Lecture 01 — Thinking in Problems

Learning objectives

• Decompose a messy requirement into precise subproblems
• State assumptions and constraints before solving
• Choose between alternative interpretations of a task

Case-based lecture: you attempt, we discuss, then we reveal.

The plan

• Four cases: cs-001, cs-002, cs-003, cs-004
• Each: 5-minute attempt → discussion → reveal → prompt
• Worked examples in between — ask anything
• Break mid-lecture · exit question at the end

Key definitions

• Problem — a goal plus obstacles: what is wanted, and what stands between the given information and that goal.
• Specification — a restatement of a problem so that two careful readers would produce the same answer for every input.
• Input — the information a solution is allowed to use. Everything not listed as input does not exist.
• Output — the exact form of the answer required (a number, a decision, a list) - agreed before solving.
• Constraint — a limit the solution must respect (time, memory, money, legal rules).
• Assumption — a statement you accept without proof to make the problem solvable; always written down.

The coffee card, modelled honestly

Expected: (20, 2, 70.0)

Cost: O(wanted) time, O(1) space; a closed form (paid = wanted - wanted // 10) exists and is worth deriving as a check.

The elevator: movement vs stops

Expected: 72

Cost: O(n) time for n requests; the *optimisation* of ordering is a different (harder) problem, previewed in Block III.

Which vending machine? Writing the rules

Expected: machine B / machine A / no purchase (per the three traces)

Cost: Constant time; the difficulty is specification completeness, not cost.

The library fine, one ambiguity at a time

wanted = 22
price = 3.50
paid = wanted - wanted // 10
free = wanted // 10
print(paid, free, paid * price)

Expected: 2.50; 8.50 (our reading); 10.00; due-date return 0.00

Cost: Constant time. The pedagogical content is the two contested readings, not the arithmetic.

• **Input:** price 3.50 per drink; the student wants 22 drinks this month
• **Output:** 20 paid drinks, total 70.00 (2 free drinks earned: the 10th and the 20th drink)
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-001 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Propose one extra rule that would change the verdict — is it an improvement?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-001 — not embedded in this deck.

• **Input:** floors requested in press order: [3, 7, 3, 1, 5]; 3 s per floor move; 6 s per stop
• **Output:** travel 3->7->3->1->5 moves 14 floors = 42 s; stops 5 = 30 s; total 72 s
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-002 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Which assumption would you attack first if you had to argue the other side?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-002 — not embedded in this deck.

• **Input:** drink 2.00; student holds coins 2.00 exactly vs drink 2.30 with coins 2.50 (needs change)
• **Output:** exact-coin machine when coins match a price; change machine otherwise; never the exact machine if it means overpaying
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-003 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Propose one extra rule that would change the verdict — is it an improvement?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-003 — not embedded in this deck.

• **Input:** 0.50/day for days 1-7, 1.00/day for days 8-30, cap 10.00; returned on day 5, on day 12, on day 40
• **Output:** 2.50; 7.00 (5 charged days: 7+5x1.00); 10.00 (capped); same-day return 0.00
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-004 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Which assumption would you attack first if you had to argue the other side?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-004 — not embedded in this deck.

Common misconceptions

• Reading the problem once and starting to write - the spec phase is the solution.
• Treating assumptions as facts: 'of course returns include the due date' - says who?
• Confusing an example with a definition (one worked instance does not define the rule).
• Choosing a model silently, then defending the arithmetic instead of the model.

Summary & exit question

• Named technique today: Thinking in Problems
• Four cases attempted — methods before answers

Exit question (choose one, 60 seconds)

1. State the four questions you must answer before attempting any case.
2. Give one example of two defensible readings of the same rule.
3. Why is the five-minute attempt not wasted time?