3n+1 Ep1: Brutal Kindergarten Math! The Collatz Conjecture

welcome to the famous 3n plus 1 conjecture i'm kevin knight there's a lot of famous unsolved problems in mathematics all deep mysteries but in this series we're going to focus on the 3n plus 1 conjecture

because in order to understand it you only need to know how to add and multiply numbers for example what's 3 times 7 plus 1

that's 21 plus 1 equals 22 and what's 22 divided by 2 right 11. okay that means we're ready to go

11. okay that means we're ready to go the three n plus one conjecture goes like this pick any number if it's even cut it in half if it's odd multiply by three and add one

then repeat okay let's pick a number to start with say 10. 10 is even so we cut it in half and

10. 10 is even so we cut it in half and get 5.

5 is odd so we multiply by 3 and add 1 which gives us 16. 16 is even so we cut it in half to 8 which we cut in half to four then two

then one we'll make it a rule to stop at one okay let's try starting with eleven instead of ten we get eleven

thirty four 17 52 cut it in half to 26 again to 13 up to 40

down to 20 and then 10 5 16 8 4 2 one okay that took a little bit longer but it also got to one the famous three n plus one conjecture

says every number goes to one is that true is that false no mathematician knows the answer so it's a deep mystery if we think the conjecture is false we can look for a

number that doesn't go to one let's see two goes to one in one step three goes to one but it takes a little longer

seven steps four goes to one five goes to one six goes to one now when we get up to something like nine it also goes to one but it goes through a long seemingly

random path of numbers see here it goes all the way up to 52 then goes all the way back down to one how about something bigger like 31 if we

start with 31 you can see in this giant original oil painting that 31 is odd so we multiply by 3 and add 1 which gives 94.

dividing this in half gives 47 and so on and so on well up to over 9000 at this point maybe this is going to keep going up forever and ever

no it starts plummeting again and finally reaches one okay we checked a bunch of numbers and they all go to one but nobody knows if every number goes to one for example there might be some number that just

wanders off into infinity or there might be a loop some number might lead back to itself and then just go round and round never reaching one if you could find a loop you could write

what would be one of the most famous mathematical papers ever it would be very short the 3n plus 1 problem has a loop by u if we start with this huge number and

iterate the 3n plus 1 operations we eventually get back to the same number this number nobody's ever checked that number should we check it right now nah not right now we don't have time

but you can imagine that mathematicians with computers have checked a lot of numbers actually they check the first billion billion numbers and they all go to one no loops you might say well then the

odds are there aren't any loops ever and in a later episode we're actually going to bizarrely calculate those odds exactly but even with the odds maybe there is a loop out there maybe we get

lucky because contrary to popular belief math is unpredictable and chaotic let me tell you the strange story of another conjecture back in 1951

a guy named lewis mordell considered a very old equation x cubed plus y cubed plus z cubed equals three

he observed that there only seem to be two solutions the first is one cubed plus one cubed plus one cubed equals three that's the easy one the second one is four

cubed plus four cubed plus negative five cubed equals three because negative five times negative five times negative five is negative 125

and 64 plus 64 is 128 minus 125 is 3.

now uh louis mordell noted that no matter what other values of xy people x y and z people tried over hundreds of years they could never get three to come out so we have a conjecture an unproven

mathematical assertion that might be true or might be false and that was that until in 2019

two guys named booker and sutherland found another solution and this one goes like this

569 quintillion 936 quadrillion 821 trillion 221 billion 962 million 380 720 cubed plus this other number cubed

plus this other number cubed equals three so they discovered a counter example and they had to go pretty far out to find it so what's the moral of this story well maybe there's a loop for the 3n

plus 1 problem maybe involving some huge numbers but nobody found it yet and we're going to look at that in upcoming episodes but first a warning the famous mathematician and professional

couchsurfer paul erdosh not only couldn't solve this problem himself he said quote mathematics is not ready for such problems unquote in other words if you try to solve this

problem you're sure to fall into a morass a pit a quagmire wasting years of your life trying to solve something that mathematics itself is not ready for and you will wind up making videos on

youtube paul ardesch would know he was one of the most collaborative mathematicians in history if you co-authored a paper with erdos you have a narrative number of one if you co-authored a paper with someone

who co-authored a paper with erdos your erdos number is two it's like the kevin bacon number if you acted in a movie with someone who acted in a movie with someone who acted in a movie with

kevin bacon then you have a bacon number of three now some people have a bacon air dose number that combines the two for example natalie portman acts in cool

movies like star wars and also writes cool papers about hemoglobin concentration in baby brains her bacon erdos number is five plus two equals seven

okay in upcoming episodes we'll look at some amazing patterns that might help us resolve the three n plus one conjecture but first we're going to build a

computer out of colored marbles and two pool cues