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