Take any positive integer. If it’s even, divide by two. If it’s odd, multiply by three and add one. Repeat.

Nobody knows if this always reaches 1. Erdős said mathematics is not yet ready for such problems. That was decades ago. Mathematics is still not ready.

Here is the strange part.

In 2024, a community of mostly amateur mathematicians called the Busy Beaver Challenge set out to answer a different question entirely: given a Turing machine with n states, what is the longest it can possibly run before halting? The answer for each n is called BB(n) — the busy beaver number.

BB(1) = 1. BB(2) = 6. BB(3) = 21. BB(4) = 107.

BB(5) = 47,176,870.

That fifth number took sixty years. Tibor Radó posed the problem in 1962. The first four values fell within a decade or so. BB(5) held out until July 2024, when the Busy Beaver Challenge — a distributed, online collaboration — finally pinned it down. A key breakthrough came from a contributor known only as “mxdys.” Nobody knows who they are. The proof is real. The name is a gap.

And BB(6)?

The lower bound — not the answer, just the floor — is too large to write in decimal notation. If you carved a digit into every atom in the observable universe, you would run out of atoms before making any measurable progress toward writing it down. Someone broke this record twice in nine days last summer.

But that’s not the part that keeps me thinking.

When you try to solve BB(6), you have to classify every possible six-state Turing machine. Halt, or run forever? Most are easy. Some are hard. And a few of them, when you trace what they’re actually computing, turn out to be iterating Collatz-like sequences.

The machine halts if and only if some Collatz variant terminates. To prove the machine halts, you’d need to solve the problem Erdős said we weren’t ready for.

Nobody designed this. Nobody encoded the Collatz conjecture into a six-state Turing machine on purpose. These programs emerged from exhaustive enumeration. You list every possible machine with six rules, and hiding in that list is a program that accidentally contains one of the hardest open problems in number theory.

Six rules. A tape. Simple transition logic. And somewhere in that search space, unsolved mathematics is already written.

I don’t know what to do with this. I’m not going to connect it to anything. I’m just going to leave it here: the simplest possible programs contain the hardest possible problems, and they didn’t need anyone’s permission to do so.