MathILy and MathILy-Er Math Jam: Views of the N-Cube
Go back to the Math Jam ArchiveAoPS Instructor and MathILy Director dr. sarah-marie belcastro leads students in exploration of the N-cube and answers questions about {MathILy, MathILy-Er}.
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Facilitator: sarah-marie belcastro
5space
2019-03-18 19:31:24
In this math jam, AoPS Instructor and MathILy Director dr. sarah-marie belcastro (smbelcas) will lead participants in an exploration of the N-cube from geometric and combinatorial viewpoints. Time will be reserved for a discussion of {MathILy, MathILy-Er} along with any questions you may have about the programs and application process.
In this math jam, AoPS Instructor and MathILy Director dr. sarah-marie belcastro (smbelcas) will lead participants in an exploration of the N-cube from geometric and combinatorial viewpoints. Time will be reserved for a discussion of {MathILy, MathILy-Er} along with any questions you may have about the programs and application process.
5space
2019-03-18 19:31:37
For now, please hold your questions -- we'll let you know when you can start asking questions. Also, due to the number of people attending tonight, we may not be able to get to every question.
For now, please hold your questions -- we'll let you know when you can start asking questions. Also, due to the number of people attending tonight, we may not be able to get to every question.
5space
2019-03-18 19:31:54
sarah-marie belcastro (smbelcas) earned her Ph.D. in mathematics from the University of Michigan back in 1997 and did her undergraduate work in mathematics and astronomy at Haverford College well before that. She currently directs the summer program MathILy, and has taught a huge variety of mathematics courses---standard and non-standard, undergraduate and graduate in level---to college students and to high-ability high-school students, at institutions including Smith College, Bowdoin College, Sarah Lawrence College, and the Hampshire College Summer Studies in Mathematics. sarah-marie's favorite research is in topological graph theory. Among her many non-pure-mathematics interests are the mathematics of knitting, pharmacokinetics, dance (principally ballet and modern), and changing the world. You may find tons of information (about her, and about other things) at her website http://www.toroidalsnark.net.
sarah-marie belcastro (smbelcas) earned her Ph.D. in mathematics from the University of Michigan back in 1997 and did her undergraduate work in mathematics and astronomy at Haverford College well before that. She currently directs the summer program MathILy, and has taught a huge variety of mathematics courses---standard and non-standard, undergraduate and graduate in level---to college students and to high-ability high-school students, at institutions including Smith College, Bowdoin College, Sarah Lawrence College, and the Hampshire College Summer Studies in Mathematics. sarah-marie's favorite research is in topological graph theory. Among her many non-pure-mathematics interests are the mathematics of knitting, pharmacokinetics, dance (principally ballet and modern), and changing the world. You may find tons of information (about her, and about other things) at her website http://www.toroidalsnark.net.
smbelcas
2019-03-18 19:32:12
Hi, everybody!
Hi, everybody!
smbelcas
2019-03-18 19:32:18
I see there are some students from my past AoPS classes here
.
I see there are some students from my past AoPS classes here
.
smbelcas
2019-03-18 19:32:26
Welcome to this combined-topic Math Jam! We are going to do some MAAAAAAAATH, and then I'll tell you a tiny bit about {MathILy, MathILy-Er}, and then I will answer questions about {MathILy, MathILy-Er}.
Welcome to this combined-topic Math Jam! We are going to do some MAAAAAAAATH, and then I'll tell you a tiny bit about {MathILy, MathILy-Er}, and then I will answer questions about {MathILy, MathILy-Er}.
fibonacci11101
2019-03-18 19:32:35
yay
yay
NoDealsHere
2019-03-18 19:32:35
hey
hey
SwimWithDolphin
2019-03-18 19:32:35
H0i
H0i
evankeri
2019-03-18 19:32:35
hi
hi
5space
2019-03-18 19:32:38
Okay, I'll now hand the room off to your discussion leader for today, sarah-marie!
Okay, I'll now hand the room off to your discussion leader for today, sarah-marie!
smbelcas
2019-03-18 19:32:45
You might like to have some scratch paper handy.
