In light of the very sad news about Ron Graham's passing, I thought I'd share an interesting tidbit about his famous constant which I only learned recently (thread).
This number has captured the imaginations of many people with its unfathomable size, myself included. I distinctly remember when I was a kid first learning about it spending hours trying to wrap my mind around it and using it as inspiration for writing ever-larger numbers.
The existence of huge numbers is not in and of itself interesting, but what captivated me was how abstraction and recursion even let us _describe_ such numbers. Honestly, it was one of the first times I appreciated how powerful good definitions could be.
Of all the videos I've made, one of my favorites topics to have covered was a proof of the inscribed rectangle problem by H. Vaughan using a Mobius strip.
But now there's a new result!
@QuantaMagazine recently did a great article about recent work by Greene and Lobb using a beefed-up version of the same idea, letting a Mobius strip encode geometric properties of the curve to solve a more general result, check it out:
From recent comments on the video above, it looks there's a little confusion where some people thought this means the inscribed square problem (i.e. the Toeplitz conjecture) has been solved, but that's not quite the case.
The birthday paradox is very famous in probability. If you take 23 people, there's about a 50/50 chance that two of them share a birthday. With 50 people, it's a 97% chance.
We could make many other fun examples to illustrate the same counterintuitive phenomenon (thread).
Choose a random card from a deck of 52 cards. Put it back, shuffle well, and choose another. Do this for only 9 draws, and more likely than not, you've pulled the same card twice.
Do it 16 times, and your chances are over 90%. Try it!
Next time you're in an event with more than 118 people, think to yourself that there's a >50% chance that two people there have phone numbers with the same last four digits (assuming those are uniformly distributed).