is where most of the partial credit lives.
5. End cleanly. "$\square$" or "$\blacksquare$" or "Hence done." or "∎".

Claim: P(n) holds for all n ≥ 1.

Proof (by induction on n).

Base case (n = 1):
    [Show P(1) directly.]
    Hence P(1) ✓.

Inductive step:
    Assume P(k) for some k ≥ 1 (inductive hypothesis).
    We show P(k+1).
        [Derivation using P(k).]
    Hence P(k+1) ✓.

By the principle of mathematical induction, P(n) holds for all n ≥ 1. ∎

Logic — topic tips

• English to logic. "Only if" puts the arrow after; "if" puts the arrow before. "$p$ unless $q$" = $\lnot q \to p$ = $p \lor q$.
• For SAT, for $\leq 4$ variables, build a truth table — it's fast and bulletproof.
• Negating quantifiers: push the $\lnot$ inward, flip each quantifier, and negate the innermost predicate.
• K-maps: always write rows/columns in Gray code (00, 01, 11, 10). Forgetting wraps around adjacencies is the #1 K-map error.
• Sudoku-style SAT problems: always write 4 kinds of clauses — cell-has-something, cell-has-only-one, row/column has all digits, block has all digits. Then add givens as unit clauses.

Proofs — topic tips

• Choose the right method.
  • "$n^2$ is even" $\to$ "$n$ is even" — contrapositive, because "$n$ is odd" is concrete.
  • "$\sqrt 2$ irrational" — contradiction, because supposing the opposite gives an equation to manipulate.
  • "Statement holds for cases A, B, C" — proof by cases.
  • "Statement involves $n$, holds for all $n$" — induction.
• For "$p$ iff $q$": prove both directions separately, labeled "($\Rightarrow$)" and "($\Leftarrow$)". Don't just chain biconditionals unless every step is genuinely a biconditional.
• For three-way or n-way equivalences: show a cycle of implications: $p_1 \to p_2 \to p_3 \to p_1$. Saves work.
• Induction base case: never skip. Even when "trivial," write the verification.
• Loop invariants: always show three things — Init, Maintenance, Termination. Termination + loop-exit condition + invariant $\Rightarrow$ postcondition.

Set Theory — topic tips

• Countability tactic: to show $X$ is countable, either (a) explicit bijection $\mathbb{N} \to X$, (b) injection $X \to \mathbb{N}$, or (c) surjection $\mathbb{N} \to X$. Pick the easiest.
• Uncountability tactic: always diagonalization unless the set is obviously equivalent to $\mathbb{R}$ or $\mathcal{P}(\mathbb{N})$.
• Encode strings as integers: any countable structure (programs, formulas, finite graphs, polynomials with integer coefficients) is a finite string over a finite alphabet, hence countable.
• Russell's paradox structure: assume the "set of all sets with property $P$" exists, ask whether it has property $P$, derive contradictions in both cases.
• Inclusion-exclusion for 3 sets is fair game: $|A\cup B\cup C| = |A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|$.

Abstract Algebra — topic tips

• Group axiom check order: closure → associativity → identity → inverse. Skip associativity proof if it's inherited from a known operation (like $\mathbb{C}^*$ multiplication).
• To show NOT a group: find ONE counterexample to ONE axiom. Don't waste time checking the others.
• Cayley tables for small finite groups are gold — they let you check closure, identify identity, and find inverses at a glance.
• Order trick: in any group, $a^m = e$ implies $\text{ord}(a) \mid m$.
• Lagrange: $\text{ord}(\text{subgroup}) \mid \text{ord}(\text{group})$. The most useful corollary: $\text{ord}(a) \mid |G|$.
• Generators of $\mathbb{Z}_n$: exactly those $k$ with $\gcd(k, n) = 1$. There are $\varphi(n)$ of them.
• Primitive root test: for each prime $q$ dividing $p - 1$, check $g^{(p-1)/q} \not\equiv 1 \pmod p$. If all checks pass, $g$ is primitive.
• Field check: commutative ring with identity, no zero divisors, every nonzero has inverse. $\mathbb{Z}_n$ is a field iff $n$ is prime.
• Diffie–Hellman walk-through: given $p, g, a, b$ small numbers, compute $A = g^a, B = g^b$, then $K = A^b = B^a$. Always cross-check by computing $K$ both ways.

Advanced Counting — topic tips

• Linear homogeneous recurrence solving template:
  1. Write characteristic equation $r^k - c_1 r^{k-1} - \cdots - c_k = 0$.
  2. Find roots, noting multiplicities.
  3. General solution: each simple root $r$ contributes $A r^n$; multiplicity-$m$ root contributes $(A_0 + A_1 n + \cdots + A_{m-1} n^{m-1})r^n$.
  4. Use initial conditions to find constants.
• For non-homogeneous, guess particular solution based on RHS form. If your guess overlaps with a homogeneous solution, multiply by $n^m$.
• Master Theorem 1-2-3:
  • $f(n)$ smaller than $n^{c^*}$ → answer $\Theta(n^{c^*})$.
  • $f(n)$ equal to $n^{c^*}$ → answer $\Theta(n^{c^*} \log n)$.
  • $f(n)$ bigger than $n^{c^*}$ (with regularity) → answer $\Theta(f(n))$.
• Generating functions cheat:
  • $\dfrac{1}{1-x} = 1 + x + x^2 + \cdots$
  • $\dfrac{1}{(1-x)^k} = \sum_n \binom{n+k-1}{k-1}x^n$
  • $(1+x)^n = \sum_k \binom{n}{k}x^k$
• Setting up counting GFs: per-object GF $\times$ per-object GF $\times \cdots$ $= $ total GF. Coefficient of $x^n$ = number of arrangements summing to $n$.

Graph Theory — topic tips

• The 5 graph families table — burn it into memory:

  Graph	$|V|$	$|E|$	Bipartite?	Hamilton?
  $K_n$	$n$	$n(n-1)/2$	$n\leq 2$	$n\geq 3$
  $C_n$	$n$	$n$	$n$ even	always
  $W_n$	$n+1$	$2n$	never	always
  $Q_n$	$2^n$	$n\cdot 2^{n-1}$	always	$n \geq 2$
  $K_{m,n}$	$m+n$	$mn$	always	iff $m=n$

• Euler exists iff: 0 odd vertices (circuit) or 2 odd vertices (path).
• Hamilton sufficient conditions: Dirac ($\deg \geq n/2$) or Ore (any nonadj. pair sums to $\geq n$).
• Handshake lemma often unlocks problems: "given $|V| = v$ and edge constraint, find degrees."
• Bipartite testing: show 2-colourable (BFS), or equivalently show no odd cycle.

Common pitfalls (and how to avoid them)

One-page revision cheat sheet