You might like to have some scratch paper handy.
smbelcas
2019-03-18 19:32:54
Those of you who have had me for class know that after I ask a question, I usually won't say anything until you have collectively responded. So don't be surprised if things look quiet for a few moments---I'm trying to give you a little bit of thinking/typing space.
Those of you who have had me for class know that after I ask a question, I usually won't say anything until you have collectively responded. So don't be surprised if things look quiet for a few moments---I'm trying to give you a little bit of thinking/typing space.
smbelcas
2019-03-18 19:33:05
Additionally, please explain your reasoning when you respond. And pay some attention to the responses that get passed into the classroom---they are chosen to help everyone's thinking.
Additionally, please explain your reasoning when you respond. And pay some attention to the responses that get passed into the classroom---they are chosen to help everyone's thinking.
smbelcas
2019-03-18 19:33:16
Finally, the math we do this evening is a tiny sample of what is done at {MathILy, MathILy-Er}. We do this same material faster and we take it much further; today we're just scratching the surface. (If you think it's slow/easy at the start, be patient---it will ramp up!) And of course because {MathILy, MathILy-Er} are face-to-face, class is all much louder and laughier and more student-run.
Finally, the math we do this evening is a tiny sample of what is done at {MathILy, MathILy-Er}. We do this same material faster and we take it much further; today we're just scratching the surface. (If you think it's slow/easy at the start, be patient---it will ramp up!) And of course because {MathILy, MathILy-Er} are face-to-face, class is all much louder and laughier and more student-run.
smbelcas
2019-03-18 19:33:29
Let's start with a picture:
Let's start with a picture:
smbelcas
2019-03-18 19:33:32
smbelcas
2019-03-18 19:33:39
What do you think comes next in this sequence? (Just the one thing that comes next...)
What do you think comes next in this sequence? (Just the one thing that comes next...)
Redragon
2019-03-18 19:34:10
cube
cube
cad314
2019-03-18 19:34:10
cube
cube
jechi7
2019-03-18 19:34:10
cube
cube
Rishi763
2019-03-18 19:34:10
a cube
a cube
EXL
2019-03-18 19:34:10
cube
cube
Mathpluspianoequalslife
2019-03-18 19:34:10
lol I mean just a cube
lol I mean just a cube
sansae
2019-03-18 19:34:10
Cube!
Cube!
Bananaman27
2019-03-18 19:34:10
a cube?
a cube?
PugLord
2019-03-18 19:34:10
cube
cube
bkim0325
2019-03-18 19:34:10
a cube
a cube
Purplegrape
2019-03-18 19:34:10
cube
cube
smbelcas
2019-03-18 19:34:13
Right, it should be a cube.
Right, it should be a cube.
smbelcas
2019-03-18 19:34:19
smbelcas
2019-03-18 19:34:23
And what do you think comes after that?
And what do you think comes after that?
gozomete
2019-03-18 19:35:15
4 dimension cube
4 dimension cube
evankeri
2019-03-18 19:35:15
4d cube? idk
4d cube? idk
Rishi763
2019-03-18 19:35:15
a 4-d cube?
a 4-d cube?
bever209
2019-03-18 19:35:15
tesseract
tesseract
bkim0325
2019-03-18 19:35:15
a 4-dimensional cube thing
a 4-dimensional cube thing
sansae
2019-03-18 19:35:15
Four dimensional cube?!
Four dimensional cube?!
SwimWithDolphin
2019-03-18 19:35:15
tessaract
tessaract
cad314
2019-03-18 19:35:15
4d "cube"
4d "cube"
monkeycalculator
2019-03-18 19:35:15
Tesseract
Tesseract
claserken
2019-03-18 19:35:15
hypercube
hypercube
haha0201
2019-03-18 19:35:15
tesseract
tesseract
AwesomeDude86
2019-03-18 19:35:15
tesseract
tesseract
Damalone
2019-03-18 19:35:15
4 dimensional cube
4 dimensional cube
gozomete
2019-03-18 19:35:15
4d cube comes next
4d cube comes next
smbelcas
2019-03-18 19:35:18
Now we run into a problem. Who knows what a tesseract or a hypercube mean? These terms aren't defined. Let's back up a little bit.
