Multi-step problems that combine small facts: parity, pigeonhole, extremal cases, counting by complementary sets, and 'must be true / could be true' logic. Work systematically, test the extreme or boundary case, and build a counterexample before trusting an intuition.
Standard tools
- Pigeonhole: to force two numbers differing by \(10\) from \(\{1, \ldots, 20\}\), pair them up; \(n =\) pairs \(+ 1\).
- Parity: the sum of degrees in a handshake graph is even; the sum of five consecutive odd numbers is \(5 \times\) the middle one.
- Factor pairs: \(x^2 - y^2 = 45\), \((x - y)(x + y) = 45\) with matching parity; \(\frac1m + \frac1n = \frac14 \Rightarrow (m - 4)(n - 4) = 16\).
- Inclusion-exclusion: numbers up to \(999\) divisible by \(2\) or \(5\): \(499 + 199 - 99\).
- Invariants: every match eliminates one player, so \(37\) players need \(36\) matches whatever the draw.
Must / could / cannot
- 'Must be true': try to build a counterexample; if none exists, prove it. 'Could be true': one example suffices. 'Cannot': show a contradiction.
- Exactly-one-true puzzles: assume each candidate is the truthful one and check consistency.
- Order/inequality statements: test negative values and values between \(0\) and \(1\) (\(ab < bc\) fails for negative \(b\)).
- Sequences with block conditions: overlapping sums (three-term sums positive, five-term sums negative) - combine blocks to compare.
Must-know (no formula booklet)
- pigeonhole: \(n\) boxes, \(n + 1\) objects
- \((x-y)(x+y)\) factor pairs
- inclusion-exclusion \(|A \cup B| = |A| + |B| - |A \cap B|\)
- test negatives, zero and fractions
Classic traps
- A sum of two primes is odd only if one of them is \(2\).
- 'Exactly two of three statements are true' - a number making all three true is excluded.
- Counting ordered pairs: \((2, 8)\) and \((8, 2)\) are different.
Worked example
Q. Smallest \(n\) so that \(n\) numbers from \(1\)-\(20\) must include two differing by \(10\)?
A. Ten pairs \(\{k, k + 10\}\); \(n = 11\).
How it has been examined
2023 · Paper 2Question 8lines dividing a triangle into right-angled trianglesquestion papersit online