Lecture 1 — Welcome & Warm-up Problems
See math1061 for staff, assessment, and the full schedule.
Warm-up problems
These preview topics for later in semester: proof techniques (number theory & methods of proof), induction/recursion, and graph theory.
Let \(p\), \(q\), \(r\) be consecutive positive integers with \(p < q < r\), and let \(a = p + r\), \(b = pr\), \(c = r - p\). Determine which of \(a, b, c\) must be even.
Four non-zero real numbers \(a, b, c, d\) are given, where \(b + d \neq 0\). If \(\frac{a}{b} + \frac{c}{d} = \frac{a+c}{b+d}\), show that \(ac < 0\).
True or False: in every group of five people, there are two people who have the same number of friends within the group.
The Tower of Hanoi: given a tower of 8 discs in decreasing size on one of three pegs, transfer the entire tower to another peg. Rules: move one disc at a time; never place a larger disc on a smaller one. Is there a solution, and what’s the fewest number of moves?
The Königsberg Bridge Problem: does a route exist through the city crossing each bridge exactly once and returning to the start (no turning around on bridges)?
Notes
- Problems 1-3 preview conditional-statements and proof-writing (the lecture slides leave the worked solutions for in-class discussion, not printed).
- Problem 4 (Tower of Hanoi) previews recursion/induction later in semester (Lectures 13-16) — the classic minimum-move count for \(n\) discs is \(2^n - 1\).
- Problem 5 (Königsberg bridges) previews graph theory (Lectures 33-34) — the classical resolution (Euler) is that no such route exists, because the underlying graph has more than two vertices of odd degree.
See also
- math1061 — assessment weights, staff and the full lecture schedule
- 2026-07-30-logical-form-and-equivalence — Lecture 2, where the symbolic logic starts properly
- practice-problems — the semester-wide Applied Class problem set, reproduced in full with solutions (Applied Classes start in Week 2; the logic sections begin at practice-problems)