Now we run into a problem. Who knows what a tesseract or a hypercube mean? These terms aren't defined. Let's back up a little bit.
smbelcas
2019-03-18 19:35:25
We understand at a glance that there is a progression here. What exactly is progressing as we go further in the sequence---what's changing?
We understand at a glance that there is a progression here. What exactly is progressing as we go further in the sequence---what's changing?
Hermain
2019-03-18 19:36:03
the dimension
the dimension
villagevinegar
2019-03-18 19:36:03
the number of dimensions
the number of dimensions
bkim0325
2019-03-18 19:36:03
one dimension is added every time
one dimension is added every time
ad46578
2019-03-18 19:36:03
the dimensions?
the dimensions?
rubixsolver
2019-03-18 19:36:03
number of dimensions
number of dimensions
PugLord
2019-03-18 19:36:03
the dimensions
the dimensions
AwesomeDude86
2019-03-18 19:36:03
the number of dimensions
the number of dimensions
spoamath321
2019-03-18 19:36:03
dimensions
dimensions
EXL
2019-03-18 19:36:03
num of dimensions
num of dimensions
Mathlete99s
2019-03-18 19:36:07
the dimension it seems
the dimension it seems
smbelcas
2019-03-18 19:36:10
Indeed, it's the dimension. What are the dimensions of each of those first four objects in the sequence?
Indeed, it's the dimension. What are the dimensions of each of those first four objects in the sequence?
MrMXS
2019-03-18 19:36:43
$0,1,2,3$
$0,1,2,3$
matharcher
2019-03-18 19:36:43
0,1,2,3
0,1,2,3
imatrashloser
2019-03-18 19:36:43
0, 1 , 2, 3
0, 1 , 2, 3
avisioner
2019-03-18 19:36:43
0 1 2 3
0 1 2 3
cad314
2019-03-18 19:36:43
0, 1, 2, 3
0, 1, 2, 3
monkeycalculator
2019-03-18 19:36:43
0, 1, 2, 3 respectively
0, 1, 2, 3 respectively
StickyWashington
2019-03-18 19:36:43
0, 1, 2, and 3
0, 1, 2, and 3
Damalone
2019-03-18 19:36:43
0,1,2,3
0,1,2,3
StanDaMan
2019-03-18 19:36:43
0, 1, 2, 3
0, 1, 2, 3
rubixsolver
2019-03-18 19:36:43
0, 1, 2, 3
0, 1, 2, 3
AwesomeDude86
2019-03-18 19:36:43
0, 1, 2, and 3
0, 1, 2, and 3
Mathpluspianoequalslife
2019-03-18 19:36:43
0,1,2,3
0,1,2,3
gozomete
2019-03-18 19:36:43
0, 1, 2, 3, 4...
0, 1, 2, 3, 4...
PugLord
2019-03-18 19:36:43
0d, 1d, 2d, 3d
0d, 1d, 2d, 3d
EXL
2019-03-18 19:36:43
0,1,2,3
0,1,2,3
smbelcas
2019-03-18 19:36:45
Without getting into the details of the definition of "dimension" (which turns out to need some fairly advanced mathematics to describe precisely), we will agree that a point has $0$ dimensions, so these four objects have dimension $0, 1, 2, 3$.
Without getting into the details of the definition of "dimension" (which turns out to need some fairly advanced mathematics to describe precisely), we will agree that a point has $0$ dimensions, so these four objects have dimension $0, 1, 2, 3$.
smbelcas
2019-03-18 19:36:56
Now, what can we say about the next object in the sequence?
Now, what can we say about the next object in the sequence?
ad46578
2019-03-18 19:37:35
4d
4d
bkim0325
2019-03-18 19:37:35
it's 4 dimensions
it's 4 dimensions
smartninja2000
2019-03-18 19:37:35
it has 4 dimensions?!
it has 4 dimensions?!
fibonacci11101
2019-03-18 19:37:35
4d
4d
kshah21
2019-03-18 19:37:35
4d
4d
MrMXS
2019-03-18 19:37:35
$4$-dimensional
$4$-dimensional
rubixsolver
2019-03-18 19:37:35
it has four dimensions!
it has four dimensions!
silverpiano
2019-03-18 19:37:35
It would be 4D?
