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• 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