Welcome to "Find Optimal Solutions"
In your Thinking Skills journey, you have already learned how to search for solutions in simple and complex situations. Now, we are taking it a step further. Finding an optimal solution isn't just about finding any answer that works—it is about finding the best possible answer under the circumstances.
Whether you are trying to find the cheapest way to travel, the quickest way to finish a task, or the way to pack a bag to fit the most items, you are looking for an "optimum." This chapter will help you master the techniques needed for Paper 1.
1. What is an "Optimal Solution"?
An optimal solution is the best outcome for a specific goal. In the Thinking Skills syllabus, "best" usually falls into two categories:
1. Maximising: Finding the highest possible value (e.g., the most profit, the most points, or the most people helped).
2. Minimising: Finding the lowest possible value (e.g., the least cost, the shortest time, or the minimum number of staff needed).
Note: To find an optimal solution, you must work within constraints. Constraints are the "rules" or "limits" given in the problem (e.g., "you only have \( \$50 \)" or "the bus cannot carry more than \( 12 \) people").
2. The Three Ingredients of an Optimization Problem
To solve these problems effectively, you need to identify three things immediately:
A. The Objective: What are you trying to make as big or as small as possible? (e.g., "Find the minimum number of tiles...")
B. The Constraints: What are the rules you cannot break? (e.g., "Tiles cannot be cut" or "The area is \( 5m \times 4m \)")
C. The Variables: What can you change to reach your goal? (e.g., The size of the tiles you choose or the pattern you lay them in).
3. Step-by-Step Strategy
Don't worry if a problem looks complicated at first! Follow these steps to stay organized:
Step 1: Simplify the Information
Read the scenario and identify the logical relationships. If the information is in a long paragraph, try putting it into a small table or a list of bullet points. (See the chapter on "Understanding Information" for more on this!)
Step 2: Test the "Extreme" Cases
Often, the optimal solution is found at the edges of the possibilities. For example, if you want to minimize cost, try using as much of the "cheapest" item as the rules allow before using more expensive items.
Step 3: Look for Patterns
In more complicated situations, you might notice a pattern. For example, if adding one more worker saves \( 10 \) minutes but costs \( \$20 \), and your goal is to save money, you can stop adding workers immediately.
Step 4: Check for "Hidden" Rules
Sometimes a constraint is implied. If you are calculating the number of buses needed for \( 50 \) people, and each bus holds \( 12 \), your answer must be a whole number. You can't have \( 4.16 \) buses; you would need \( 5 \).
4. Common Techniques for the Exam
The "Greedy" Approach
This is a simple trick where you always pick the "best" immediate option.
Example: If you are asked to provide \( 37 \) cents using the fewest coins possible, you would start with the largest possible coin (\( 25 \) cents), then the next largest (\( 10 \) cents), and so on.
\( 37 = 25 + 10 + 1 + 1 \) (Total: \( 4 \) coins).
Warning: This doesn't always work for every problem, so always double-check if a different combination might be better!
The "Trial and Improvement" Method
If you aren't sure where to start, pick a reasonable starting value and see if it works. If it does, try to "push" it further to see if you can get an even better result. If it doesn't work, adjust it until it does.
Quick Review: Key Terms
Constraint: A limit or rule you must follow.
Feasible Solution: Any answer that follows all the rules (but might not be the "best" one).
Optimal Solution: The absolute best feasible solution.
5. Avoiding Common Mistakes
Even the best students can make these errors. Watch out for:
1. Forgetting a constraint: Always re-read the rules after you find your answer to make sure you didn't break one.
2. Stopping too soon: Just because you found an answer that works doesn't mean it's the optimal one. Ask yourself: "Can I make this even smaller/larger?"
3. Calculation errors: Use your calculator! The syllabus (especially for 2028) strongly recommends using a calculator for Paper 1.
4. Missing units: From 2028 onwards, the examiners explicitly want you to label your values. Instead of just writing \( 50 \), write \( \$50 \) or \( 50 \text{ kg} \).
6. Summary and Key Takeaways
Finding optimal solutions is like solving a puzzle where the goal is perfection. Remember these three things:
Identify: Find the goal (Max or Min) and the rules (Constraints).
Organize: Use tables or lists to manage the data.
Check: Ensure your final answer is "feasible" (it works) and "optimal" (nothing else is better).
Pro-Tip: If you find yourself stuck, look back at your working for "Searching for solutions in more complicated situations." Often, the optimal solution is just the best result from a search you've already performed!