It would be 4D?
evankeri
2019-03-18 19:37:35
it is 4d, or four-dimensional
it is 4d, or four-dimensional
sansae
2019-03-18 19:37:35
Its has dimension 4
Its has dimension 4
cad314
2019-03-18 19:37:35
it is 4d
it is 4d
avisioner
2019-03-18 19:37:35
it is 4d
it is 4d
phanithans1
2019-03-18 19:37:35
It will have 4-dimensions
It will have 4-dimensions
Damalone
2019-03-18 19:37:35
4 dimensions
4 dimensions
StickyWashington
2019-03-18 19:37:35
It will apparently have 4 dimensions according to the sequence
It will apparently have 4 dimensions according to the sequence
villagevinegar
2019-03-18 19:37:35
it should have 4 dimensions and equal length along each dimension
it should have 4 dimensions and equal length along each dimension
smbelcas
2019-03-18 19:37:37
Right, it should be $4$-dimensional. This sets us up for thinking more deeply:
Right, it should be $4$-dimensional. This sets us up for thinking more deeply:
smbelcas
2019-03-18 19:37:38
How do we form this sequence? That is, how do we go from one object in the sequence to the next one?
How do we form this sequence? That is, how do we go from one object in the sequence to the next one?
smartninja2000
2019-03-18 19:38:32
Increase the number of dimensions by 1
Increase the number of dimensions by 1
gozomete
2019-03-18 19:38:32
add 1d
add 1d
MathGenius_
2019-03-18 19:38:32
add a dimension
add a dimension
kshah21
2019-03-18 19:38:32
We keep on going up and up in dimension
We keep on going up and up in dimension
del-math
2019-03-18 19:38:32
How do we add a 4th dimension?
How do we add a 4th dimension?
smbelcas
2019-03-18 19:38:34
Somehow we're going up a dimension. But how, precisely, do we do that?
Somehow we're going up a dimension. But how, precisely, do we do that?
imatrashloser
2019-03-18 19:39:10
here's the thing: points determine lines, lines determine squares, squares determine cubes, cubes determine tesseracts
here's the thing: points determine lines, lines determine squares, squares determine cubes, cubes determine tesseracts
fishy15
2019-03-18 19:39:10
take 2 copies of the object and connect the corresponding vertices
take 2 copies of the object and connect the corresponding vertices
SwimWithDolphin
2019-03-18 19:39:10
You pull the last one out into a new, orthogonal dimension
You pull the last one out into a new, orthogonal dimension
monkeycalculator
2019-03-18 19:39:10
Duplicate the object. Connect pairwise corresponding points
Duplicate the object. Connect pairwise corresponding points
cad314
2019-03-18 19:39:10
extending lines from points and planes from lines
extending lines from points and planes from lines
villagevinegar
2019-03-18 19:39:10
we add one more copy in the same dimension, then connect the two copies
we add one more copy in the same dimension, then connect the two copies
smbelcas
2019-03-18 19:39:14
These are reasonable general descriptions. Let's look at the process one step at a time. What do we do to go from a point to a line segment?
These are reasonable general descriptions. Let's look at the process one step at a time. What do we do to go from a point to a line segment?
ad46578
2019-03-18 19:40:10
add a point and draw a line connecting the points
add a point and draw a line connecting the points
DOGSTREET1
2019-03-18 19:40:10
we draw another point and connect the points
we draw another point and connect the points
MrMXS
2019-03-18 19:40:10
copy the point then connect the two
copy the point then connect the two
villagevinegar
2019-03-18 19:40:10
we add one more point, then connect the two points
we add one more point, then connect the two points
Damalone
2019-03-18 19:40:10
we double the number of vertices and connect them with a line
we double the number of vertices and connect them with a line
phanithans1
2019-03-18 19:40:10
add another point and connect it
add another point and connect it
ChickenAgent2227-_-
2019-03-18 19:40:10
sweep it across the next dimension?
sweep it across the next dimension?
bkim0325
2019-03-18 19:40:10
add a point and connect the original and the new one
add a point and connect the original and the new one
StickyWashington
2019-03-18 19:40:10
Add a new line at a right angle and duplicate the object along it
Add a new line at a right angle and duplicate the object along it
smbelcas
2019-03-18 19:40:12
We moosh the point one unit over, and keep the trail it leaves.
