Generating Constraint Lists
Advanced problems often ask you to track recurring events (like rest stops) over a long journey. The first step is to generate a list of every single time the event happens. You must ensure you unify your units (everything must be in distance) before making your list.
Example: Distance List
"A 420 km journey. A driver stops every 100 km."
1. List the multiples: Write down every multiple of $100$ that is strictly less than the destination.
List A: 100, 200, 300, 400
(Note: We stop at 400 because 500 is past the destination).
Example: Time-Converted List
"A 420 km journey at 80 km/h. A driver stops every 2 hours."
1. Unify the Units: Convert the time constraint into a distance constraint.
$$ 2 \text{ hours} \times 80 \text{ km/h} = 160 \text{ km} $$2. List the multiples: Write down every multiple of $160$.
List B: 160, 320
Knowledge Check 1
Marking Schedule:
Q1: Multiples of 150 below 500: $150, 300, 450$. This is $3$ stops.
Q2: Distance per stop $= 1.5 \text{ hrs} \times 120 \text{ km/h} = 180 \text{ km}$. Multiples below 300: just $180$. This is $1$ stop.
Merging & Deduplication (The Overlap)
When a problem says you must stop for Constraint A AND Constraint B, you must combine both lists. The trap is the overlap. If both constraints land on the exact same distance (their Lowest Common Multiple), the driver only takes one rest stop, not two.
Visualizing the "Drop-Down" Method
If you don't have colored pens, draw two separate parallel lines. Mark the stops on each. Then, drop vertical lines down to a final "Merged" line. If two stops line up vertically, they merge into a single dot.
Raw Combination: $60, 120, 180, 240$ AND $90, 180, 270$ (7 stops total)
Deduplicated Set: $60, 90, 120, \mathbf{180}, 240, 270$ (Only 6 unique stops!)
Knowledge Check 2
Marking Schedule:
Q1: Combined set: $40, 60, 80, 120$. Total $= 4$ unique stops.
Q2: Combined set: $50, 75, 100, 150, 225$. Total $= 5$ unique stops.
The Complete Adaptive Method
Challenge Problem (JMC 2016 Variant)
"Travelling at 80 km/h on a 420 km journey, a driver must stop to rest every 100 km AND every 2 hours. How many unique times does he stop before reaching the end?"
Do not guess the overlaps. Run the process strictly step-by-step.
Step 1: Distance List
Write out every multiple of the distance constraint until you reach the end of the journey ($420 \text{ km}$).
Step 2: Time-Converted List
Convert the time constraint into a distance using $d = v \times t$. Then list the multiples.
Conversion: $2 \text{ hours} \times 80 \text{ km/h} = 160 \text{ km}$
List B (160km): 160, 320
Step 3: Merge, Deduplicate, & Truncate
Combine the lists. Remove duplicates. Finally, ensure no stop is AT or PAST the $420 \text{ km}$ destination (a stop at the destination is the end, not a rest stop).
Combined Raw List: $100, 200, 300, 400, 160, 320$
Sorted Unique List: $100, 160, 200, 300, 320, 400$
Total Unique Stops = 6
Knowledge Check 3
Marking Schedule:
List A: $120, 240$
Time Conversion: $1.5 \text{ hrs} \times 60 \text{ km/h} = 90 \text{ km}$.
List B: $90, 180, 270$.
Merged Unique List: $90, 120, 180, 240, 270$. Total = $5$ stops.
Synthesis Task
Combine all skills. Complete 18 mixed-context questions. You have 2 attempts. (Use exact numbers).
Section A: Single Constraint Lists
Section B: Deduplication & Overlaps
Section C: Dual Lists & Merging
Section D: Full Method Challenge
Student Mastery Report
Take a screenshot of this report to submit to Mr. Shen.