What concepts or facts do you know from math that is mind blowing, awesome, or simply fascinating?
Here are some I would like to share:
- Gödel's incompleteness theorems: There are some problems in math so difficult that it can never be solved no matter how much time you put into it.
- Halting problem: It is impossible to write a program that can figure out whether or not any input program loops forever or finishes running. (Undecidablity)
The Busy Beaver function
Now this is the mind blowing one. What is the largest non-infinite number you know? Graham's Number? TREE(3)? TREE(TREE(3))? This one will beat it easily.
- The Busy Beaver function produces the fastest growing number that is theoretically possible. These numbers are so large we don't even know if you can compute the function to get the value even with an infinitely powerful PC.
- In fact, just the mere act of being able to compute the value would mean solving the hardest problems in mathematics.
- Σ(1) = 1
- Σ(4) = 13
- Σ(6) > 10^10^10^10^10^10^10^10^10^10^10^10^10^10^10 (10s are stacked on each other)
- Σ(17) > Graham's Number
- Σ(27) If you can compute this function the Goldbach conjecture is false.
- Σ(744) If you can compute this function the Riemann hypothesis is false.
Sources:
- YouTube - The Busy Beaver function by Mutual Information
- YouTube - Gödel's incompleteness Theorem by Veritasium
- YouTube - Halting Problem by Computerphile
- YouTube - Graham's Number by Numberphile
- YouTube - TREE(3) by Numberphile
- Wikipedia - Gödel's incompleteness theorems
- Wikipedia - Halting Problem
- Wikipedia - Busy Beaver
- Wikipedia - Riemann hypothesis
- Wikipedia - Goldbach's conjecture
- Wikipedia - Millennium Prize Problems - $1,000,000 Reward for a solution
There are more ways to arrange a deck of 52 cards than there are atoms on Earth.
52 Factorial
I feel this one is quite well known, but it's still pretty cool.
An extension of that is that every time you shuffle a deck of cards there's a high probability that that particular arrangement has never been seen in the history of mankind.
With the caveat that it's not the first shuffle of a new deck. Since card decks come out of the factory in the same order, the probability that the first shuffle will result in an order that has been seen before is a little higher than on a deck that has already been shuffled.
Since a deck of cards can only be shuffled a finite number of times before they get all fucked up, the probability of deck arrangements is probably a long tail distribution
The most efficient way is not to shuffle them but to lay them all on a table, shift them around, and stack them again in arbitrary order.
assuming a perfect mechanical shuffle, I think the odds are near zero. humans don't shuffle perfectly though!
What's perfect in this context? It's maybe a little counterintuitive because I'd think a perfect mechanical shuffle would be perfectly deterministic (assuming no mechanical failure of the device) so that it would be repeatable. Like, you would give it a seed number (about 67 digits evidently) and the mechanism would perform a series of interleaves completely determined by the seed. Then if you wanted a random order you would give the machine a true random seed (from your wall of lava lamps or whatever) and you'd get a deck with an order that is very likely to never have been seen before. And if you wanted to play a game with that particular deck order again you'd just put the same seed into the machine.
Perfect is the sense that you have perfect randomness. Like the Fisher-Yates shuffle.
In order to have a machine that can "pick" any possible shuffle by index (that's all a seed really is, a partial index into the space of random numbers), you'd need a seed 223 bits long.
But you wouldn't want perfect mechanical shuffles though because 8 perfect riffles will loop the deck back to it's original order! The minor inaccuracies are what makes actual shuffling work.
I'd probably have the machine do it all electronically and then sort the physical deck to match, not sure you could control the entropy in a reliable way with actual paper cards otherwise.