We moosh the point one unit over, and keep the trail it leaves.
smbelcas
2019-03-18 19:40:20
(Points to ChickenAgent for the best answer.)
(Points to ChickenAgent for the best answer.)
smbelcas
2019-03-18 19:40:23
smbelcas
2019-03-18 19:40:29
What do we do to go from a line segment to a square?
What do we do to go from a line segment to a square?
mgrimalo
2019-03-18 19:41:18
sweep the whole segment
sweep the whole segment
fishy15
2019-03-18 19:41:18
same process but upwards for the entire line
same process but upwards for the entire line
cad314
2019-03-18 19:41:18
move the line up one unit and keep the trail
move the line up one unit and keep the trail
Damalone
2019-03-18 19:41:18
sweep the line up through the second dimension
sweep the line up through the second dimension
ad46578
2019-03-18 19:41:18
scoot the line segment across to another dimension and keep its trail
scoot the line segment across to another dimension and keep its trail
ChickenAgent2227-_-
2019-03-18 19:41:18
moosh the line segment over a bit
moosh the line segment over a bit
jumpmonkey
2019-03-18 19:41:18
stratch the line
stratch the line
villagevinegar
2019-03-18 19:41:18
we sweep the line segment over by one unit
we sweep the line segment over by one unit
smbelcas
2019-03-18 19:41:20
smbelcas
2019-03-18 19:41:24
We moosh the segment one unit over, and keep the trail it leaves. But wait---what direction do we moosh in? Does it matter?
We moosh the segment one unit over, and keep the trail it leaves. But wait---what direction do we moosh in? Does it matter?
Mathpluspianoequalslife
2019-03-18 19:42:13
perpendicular
perpendicular
Captain_Crush
2019-03-18 19:42:13
the direction does matter
the direction does matter
monkeycalculator
2019-03-18 19:42:13
in any orthogonal direction
in any orthogonal direction
StickyWashington
2019-03-18 19:42:13
At a right angle to the other dimension...s...
At a right angle to the other dimension...s...
Allen31415
2019-03-18 19:42:13
it is perpendicular to all other directions.
it is perpendicular to all other directions.
lcalvert99
2019-03-18 19:42:13
Perpendicular
Perpendicular
fishy15
2019-03-18 19:42:13
as long as its a new dimension perpendicular to the line
as long as its a new dimension perpendicular to the line
JCN360
2019-03-18 19:42:13
it is moved perpendicularly to the line segment.
it is moved perpendicularly to the line segment.
smbelcas
2019-03-18 19:42:15
Yes, it matters a lot. We have to moosh perpendicular to the line segment, or else we'll get a parallelogram instead of a square.
Yes, it matters a lot. We have to moosh perpendicular to the line segment, or else we'll get a parallelogram instead of a square.
smbelcas
2019-03-18 19:42:32
(Or just a line if we choose really badly.)
(Or just a line if we choose really badly.)
smbelcas
2019-03-18 19:42:36
(If you're worried about what moosh really means, it is a type of Cartesian product. Of course, for that to make sense, you have to already know what a Cartesian product is...)
(If you're worried about what moosh really means, it is a type of Cartesian product. Of course, for that to make sense, you have to already know what a Cartesian product is...)
smbelcas
2019-03-18 19:42:48
Okay. What about going from a square to a cube?
Okay. What about going from a square to a cube?
