Collatz Conjecture : The Simplest Math Problem No One Can Solve - Veritasium info

Explore the mystery of the 3x+1 problem, also known as the Collatz Conjecture — a deceptively simple mathematical question that remains unsolved despite decades of effort.

Why is the Collatz Conjecture still unsolved? Discover the mystery behind the 3x+1 problem, its rules, patterns, and why it puzzles even top mathematicians.

Why the Collatz Conjecture Is One of the Most Dangerous Unsolved Problems in Mathematics

The Collatz conjecture is considered one of the most mysterious and unsolved math problems in history. It’s so deceptively simple that anyone can understand the rules, yet not even the greatest mathematicians have been able to prove it. In fact, many experts in the field warn young mathematicians not to waste their time on it.

This unsolved problem in mathematics, also known as the 3x+1 problem, has intrigued and frustrated mathematicians for decades. The legendary Paul Erdős once remarked, “Mathematics is not yet ripe for such questions.” So what is the Collatz conjecture, and why is it so infamous?

What Is the Collatz Conjecture? | Simple Explanation of 3x+1

Here's how it works:

Pick any positive whole number. Let’s say 7.

Now apply two simple rules:

1. If the number is odd, multiply it by three and add one.

2. If the number is even, divide it by two.

So:

• 7 is odd → 3×7 + 1 = 22

• 22 is even → 22 ÷ 2 = 11

• 11 is odd → 3×11 + 1 = 34

• 34 ÷ 2 = 17, and so on...

Eventually, this sequence always seems to fall into a loop:

4 → 2 → 1 → 4 → 2 → 1...

This cycle is often referred to as the “4-2-1 loop.”

Why Is the 3x+1 Problem Famous (or Infamous)?

The Collatz conjecture, also called the Ulam conjecture, Syracuse problem, or simply the 3N+1 problem, proposes that every positive integer eventually reaches 1 when subjected to these rules.

Despite its simplicity, this unsolved mathematical problem has resisted proof since it was introduced by Lothar Collatz in the 1930s. Mathematicians have thrown massive computational power at it, verifying its truth for numbers up to 2⁶⁸ — that’s over 295 quintillion numbers — yet a formal proof remains elusive.

What Are Hailstone Numbers? | Collatz Sequence Behavior

The numbers in a Collatz sequence are called hailstone numbers, because their behavior mimics hailstones: rising and falling erratically before eventually settling. For example:

• Starting with 26, the sequence peaks at 40 before dropping to 1 in 10 steps (total stopping time).

• Starting with 27, the sequence skyrockets to 9,232, taking 111 steps before it reaches the 4-2-1 loop.

This unpredictable, chaotic behavior in sequences of nearby numbers has led many to call this the most dangerous problem in mathematics.

Randomness, Stock Markets, and Brownian Motion

The wild fluctuations in Collatz sequences resemble the random walk of stock market prices. In fact, if you plot the logarithm of the values, they resemble geometric Brownian motion, the same model used in financial mathematics.

Each step in a 3x+1 sequence is like flipping a coin:

• Heads = up

• Tails = down

Although the process is deterministic, its behavior appears statistically random — a feature that has fascinated both number theorists and chaos theorists alike.

Benford's Law and the Collatz Conjecture

One fascinating pattern found in Collatz sequences is the distribution of leading digits. When analyzing sequences starting from different numbers, a pattern emerges that matches Benford’s Law — a well-known distribution in statistics and forensic accounting.

According to Benford’s Law:

• About 30% of numbers start with the digit 1

• 17.5% start with 2

• Less than 5% start with 9

This same pattern appears across data sets from natural phenomena, population sizes, and even financial reports. While Benford’s Law doesn’t help solve the conjecture, it adds to the mystery and suggests deeper underlying mathematical principles at play.

Why Doesn’t the 3x+1 Function Grow Forever?

At first glance, it seems like the 3x+1 function should grow exponentially — after all, multiplying odd numbers by 3 and adding 1 seems to inflate values. But there’s a catch.

Every time you apply the "multiply by 3 and add 1" rule to an odd number, the result becomes even — and therefore gets divided by 2 (and possibly again and again).

So instead of multiplying by 3, you're often multiplying by an average factor of less than 1, specifically 3/4 when you consider the entire sequence. This means that, on average, the sequence shrinks over time, though its individual steps might jump wildly.

Visualizing the Collatz Graph

Mathematicians have built directed graphs to visualize how each number connects to the next in its sequence. These graphs resemble trees, with all paths converging into the “river” of the 4-2-1 loop.

Some have added artistic flair by rotating the sequence path:

• Clockwise for even numbers

• Counterclockwise for odd numbers

This creates stunning coral-like patterns — a beautiful intersection of math and art.

Could the Collatz Conjecture Be False?

There are only two possibilities if this problem is ever proven false:

1. There exists a number whose sequence grows infinitely large — defying gravity.

2. A non-trivial loop exists — a set of numbers that cycle endlessly without reaching 1.

Yet, despite extensive computational testing and mathematical analysis, no such example has been found.

Conclusion: Why Mathematicians Avoid the Collatz Conjecture

Despite its accessibility, the Collatz conjecture is notoriously resistant to resolution. Mathematicians like Jeffrey Lagarias and Alex Kontorovich, some of the top minds in the field, have cautioned others about spending too much time on it.

Still, the 3x+1 problem continues to captivate both amateurs and professionals. It sits at the crossroads of number theory, chaos theory, computational mathematics, and randomness — making it one of the most fascinating open questions in all of mathematics.

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