Divide by 2
If the number is even, divide it by 2.
n → n ÷ 2
Example: 18 ÷ 2 = 9
The Busy Brains · Tuition Classes
Class 7 Mathematics · Number Patterns
The Collatz Conjecture
A remarkably simple number game that leads to one of mathematics’ most intriguing unsolved questions.
01 · The Central Idea
What is the Collatz Conjecture?
Choose any positive whole number. Apply a simple rule to it, then apply the same rule to the new number. Keep repeating the process.
The Collatz Conjecture says that, no matter which positive whole number we start with, this process will eventually reach 1.
Once the sequence reaches 1, it enters a repeating cycle:
1 → 4 → 2 → 1 → 4 → 2 → 1 → …
Important mathematical point: The conjecture has been tested for an enormous number of starting values, but no proof is known that it works for every positive whole number. It remains an unsolved problem in mathematics.
02 · A Mathematician’s Name
Why is it called the Collatz Conjecture?
The problem is named after Lothar Collatz, a German mathematician who introduced the problem in 1937. It is also known as the 3n + 1 problem, because one of its rules is to multiply an odd number by 3 and add 1.
1937
Problem introduced
Problem introduced
The word conjecture means a mathematical statement that appears to be true based on evidence, but has not yet been proved.
03 · The Two Rules
How does the number game work?
At every step, look at the number you currently have. There are only two possibilities:
Multiply by 3, then add 1
If the number is odd, multiply it by 3 and add 1.
n → 3n + 1
Example: 7 × 3 + 1 = 22
Remember: After finding each new number, check whether it is even or odd. Then apply the appropriate rule again. Do not stop until the sequence reaches 1.
04 · Learn by Example
Worked Examples
Example 1 · Start with 6
Even and Odd Steps
We begin with 6. Apply the correct rule at each step:
- 6 is even: 6 ÷ 2 = 3.
- 3 is odd: (3 × 3) + 1 = 10.
- 10 is even: 10 ÷ 2 = 5.
- 5 is odd: (5 × 3) + 1 = 16.
- 16, 8 and 4 are even, so keep dividing by 2.
6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1
Conclusion: Starting with 6, the sequence reaches 1 after 8 steps.
Example 2 · Start with 7
A Longer Sequence
Apply the rules repeatedly, checking each new number:
7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
Conclusion: Starting with 7, the sequence also reaches 1. Notice that the numbers do not simply decrease at every step. Some become larger before eventually falling to 1.
05 · Observe the Pattern
What can we notice?
| Starting Number | Steps to Reach 1 | Highest Number |
|---|---|---|
| 2 | 1 | 2 |
| 3 | 7 | 16 |
| 6 | 8 | 16 |
| 7 | 16 | 52 |
Different starting numbers can produce sequences of different lengths and reach different highest values. Yet these examples all reach 1. The conjecture asks whether this is true for every positive whole number.
06 · Practice Worksheet
Now It Is Your Turn
Instructions: Start with the given positive whole number. Write every term in the sequence, show the rule used, and stop when you reach 1.
A. Generate the Sequences
- Start with 4. Write the complete Collatz sequence.
- Start with 5. Write the complete Collatz sequence.
- Start with 9. Write the complete Collatz sequence.
- Start with 10. Write the complete Collatz sequence.
- Start with 12. Write the complete Collatz sequence.
B. Think and Calculate
- What is the next number after 14? Explain which rule you used.
- What is the next number after 19? Show the calculation.
- For the starting number 7, identify the highest number in its sequence.
- How many steps does the sequence starting at 6 take to reach 1?
- Can two different starting numbers have the same highest number in their sequences? Use the examples in this lesson to support your answer.
C. Challenge Questions
- A student says, “The Collatz sequence always becomes smaller.” Is this statement correct? Give an example to justify your answer.
- Write the first five terms of the sequence that starts at 15.
- Explain, in your own words, why the Collatz statement is called a conjecture and not a theorem.
Teacher’s Answer Check (Open After Attempting)
1. 4 → 2 → 1
2. 5 → 16 → 8 → 4 → 2 → 1
3. 9 → 28 → 14 → 7 → 22 → 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1
4. 10 → 5 → 16 → 8 → 4 → 2 → 1
5. 12 → 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1
6. 7 (14 ÷ 2). 7. 58 (19 × 3 + 1). 8. 52. 9. 8 steps. 10. Yes. For example, 3 and 6 both reach a highest value of 16.
11. No. For example, 3 → 10, so the value increases. 12. 15 → 46 → 23 → 70 → 35. 13. It has evidence from many examples, but it has not been proved for all positive whole numbers.
Try the process with a number of your choice. Does it reach 1? Can you prove that every positive whole number must eventually reach 1? Testing examples is useful, but a mathematical proof must explain why the statement is true for all permitted starting numbers.
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