Mathpluspianoequalslife
2019-03-18 19:43:42
moosh it perpendicular to the square (upwards)
moosh it perpendicular to the square (upwards)
ad46578
2019-03-18 19:43:42
moosh the square into a perpendicular dimension and keep the trail it leaves
moosh the square into a perpendicular dimension and keep the trail it leaves
EXL
2019-03-18 19:43:42
moosh perpendicularly
moosh perpendicularly
cad314
2019-03-18 19:43:42
extend the square out and keep the trail
extend the square out and keep the trail
MathGenius_
2019-03-18 19:43:42
push the entire square over forward and keep the trail
push the entire square over forward and keep the trail
villagevinegar
2019-03-18 19:43:42
moosh the square by one unit
moosh the square by one unit
StickyWashington
2019-03-18 19:43:42
Moosh the square 'backwards'
Moosh the square 'backwards'
ChickenAgent2227-_-
2019-03-18 19:43:42
moosh it along the z-axis
moosh it along the z-axis
mgrimalo
2019-03-18 19:43:42
sweep the square in a perpendicular manner to get a 3d object
sweep the square in a perpendicular manner to get a 3d object
kshah21
2019-03-18 19:43:42
push the square in a direction perpendicular to the square and keep the path
push the square in a direction perpendicular to the square and keep the path
smbelcas
2019-03-18 19:43:44
Right, we moosh the square one unit in a direction perpendicular to the square, and keep the trail it leaves.
Right, we moosh the square one unit in a direction perpendicular to the square, and keep the trail it leaves.
smbelcas
2019-03-18 19:43:47
smbelcas
2019-03-18 19:43:58
And what comes next?
And what comes next?
EXL
2019-03-18 19:44:58
moosh again
moosh again
avisioner
2019-03-18 19:44:58
moosh outwards
moosh outwards
ad46578
2019-03-18 19:44:58
moosh the cube one unit perpendicular to the cube and keep the trail it leaves
moosh the cube one unit perpendicular to the cube and keep the trail it leaves
MrMXS
2019-03-18 19:44:58
move it in another direction perpendicular to those three
move it in another direction perpendicular to those three
cad314
2019-03-18 19:44:58
moosh the cube one unit in a perpendicular direction and keep the trail
moosh the cube one unit in a perpendicular direction and keep the trail
Allen31415
2019-03-18 19:44:58
moosh the cube into a direction perpendicular to it?
moosh the cube into a direction perpendicular to it?
JCN360
2019-03-18 19:44:58
Just moosh the cube into the fourth dimension (which we cannot picture).
Just moosh the cube into the fourth dimension (which we cannot picture).
StickyWashington
2019-03-18 19:44:58
...We moosh it at a right angle to a cube 'towards the fourth dimension' 0.O
...We moosh it at a right angle to a cube 'towards the fourth dimension' 0.O
rubixsolver
2019-03-18 19:44:58
we moosh the cube one unit perpendicular to the cube and keep the trail
we moosh the cube one unit perpendicular to the cube and keep the trail
StanDaMan
2019-03-18 19:44:58
move it perpendicular again??? Which way even is perpendicular?
move it perpendicular again??? Which way even is perpendicular?
smbelcas
2019-03-18 19:45:00
Exactly. We moosh the cube one unit in a direction perpendicular to the cube, and keep the trail it leaves.
Exactly. We moosh the cube one unit in a direction perpendicular to the cube, and keep the trail it leaves.
smbelcas
2019-03-18 19:45:01
smbelcas
2019-03-18 19:45:19
We have agreed that the resulting object is $4$-dimensional, and it's certainly cube-like, so let's call it a 4-cube.
We have agreed that the resulting object is $4$-dimensional, and it's certainly cube-like, so let's call it a 4-cube.
smbelcas
2019-03-18 19:45:36
Here are all the cubes we've made so far, for reference:
Here are all the cubes we've made so far, for reference:
smbelcas
2019-03-18 19:45:38
smbelcas
2019-03-18 19:46:04
How do we make a $5$-cube? Can you draw one (on paper, not on screen---that would take too long)?
How do we make a $5$-cube? Can you draw one (on paper, not on screen---that would take too long)?
StickyWashington
2019-03-18 19:47:18
You would moosh the 4-cube
You would moosh the 4-cube
mandrake41
2019-03-18 19:47:18
moosh it perpendicular again
moosh it perpendicular again
Allen31415
2019-03-18 19:47:18
moosh the 4-cube perpendicular to it.
moosh the 4-cube perpendicular to it.
kshah21
2019-03-18 19:47:18
We push it to the right and leave the trail it left behind.
