(Roughly) Daily

In 1937, Lother Collatz proposed that, when starting with any natural number (positive integer) n, then either dividing it by 2 if n is even, or multiplying n by 3 and adding 1 if n is odd, then continuing this process indefinitely, one will always reach the number 1.

The proposition is pretty broadly known as “The Collatz conjecture”; but as it commanded the energies of a number of other mathematicians, it is also sometimes called “the  Ulam conjecture” (after Stanislaw Ulam), “Kakutani’s problem” (after Shizuo Kakutani), “the Thwaites conjecture” (after Sir Bryan Thwaites), “Hasse’s algorithm” (after Helmut Hasse), or “the Syracuse problem.”  More generically, it’s often just called the “3n+1 conjecture”…  and the sequence that it generates, “the wondrous numbers.”

But as wondrous as the sequence may be, it is no pushover to prove.  Indeed, legendary (and legendarily “homeless”) mathematician Paul Erdos– the source of this post’s title quotation– posted a $500 prize for a proof… an offer he never had to make good.  Still, most mathematicians who have played with the problem believe the conjecture to be true,as both experimental evidence and heuristic arguments support it.

It is in any case, fascinating– and beautiful.  London-based designer Jason Davies has created a visualization of the Wondrous Numbers, an animation that runs the Collatz logic in reverse— starting with 1 and “growing” a tree of natural numbers:

click here to watch this grow from a single digit

As we remark that there’s a fundamental propriety to everything in the world ultimately coming back to one, we might recall that it was on this date in 1940, at approximately 11:00 am, that the first Tacoma Narrows suspension bridge collapsed due to wind-induced vibrations. Situated on the Tacoma Narrows in Puget Sound, near the city of Tacoma, Washington, the bridge had only been open for traffic a few months.  For those who missed it in high school physics: