The Simplest Problem No One Has Solved Yet
A century-old puzzle worth $800,000, still mocking the brightest mathematicians alive.
☕ The Deep Brew - Edition 1
The Collatz Conjecture: When Simple Turns Impossible
Welcome to the very first edition of The Math Café!
I’m so glad to see you here.
On the menu today: a deceptively simple puzzle that has stumped some of the brightest mathematical minds of our time.
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The Story
Eighty-eight years ago, a young mathematics assistant named Lothar Collatz noticed a curious pattern:
Take any positive number.
If it’s even - divide it by 2.
If it’s odd - triple it and add 1.
Repeat.
No matter where you start, you seem to always end up at 1.
For example:
Start with 21 → 64 → 32 → 16 → 8 → 4 → 2 → 1.
Start with 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.
Recommended: Try it with your birthday month. But beware - it might take longer than you expect (up to 19 steps)!
For example, if you were born in November (11): 11 → 34 → 17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.
Some Good (and Bad) News
Looks simple enough, right?
The good news: If you manage to prove that this process always reaches 1, you could earn over $800,000 USD - just publish your proof in a recognized journal before 2031 and claim your prize from the Math Prize.
The bad news: No one has managed to prove it yet. No one can even be *sure* the conjecture is true.
Even Terence Tao, one of the greatest problem-solvers alive and a Fields Medalist, tried his hand at it. In 2019, he proved that the conjecture holds for almost all numbers (refining previous results by mathematicians Krasikov and Lagarias, Terras, and Korec). This extraordinary step forward involved sophisticated statistics tools, and great insights. But a complete proof, valid for all numbers, remains elusive.
On the other hand, massive computer searches have verified it for numbers up to sextillions (that’s billions of trillions!). Yet pure computation can never prove it true for every number, and only fail to find a counterexample.
Up to this date, numerous attempt involving AI-assisted formal proofs remain unsuccessful.
Why Is It So Hard?
The Collatz Conjecture lives in that tricky space where numbers behave chaotically.
Every simple step (divide by 2, or 3x+1) feels predictable - but the long-term outcome is almost impossible to foresee.
This makes it a perfect example of how complexity can emerge from simplicity, a recurring theme in mathematics.
What Became of Collatz?
Lothar Collatz went on to make important contributions to functional analysis, differential equations, and numerical analysis. Yet, to this day, he’s best remembered for this playful riddle from his early career - the one we now call the 3x+1 problem or the Collatz Conjecture.
So next time you face a puzzle that seems “too simple,” remember:
You might just be having your own Collatz moment. 🙂
☕ Your Turn
💭 What “deceptively simple” problem puzzled you this month?
Take a moment to reflect: sometimes, the hardest puzzles teach us the most 🪞.
Next week
Your first Espresso Shot: sharp puzzles to sharpen your logic.
Looking forward to seeing you there!
With curiosity and love, 🌿
Sybille for The Math Café