We push it to the right and leave the trail it left behind.
cad314
2019-03-18 19:47:18
moosh the 4-cube in a perpendicular direction and keep the trail
moosh the 4-cube in a perpendicular direction and keep the trail
shrungpatel
2019-03-18 19:47:18
We moosh the 4-cube one unit in a direction perpendicular to the 4-cube, and keep the trail it leaves.
We moosh the 4-cube one unit in a direction perpendicular to the 4-cube, and keep the trail it leaves.
ad46578
2019-03-18 19:47:18
moosh the 4-cube one unit perpendicular to the 4-cube and keep the trail it leaves. drawing it is perhaps possible but not pretty
moosh the 4-cube one unit perpendicular to the 4-cube and keep the trail it leaves. drawing it is perhaps possible but not pretty
Mathlete99s
2019-03-18 19:47:18
Move the 4 cubes perpendicular with respect to the 5th dimension i think....
Move the 4 cubes perpendicular with respect to the 5th dimension i think....
smbelcas
2019-03-18 19:47:22
Just as we have done for earlier dimensions, we moosh the $4$-cube one unit in a direction perpendicular to the $4$-cube, and keep the trail it leaves.
Just as we have done for earlier dimensions, we moosh the $4$-cube one unit in a direction perpendicular to the $4$-cube, and keep the trail it leaves.
smbelcas
2019-03-18 19:47:29
How do we make an $n$-cube?
How do we make an $n$-cube?
ChickenAgent2227-_-
2019-03-18 19:48:32
moosh an $n-1$-cube
moosh an $n-1$-cube
SwimWithDolphin
2019-03-18 19:48:32
You moosh the $n-1$cube
You moosh the $n-1$cube
shrungpatel
2019-03-18 19:48:32
We moosh the (n-1)cube one unit in a direction perpendicular to the (n-1)cube, and keep the trail it leaves.
We moosh the (n-1)cube one unit in a direction perpendicular to the (n-1)cube, and keep the trail it leaves.
ad46578
2019-03-18 19:48:32
moosh the (n-1)-cube one unit perpendicular to the (n-1) cube and keep the trail
moosh the (n-1)-cube one unit perpendicular to the (n-1) cube and keep the trail
MrMXS
2019-03-18 19:48:32
take an $(n-1)$-cube and move it in a new perpendicular direction
take an $(n-1)$-cube and move it in a new perpendicular direction
Rishi763
2019-03-18 19:48:32
we moosh an n-1 cube
we moosh an n-1 cube
evankeri
2019-03-18 19:48:32
mush an n-1 cube into the nth dimension, keep the trail
mush an n-1 cube into the nth dimension, keep the trail
Allen31415
2019-03-18 19:48:32
moosh an $(n-1)$ cube perpendicular to it.
moosh an $(n-1)$ cube perpendicular to it.
cad314
2019-03-18 19:48:32
moosh the (n-1)-cube in a perpendicular direction and keep the trail
moosh the (n-1)-cube in a perpendicular direction and keep the trail
smbelcas
2019-03-18 19:48:33
Yup. We moosh a $(n-1)$-cube one unit in a direction perpendicular to the $(n-1)$-cube, and keep the trail it leaves.
Yup. We moosh a $(n-1)$-cube one unit in a direction perpendicular to the $(n-1)$-cube, and keep the trail it leaves.
EXL
2019-03-18 19:48:57
moosh a point n times
moosh a point n times
gozomete
2019-03-18 19:48:57
moosh it one unit perpendiculary and keep the trail
moosh it one unit perpendiculary and keep the trail
avisioner
2019-03-18 19:48:57
moosh one unit in a perpendicular direction n times
moosh one unit in a perpendicular direction n times
smbelcas
2019-03-18 19:48:58
Another way to do this is to start with a point and moosh $n$ times.
Another way to do this is to start with a point and moosh $n$ times.
smbelcas
2019-03-18 19:49:01
Some of you have wondered where these extra perpendicular directions are.
