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Number Play - An Unsolved Mystery — the Collatz Conjecture!

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Collatz Conjecture is a mathematical hypothesis that involves a sequence of numbers starting from any positive integer nn.

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If the current number nn is even, the next number is calculated by dividing it by 22, represented as n2\frac{n}{2}.

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If the current number nn is odd, the next number is calculated by multiplying it by 33 and adding 11, represented as 3n+13n + 1.

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The conjecture states that regardless of the starting value of nn, the sequence will always reach the number 11.

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Once the sequence reaches 11, it enters a repeating loop: 1→4→2→11 \to 4 \to 2 \to 1.

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This problem is also known as the 3n+13n + 1 problem or the Ulam conjecture.

📐Formulae

f(n)=n2 (if n is even)f(n) = \frac{n}{2} \text{ (if } n \text{ is even)}

f(n)=3n+1 (if n is odd)f(n) = 3n + 1 \text{ (if } n \text{ is odd)}

💡Examples

Problem 1:

Apply the Collatz rules to the starting number n=6n = 6 and find the sequence until it reaches 11.

Solution:

6→3→10→5→16→8→4→2→16 \to 3 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1

Explanation:

  1. 66 is even, so 6÷2=36 \div 2 = 3.
  2. 33 is odd, so (3×3)+1=10(3 \times 3) + 1 = 10.
  3. 1010 is even, so 10÷2=510 \div 2 = 5.
  4. 55 is odd, so (3×5)+1=16(3 \times 5) + 1 = 16.
  5. 1616 is even, so 16÷2=816 \div 2 = 8.
  6. 88 is even, so 8÷2=48 \div 2 = 4.
  7. 44 is even, so 4÷2=24 \div 2 = 2.
  8. 22 is even, so 2÷2=12 \div 2 = 1.

Problem 2:

Start with the number n=5n = 5. How many steps does it take to reach the number 11?

Solution:

The sequence is 5→16→8→4→2→15 \to 16 \to 8 \to 4 \to 2 \to 1. It takes 55 steps to reach 11.

Explanation:

Step 1: 55 is odd, 3(5)+1=163(5) + 1 = 16. Step 2: 1616 is even, 16/2=816/2 = 8. Step 3: 88 is even, 8/2=48/2 = 4. Step 4: 44 is even, 4/2=24/2 = 2. Step 5: 22 is even, 2/2=12/2 = 1.