Keith Numbers - Numberphile

Key Takeaways

so the number 197 and what i'm going to do is i'm going to add the three digits so 1 plus 9 plus 7 is 17. now what i'm going to do is i'm going to form a number out of the last two digits of the original number and the the new number nine plus seven plus seventeen and that is thirty-three and now i'm going to do the same again so it's seven plus seventeen plus 33 and that gives me 57. you're beginning to get the idea 17 plus 33 plus 57 and that gives me 107. and now brady you've been very patient the keith number is about to emerge 33 plus 57 plus 107 and look what i get when i do that i get 197. that's quite interesting like i can't decide if these would be really really rare or really really common i intuitively it strikes me this would be rare they're very rare this is a number with three digits there are only 80 of order 80 numbers 80 keith numbers with less than of or 80 hang on i don't like this of order there are in fact 84 84 keith numbers with um less than uh 26 digits once you get a digit once you get more than three digits how do you start this process say you talk with two seven eight five for example then you would begin your process with two plus seven plus eight plus five which is equal to 10 17 22 and the next stage along the line is you take the the preceding three digits and add the fourth from the new number so it'd be seven plus eight plus 5 plus 22. there are for example two digit numbers okay so if i take d equals 2 then 14 is a keith number you go 1 plus 4 is 5. 4 plus 5 is nine five plus nine is fourteen mike keith who's a mathematician uh and discovered them in 1987. for me this is recreational mathematics it's it's how i got into mathematics uh my hero wasn't a physicist early on it was actually a guy called martin gardner i'd never heard of euler sadly earlier on when i was at school and gauss but martin gardner was this guy brilliant mathematician that worked i knew him as a as the guy working for scientific american and i subscribed to scientific american and i never understood a single article i read i kept trying to understand i'd re-read it and i'd just be it'd be a blur but the mathematics he had this mathematics puzzles at the back and then i i generally couldn't do them either but i was fascinated by them there were little numerology things and then i went and bought some of his books and just got into it just playing around with numbers and you just get to love them so the people who watch number file and get a bit sniffy about our recreational maths videos what do you say to them oh you shouldn't you should just enjoy it embrace it and just have fun just have fun playing around with them there's plenty of occasions when you're doodling thinking what shall i do and just play around with numbers and you'll be amazed some of the coincidences that can come up and uh and and out of some of these things come major breakthroughs we're not quite finished there's just a couple more things which uh um are around there there's one called the the reverse keith and this is where so again it's best just to show you an example of a reverse keith number it's where the reverse number appears in the sequence so if i begin say with 12 okay so we just follow the prescription that we've had so we go one plus two two plus three three plus five five plus eight eight plus thirteen and that's equal to twenty one and that's your reverse keith number you've reversed the order of the uh the integers in in in the original there's even fewer of what you could call keith clusters keith has got a conjecture about these keith numbers his numbers he's allowed to have a conjecture about his own numbers of course and they're to do with the clustering of these numbers and what he's done is he's said how many ways can you cluster the keith numbers such that in any cluster which could have of course the cluster has to have more than one number so it could have two numbers three numbers four numbers each number in that cluster is an integer multiple of the lowest number this conjecture is the following here there are three that are known just three and he says that's he believes that's all there are so i think it's quite an interesting challenge for some of the number four people to go and maybe find some more so um so here's here's an example of a keith cluster so the number 14 and 28 are both keith numbers and of course 28 is twice 14. and another keith cluster is the number 1004 well one thousand one hundred and four and then if i double that two thousand two hundred and eight we've also got to have the same number of digits in the sequence there is remarkably one cluster of three keith numbers which which is three one three three one of course six two six six two nine three nine nine three i think that's amazing each of those are keith numbers they all satisfy the properties that we've been describing and they form this cluster so the conjecture is that's it i used to i used to do these puzzles on the bus going to school i had to get two buses and uh i'd just try and do well whatever i could quite usually i got stuck especially with the geometrical ones he has geometrical puzzles as well as sort of basic algebraic ones and then he has all puzzles on paradoxes you know the how can you get uh infinite number of people in a hotel with an infinite number of rooms and then double the number of people and things it was great