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  • What's this, Brady?

  • It's like a graph, or joined up circles?

  • No, this is an ant. You can see the antenna up there, and then this is the body of the ant.

  • Oh yeah, OK.

  • We're going to place odd consecutive integers starting with one into this ant.

  • So, what are those odd consecutive integers starting with one?

  • Well, we need one, three, five, seven, and nine.

  • We're going to be placing them in here.

  • So, what number do you want here?

  • Seven.

  • Seven, and here?

  • Three.

  • And here?

  • One.

  • Let's put nine.

  • And?

  • Five.

  • OK, so this is an example of a failure.

  • We're going to look at five minus one; that is four.

  • 4 9 minus 1 that's 8 3 minus 1 that's 2

  • everything looks different so far looking good

  • Sadly 7 minus 3 is also

  • 4 so we failed this is an example of a failure your job, is to try to make all of those differences

  • different if I go seven three one nine five

  • then we have 4 & 8 &

  • 6 &

  • 4. Oh wait no this is still a disaster I wonder if this is even possible to solve this at your you're okay to fail

  • but I I'm supposed to be succeeding here okay um

  • let's go 9 1

  • No, maybe I'm gonna put the 7 up

  • Here and the 5 here and a 3 here does this work

  • Yeah yeah yeah yeah this looks good so this is nine minus one that's eight seven minus one

  • six

  • four and two

  • We're all different

  • we are successful we get to ride the end

  • So this is part of an unsolved problem from

  • 1967 called the graceful tree conjecture but it belongs in every elementary school's

  • curriculum, whenever they're learning subtraction so the general problem is given any number of connected circles so they have to be

  • connected any

  • number that you want they have to be connected and there can be no loops so this would be a fine

  • Insectoid to solve let me show you a loop so you could ask can this be solved so this might be solvable it might not

  • But it's certainly not the graceful tree conjecture the graceful tree conjecture means that there's no loops, okay?

  • okay, but why don't we try to solve this so now we need 1 3 5 7 &

  • 9 in 2 & 2 here can this be solved

  • Consider how many even numbers you can get what's the biggest even number that you can get as a difference between

  • these right 8 yeah 9 and 1 the biggest you can get is 8

  • what's the second biggest number that you can get 6 yes and

  • The third biggest number 4 and then the last one is 2 there are only

  • 4 different numbers that you can have but how many lines do we have to satisfy we have got

  • 5 lines we've only got 4 even numbers to distribute between these five lines

  • So no matter how you distribute

  • these odd numbers

  • You are always going to end up with a duplicate so here we have 8

  • 6

  • 4 2

  • & 4 again

  • We've ended up with a duplicate over here so this starfish

  • Cannot be solved it will fail every time if we have n circles and n lines connecting them

  • We will never be able to solve that we can potentially solve for M

  • circles in this case 5 circles and n minus 1

  • Maximum of four lines, we?

  • Sometimes can solve for that so five circles where you you could solve it so here's a butterfly

  • There are five lines and one less connector so this could be solved what about this

  • Five circles and four connectors I don't know if this can be solved but this might be able to be solved the graceful tree

  • Conjecture doesn't include loops but this is still an

  • Interesting question I I don't know the solution to this so the graceful tree conjecture is one where you have all of your circles connected

  • So you don't have an outlier here and there's no loops

  • So that is guaranteed if you've got n circles you're going to have n minus 1 connectors

  • so here we've got 1 2 3 4 5 6 7 circles 1 2 3 4 5 6

  • lines and this is what is unsolved whether 4 n

  • circles and n minus 1 lines

  • Where there are no loops

  • Can you always put in there odd?

  • consecutive integers

  • such that the difference between

  • all connected circles all of those differences

  • Are different what's it's not proven for that surely you mean for old designs of three

  • oh

  • yeah for all

  • insectoids or all trees that you would care to design so this one for seven has been solved this is an example of

  • Seven circles and six connecting lines there are actually 11 different ways

  • that you can do that so here we have the 11 different ways to have seven circles all

  • connected no loops

  • What is unsolved is?

  • going for more circles if you go into

  • 30 or 40 circles

  • Are these all solvable this has

  • been an open conjecture since 1967 so presumably God if I start having more circles like 30 circles of 40 per calls the number of

  • insectoids possible must just get metal it explodes

  • Exactly right up to what number of circles is it solved for there are

  • individual cases that have been solved for example I can put Circle in the middle and I can add any number of

  • Circles on the outside that you want and this is always

  • solvable and I can prove that quite easily I can put a 1 in the middle then I can go 3 5 7 9 11

  • 13

  • 15 17 19

  • 21 23

  • And you can imagine I can keep on going here and this is always solvable because these differences are always different I could also put

  • The largest number in the middle 23 could go in the middle and then I could distribute

  • The other numbers all around all we solvable another example of a species that is always solvable are

  • Snakes so you could imagine a snake of any length that you want

  • and these are always solvable you might want to try this you can start with one and then you can go to your

  • largest one I'm actually going to just skip I'm gonna go 1 3

  • 5 7 and then I'm gonna come back 9

  • 11 13

  • here, we have a difference of

  • 12 a difference of 10 a difference of 8 a

  • difference of 6 a difference of 4 and a difference of 2

  • All snakes can be solvable just by replicating that pattern there's lots of species out there

  • Not everything is a snake not everything is a sea star

  • There's lots of species out there you could very easily create and explore new species for example what happens if we have

  • Something crazy that you know to sea stars joined with the long dendritic chain

  • So is that solvable you can imagine that these can get very very complex very very quickly and

  • Most species by the time you get to 40 circles most of those insectoids

  • Very difficult to solve so presumably good if this is still an unanswered question

  • That means there's yet to be a species of tree found that is definitely

  • Unsolvable because that would that would ruin it that's correct

  • We have not found an example that doesn't work but that doesn't mean that it doesn't exist

  • there are plenty of mathematicians who believe that that tree

  • Exists that that there is a tree out there that cannot be solved would you be happy or sad if one was found oh

  • I would be thrilled if one was find because I do this with my elementary school kids and

  • after a week of them struggling to find one that can't be solved I would love to say and

  • Here, we have

  • but it feels like it

  • feels like nature is telling us they can always be solved like it feels like a

  • Beautiful thing they're also about it feels like it would break something beautiful

  • Well it's already broken like if you look beyond the graceful tree conjecture to allowing loops so here

  • we have seven circles

  • Six connectors so these ones cannot be solved so in a way it's already broken but it's interesting which

  • Species work which species fail

  • I like that I sort of see them as

  • abominations and therefore they should

  • It's fair enough that they can't be solved but there's something perfect about the tree that has no loops and no islands

  • Well that that's where the crux is well where does a dividing line between

  • Species that are always solvable and species that aren't that's what becomes interesting

  • Are, you right are all of the trees really beautiful, well we'll have to wait and find out

What's this, Brady?

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A2 初級

優雅な木の問題 - Numberphile (The Graceful Tree Problem - Numberphile)

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    林宜悉 に公開 2021 年 01 月 14 日
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