×
π’ The Collatz Conjecture
The Collatz Conjecture, also known as the 3n+1 problem, is one of the most famous unsolved problems in mathematics. Despite its simple rules, it has puzzled mathematicians for decades.
π History
Named after German mathematician Lothar Collatz who introduced it in 1937, this conjecture has been studied by mathematicians worldwide. It's also known by various names including the Ulam conjecture, Kakutani's problem, the Thwaites conjecture, Hasse's algorithm, and the Syracuse problem.
π¬ The Problem
Starting with any positive integer n, repeatedly apply these simple rules:
If n is even: n β n Γ· 2
If n is odd: n β 3n + 1
The conjecture states that no matter what positive integer you start with, you will always eventually reach 1. Once you reach 1, the sequence enters a cycle: 1 β 4 β 2 β 1.
π Examples
Starting with 7:
7 β 22 β 11 β 34 β 17 β 52 β 26 β 13 β 40 β 20 β 10 β 5 β 16 β 8 β 4 β 2 β 1
(16 steps to reach 1)
Starting with 27:
This sequence reaches a peak of 9,232 before eventually reaching 1 after 111 steps!
π€ Why Is It Unsolved?
While the conjecture has been verified for:
- All numbers up to 2βΆβΈ (approximately 295 quintillion)
- Various mathematical patterns and special cases
- Countless computer simulations
No one has been able to prove that it works for all positive integers. The challenge lies in the unpredictable nature of the sequence - some numbers take just a few steps, while others (like 27) can have very long paths with surprisingly high peaks.
π― Mathematical Significance
The Collatz Conjecture demonstrates several important mathematical concepts:
- Computational complexity: Simple rules can lead to complex behavior
- Number theory: Relationships between odd and even numbers
- Dynamical systems: How sequences evolve over time
- Proof techniques: The difficulty of proving universal statements
π The Prize
While not one of the official Millennium Prize Problems, the Collatz Conjecture has captured the imagination of mathematicians and computer scientists worldwide. Jeffrey Lagarias offered a $500 prize for its proof in 1985, and various other bounties have been proposed over the years.
π‘ Fun Facts
- Paul ErdΕs said about the Collatz conjecture: "Mathematics may not be ready for such problems."
- The number 27 holds the record for the longest known sequence under 1,000
- Some sequences can reach peaks over 1 million before returning to 1
- The conjecture has inspired art, music, and recreational mathematics
Use this visualizer to explore different starting numbers and see how the Collatz sequence behaves. Try small numbers first, then experiment with larger ones to see the fascinating patterns that emerge!