Weird Algorithm

Problem Overview

Learning Goals

After solving this problem, you will be able to:

• Understand the Collatz conjecture and its properties
• Implement a simple simulation algorithm
• Handle potential integer overflow with large numbers
• Format output correctly with space-separated values

Problem Statement

Problem: Given a positive integer n, simulate the following algorithm and print all values in the sequence:

• If n is even, divide it by 2
• If n is odd, multiply it by 3 and add 1
• Repeat until n equals 1

Input:

• A single integer n

Output:

• Print all values of n during the algorithm, including the initial value and the final 1

Constraints:

• 1 <= n <= 10^6

Example

Input:
3

Output:
3 10 5 16 8 4 2 1

Explanation: Starting with n=3:

• 3 is odd, so 3*3+1 = 10
• 10 is even, so 10/2 = 5
• 5 is odd, so 5*3+1 = 16
• 16 is even, so 16/2 = 8
• 8 is even, so 8/2 = 4
• 4 is even, so 4/2 = 2
• 2 is even, so 2/2 = 1
• We reached 1, so we stop

Intuition: How to Think About This Problem

The Collatz Conjecture

Key Insight: This problem is based on the famous Collatz conjecture (also known as the 3n+1 problem), which states that this sequence always eventually reaches 1 for any positive starting integer.

The Collatz conjecture is one of the most famous unsolved problems in mathematics. Despite its simple rules, no one has proven that the sequence always reaches 1 for all positive integers. However, it has been verified computationally for all numbers up to very large values.

Breaking Down the Problem

1. What are we looking for? The complete sequence from n down to 1
2. What information do we have? The starting value n
3. What’s the relationship between input and output? Each value determines the next via a simple rule

Analogies

Think of this problem like a ball rolling down a hill with two different slopes:

• On even terrain (even numbers), the ball rolls gently downhill (divide by 2)
• On odd terrain (odd numbers), the ball bounces up briefly (multiply by 3, add 1) before continuing down
• Eventually, the ball always reaches the bottom (value 1)

Solution: Simulation

Idea

This is a straightforward simulation problem. We simply apply the rules repeatedly until we reach 1, printing each value along the way.

Algorithm

1. Print the initial value n
2. While n is not equal to 1:
   • If n is even, set n = n / 2
   • If n is odd, set n = n * 3 + 1
   • Print n
3. Done

Dry Run Example

Let’s trace through with input n = 7:

Initial: n = 7
Print: 7

Step 1: n = 7 (odd)
  n = 7 * 3 + 1 = 22
  Print: 22

Step 2: n = 22 (even)
  n = 22 / 2 = 11
  Print: 11

Step 3: n = 11 (odd)
  n = 11 * 3 + 1 = 34
  Print: 34

Step 4: n = 34 (even)
  n = 34 / 2 = 17
  Print: 17

Step 5: n = 17 (odd)
  n = 17 * 3 + 1 = 52
  Print: 52

Step 6: n = 52 (even)
  n = 52 / 2 = 26
  Print: 26

Step 7: n = 26 (even)
  n = 26 / 2 = 13
  Print: 13

Step 8: n = 13 (odd)
  n = 13 * 3 + 1 = 40
  Print: 40

Step 9: n = 40 (even)
  n = 40 / 2 = 20
  Print: 20

Step 10: n = 20 (even)
  n = 20 / 2 = 10
  Print: 10

Step 11: n = 10 (even)
  n = 10 / 2 = 5
  Print: 5

Step 12: n = 5 (odd)
  n = 5 * 3 + 1 = 16
  Print: 16

Step 13-16: 16 -> 8 -> 4 -> 2 -> 1

Final output: 7 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1

Visual Diagram

n = 3:

3 (odd)
|
v  *3+1
10 (even)
|
v  /2
5 (odd)
|
v  *3+1
16 (even)
|
v  /2
8 (even)
|
v  /2
4 (even)
|
v  /2
2 (even)
|
v  /2
1 (STOP)

Output: 3 10 5 16 8 4 2 1

Code (Python)

def solve():
  """
  Weird Algorithm - Collatz sequence simulation.

  Time: O(?) - unknown, but finite for all tested values
  Space: O(1) - no extra storage needed
  """
  n = int(input())

  result = [n]
  while n != 1:
    if n % 2 == 0:
      n = n // 2
    else:
      n = n * 3 + 1
    result.append(n)

  print(' '.join(map(str, result)))

solve()

Complexity

Common Mistakes

Mistake 1: Integer Overflow

Problem: When n is large and odd, multiplying by 3 and adding 1 can exceed the 32-bit integer limit. Fix: Use long long in C++ or Python’s built-in arbitrary precision integers.

Mistake 2: Output Format - Extra Space

# WRONG - extra space at the end
while n != 1:
  print(n, end=' ')
  # ...
print(n)  # prints "1" but there's trailing space before it

# CORRECT - join with spaces
result = [n]
while n != 1:
  # ...
  result.append(n)
print(' '.join(map(str, result)))

Problem: Trailing or leading spaces can cause wrong answer. Fix: Collect all values first, then join with single spaces.

Mistake 3: Forgetting to Print Initial Value

# WRONG
while n != 1:
  if n % 2 == 0:
    n = n // 2
  else:
    n = n * 3 + 1
  print(n, end=' ')  # Missing initial value!

# CORRECT
print(n, end=' ')  # Print initial value first
while n != 1:
  # ...

Problem: The sequence should include the starting value. Fix: Print the initial value before entering the loop.

Edge Cases

When to Use This Pattern

Use This Approach When:

• The problem asks you to simulate a process step by step
• There are clear rules for state transitions
• The termination condition is well-defined
• No optimization beyond direct simulation is needed

Don’t Use When:

• The number of steps could be astronomically large (need memoization or math)
• You need to analyze properties without computing the full sequence
• Pattern detection could shortcut the computation

Pattern Recognition Checklist:

• Are there clear rules for what happens next? -> Simulation
• Is the sequence deterministic? -> Simulation
• Do we need all intermediate values? -> Simulation with storage

Related Problems

Easier (Do These First)

Similar Difficulty

Harder (Do These After)

Key Takeaways

1. The Core Idea: Simulate the Collatz sequence by repeatedly applying the transformation rules until reaching 1
2. Overflow Risk: The sequence values can grow large before eventually decreasing, so use 64-bit integers
3. Output Format: Pay attention to space-separated output requirements
4. Termination: While the Collatz conjecture is unproven mathematically, the sequence terminates for all practical inputs

Practice Checklist

Before moving on, make sure you can:

• Solve this problem without looking at the solution
• Explain why long long is needed in C++
• Handle the output format correctly
• Identify this as a simulation problem pattern

Additional Resources

• Wikipedia: Collatz Conjecture
• CSES Weird Algorithm - Collatz sequence
• Numberphile: The Collatz Conjecture