1. A number recycling machine follows this pattern: Input 2 → Output 6, Input 4 → Output 12, Input 6 → Output 18. Lucy thinks that if she inputs 10, the output will be 24. Explain why Lucy's reasoning is incorrect and calculate the correct output.

2. The recycling machine has been modified. Now: Input 3 → Output 15, Input 5 → Output 35, Input 7 → Output 63. If the pattern continues, what would be the input value needed to produce an output of 143? Show your working and explain your strategy.

3. Two students are arguing about a recycling machine pattern. Student A says "The machine multiplies by 4 then subtracts 2" while Student B says "The machine adds 6 then multiplies by 2". Given these outputs: Input 1 → Output 2, Input 3 → Output 10, Input 5 → Output 18, who is correct? Justify your answer mathematically.

4. A recycling centre processes bottles using three machines in sequence. Machine A increases bottles by 50%, Machine B doubles the amount, and Machine C removes 25% for quality control. If 120 bottles enter Machine A, how many bottles exit Machine C? Lucy calculated 180 bottles as her final answer. Identify her likely error and provide the correct solution.

5. Challenge Problem: A special recycling machine processes materials in batches. The relationship between input weight (kg) and processing time (minutes) follows the rule: Time = (Weight × 3) + (Weight²/4). If it takes 19 minutes to process a 4kg batch, verify this rule works. Then determine what weight would require exactly 31 minutes to process. Explain your problem-solving approach.

6. A recycling plant's efficiency varies throughout the day. In the morning, it processes 240 tonnes in 4 hours. In the afternoon, due to higher temperatures, efficiency drops by 25%. If the plant needs to process 500 tonnes total in one day, and morning shift is 4 hours, how long must the afternoon shift work? Show all calculations and explain any assumptions you make.

7. Extension: Create your own recycling machine pattern that involves both multiplication and addition/subtraction. Provide three input-output examples, then write a word problem that someone else could solve using your pattern. Include the solution.