Introduction to Searching in Complicated Situations

Welcome! In the previous chapter, we looked at searching for solutions in simple situations. Now, we are stepping up the challenge. In the Thinking Skills (9694) syllabus, "complicated" doesn't mean "impossible"—it just means there are more pieces to the puzzle.

In Paper 3 (Problem Analysis and Solution), you will encounter scenarios that are longer and contain more detailed information. Instead of just one or two rules, you might have five or six constraints to juggle at once. This chapter will teach you how to stay organized and find a valid solution even when the problem feels overwhelming.

Note: If you are looking for the absolute "best" or "cheapest" solution, check out our chapter on Find optimal solutions (3.5). Here, we focus on the process of finding any solution that fits all the rules.

What Makes a Search "Complicated"?

A search becomes "complicated" when you cannot see the answer immediately. You might face:

  • Multiple Constraints: For example, a delivery truck must visit \( 5 \) cities, but City A must be visited before City B, and the total distance must be less than \( 200 \) km.
  • Interdependent Variables: Changing one part of your solution (like the time a meeting starts) affects everything else (like who is available to attend).
  • Large Data Sets: You might have a large table of information to filter through before you even start calculating.

The biggest mistake students make is "guessing and checking" randomly. In complicated situations, you must be systematic. This means following a logical order so you don't miss anything and don't repeat work.

Step-by-Step Approach:

1. List your constraints: Before calculating, write down a checklist of every rule the solution must follow.

2. Choose a starting point: Pick the most "restrictive" element. For example, if a person can only work on Tuesday, start your schedule by placing them on Tuesday first.

3. Exhaust one branch at a time: Try one possibility to its conclusion. If it fails, move to the next logical possibility.

Example: If you are trying to find a \( 3 \)-digit code using numbers \( 1, 2, 3 \) where no number repeats, a systematic list would look like this:

\( 123, 132, 213, 231, 312, 321 \)

By keeping the first digit the same until you've used all options, you ensure you haven't missed any.

Strategy 2: Trial and Improvement

In Paper 3, you are often asked to "show your working." This is great news! Even if your final answer is wrong, the examiner can give you credit for the steps you took. Trial and Improvement is a formal way of "guessing" where each guess gets you closer to the truth.

How to do it effectively:

1. Make a "First Pass": Create a solution that fits most of the rules. Don't worry if it's not perfect yet.

2. Identify the Conflict: See which rule your solution broke. (e.g., "The total cost is \( \$10 \) too high.")

3. Adjust and Repeat: Change one variable to fix that specific conflict and see what happens to the others.

Memory Aid: Think of it like tuning a guitar. You tighten a string, check the sound, and then adjust it a little bit more until it's just right.

Handling "Refined" Solutions

A common feature of the 9694 exam is that a problem will develop. Part (a) might be a simple search, but part (c) will add a new rule that changes everything.

Key Tip: Always look back at your previous working. Usually, the "complicated" solution is just a "refined" version of the simple one. Don't start from scratch unless the problem has changed completely! You can adapt a method of solution (Skill 3.6) that you already used in an earlier part of the question.

Common Pitfalls to Avoid

1. The "Logic Trap": Don't assume a constraint exists if it isn't written down. If the problem doesn't say "you cannot use the same road twice," then you can use it twice!

2. Forgetting Units: The 2028 syllabus explicitly asks for values to be labelled with units and words. If you find a solution is \( 45 \), write \( 45 \text{ litres} \) or \( \$45 \text{ profit} \).

3. Losing Track of "Necessary and Sufficient" Conditions: A necessary condition is something that must be true for the solution to work. If your solution doesn't meet a necessary condition, stop immediately and try a different path—don't waste time finishing the calculation.

Quick Review: How to Tackle the Exam Question

When you turn the page and see a massive table or a long list of rules, don't panic. Follow these steps:

  1. Scan for "No-Go" zones: Identify the rules that limit your options the most.
  2. Organize: Use a table, a list, or a simple diagram to keep track of your attempts.
  3. Label everything: As the 2028 syllabus suggests, label your calculations (e.g., \( \text{Total cost} = 5 \times \$10 = \$50 \)).
  4. Check against ALL criteria: Once you think you have the answer, go back through the text one last time. Does it meet every single rule?
Key Takeaway

Searching in complicated situations is about organization, not just math. Break the big problem into smaller rules, use a systematic approach to test possibilities, and always keep your working clear so you can pick up partial marks!