Is it true that “any number eventually becomes 1”? ── Playing with the Collatz Conjecture

Hello! I’m kenji.

Suddenly, but if you hear “a rule where any number eventually becomes 1”, isn’t it a bit mysterious?

For example, 19, or 87, or even 1000000. If you tweak the numbers according to appropriate rules, for some reason it converges to “1” at the end.

Such a dream-like story is the Collatz Conjecture.

What is the Collatz Conjecture anyway?

First, let me introduce the rules.

Start: Choose any positive integer
Operation:
- If it is even → Halve it (n → n / 2)
- If it is odd → Triple it and add 1 (n → 3n + 1)

If you repeat this forever, the conjecture says that any number will eventually reach 1.

For example, starting from 6:

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6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1

It properly became “1”. Welcome back!

Let’s do it in code: Collatz in Python

Now, in times like this, it’s faster to try it in code! Let’s output the “Collatz sequence” in Python.

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

# Example: Starting from 19
print(collatz(19))

When you execute it:

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[19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]

It splendidly reaches 1. Even though it takes quite a detour, it firmly reaches the goal at the end!

By the way, even if you start from 29, it reaches 1 in the same way.

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print(collatz(29))

When you execute it:

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[27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242,
121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350,
175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167,
502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479,
1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911,
2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732,
866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35,
106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1]

Surprisingly, it takes 111 steps!

Moreover, there are scenes where it balloons to over 9000 along the way. It’s a pattern that takes a huge detour before reaching the goal.

So, what’s amazing about it in the end?

What’s amazing about this conjecture is,

Even though it hasn’t been proven, it seems to become 1 no matter what number you use

That’s the point.

Eh? Then, what about 1 trillion, or 10 quadrillion…?

If you thought that, you are sharp. Actually, it has been verified up to about “2 to the 68th power” using computers, and all have reached 1. Unbelievable…

But, it hasn’t been theoretically proven that “it always happens”. This is what they call an “unsolved problem” in the world of mathematics.

Who is Mr. Collatz?

So, reading this far, you might wonder “who is Collatz anyway?”. Let me introduce him properly!

Name: Lothar Collatz
Nationality: Germany
Year of birth: 1910 - 1990
Title: Mathematician (Active in the fields of functional analysis and number theory)

He proposed this conjecture in 1937, and since then, for over 80 years, no one has been able to prove or disprove it.

By the way, this problem is so simple yet so deep that even Paul Erdős (a super famous mathematician) is said to have said this:

“Mathematics may not be ready for such problems.”

In other words, the theory that human mathematics hasn’t caught up with this mystery yet…

No “complex math formulas” are necessary

The good thing about the Collatz Conjecture is that anyone can play with it.

You can do it if you have paper and pen. If you write code in Python, you can test it automatically. And yet, cutting-edge mathematicians are seriously challenging it.

Doesn’t it make you excited?

Bonus: Code to test it all at once

I’ll also include code to test various numbers all at once.

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for n in range(1, 21):
    steps = collatz(n)
    print(f"{n}: {steps} (Steps: {len(steps)-1})")

This outputs the Collatz sequences from “1 to 20” all at once.

Conclusion: This world is indeed mysterious

So, that’s the Collatz Conjecture.

Even though it’s super simple
No one can prove it
It’s a huge problem in the math community

It’s an existence like a cluster of mysteries.

Even programming beginners can try it, so please definitely play with it~!