Some of you have wondered where these extra perpendicular directions are.
smbelcas
2019-03-18 19:49:08
One answer: Usually when you graph things, you use the $x$ axis and the $y$ axis, and for $3$-dimensional things, the $z$ axis. Here, we also use the $w$ axis, and for the $5$-cube, the $v$ axis.
One answer: Usually when you graph things, you use the $x$ axis and the $y$ axis, and for $3$-dimensional things, the $z$ axis. Here, we also use the $w$ axis, and for the $5$-cube, the $v$ axis.
smbelcas
2019-03-18 19:49:19
The $w$ axis is perpendicular to the $x, y$, and $z$ axes. The $v$ axis is perpendicular to the $x, y, z$, and $w$ axes.
The $w$ axis is perpendicular to the $x, y$, and $z$ axes. The $v$ axis is perpendicular to the $x, y, z$, and $w$ axes.
smbelcas
2019-03-18 19:49:27
Just like you can't draw the $z$ axis as perpendicular to the $x$ and $y$ axes on paper, but you can still understand where it goes in $3$ dimensions...
Just like you can't draw the $z$ axis as perpendicular to the $x$ and $y$ axes on paper, but you can still understand where it goes in $3$ dimensions...
smbelcas
2019-03-18 19:49:43
...you can't place the $w$ axis as perpendicular to the $x, y$, and $z$ axes in regular space, but you can still understand where it goes in $4$ dimensions. There isn't room for the $w$ axis in regular space, but there is enough room in your head. With practice, you can visualize it pretty clearly.
...you can't place the $w$ axis as perpendicular to the $x, y$, and $z$ axes in regular space, but you can still understand where it goes in $4$ dimensions. There isn't room for the $w$ axis in regular space, but there is enough room in your head. With practice, you can visualize it pretty clearly.
smbelcas
2019-03-18 19:50:10
Anyway: $5$-cube!
Anyway: $5$-cube!
smbelcas
2019-03-18 19:50:11
imatrashloser
2019-03-18 19:50:31
my eyes
my eyes
cad314
2019-03-18 19:50:31
Wow!
Wow!
SwimWithDolphin
2019-03-18 19:50:31
Holy guacamole
Holy guacamole
Nami24
2019-03-18 19:50:31
WOW
WOW
gozomete
2019-03-18 19:50:31
WHOA!!
WHOA!!
avisioner
2019-03-18 19:50:31
what?
what?
mandrake41
2019-03-18 19:50:40
wat
wat
Purplegrape
2019-03-18 19:50:40
coool
coool
smbelcas
2019-03-18 19:50:41
$6$-cube!
$6$-cube!
smbelcas
2019-03-18 19:50:43
DOGSTREET1
2019-03-18 19:51:05
too many lines!!! brain overload
too many lines!!! brain overload
ChickenAgent2227-_-
2019-03-18 19:51:05
help me please
help me please
Mathlete99s
2019-03-18 19:51:05
WOW
WOW
Mathlete161
2019-03-18 19:51:05
Holy goly godapppers
Holy goly godapppers
PracticingMath
2019-03-18 19:51:05
Ouch my brain
Ouch my brain
Nami24
2019-03-18 19:51:05
wut is life
wut is life
MathGenius_
2019-03-18 19:51:05
omg
omg
kshah21
2019-03-18 19:51:05
(X_X)
(X_X)
avisioner
2019-03-18 19:51:05
cough cough
cough cough
bkim0325
2019-03-18 19:51:05
wow
wow
Allen31415
2019-03-18 19:51:05
It looks complicated...
It looks complicated...
ad46578
2019-03-18 19:51:05
my brain hurts
my brain hurts
smbelcas
2019-03-18 19:51:07
Okay, that was just for fun. Sometimes I get a bit excited.
Okay, that was just for fun. Sometimes I get a bit excited.
karthik_malasani
2019-03-18 19:51:15
Looks great! How did you draw these?
Looks great! How did you draw these?
spoamath321
2019-03-18 19:51:15
How did you do it?
How did you do it?
avisioner
2019-03-18 19:51:15
wowowowowowow where did you get this
wowowowowowow where did you get this
smbelcas
2019-03-18 19:51:25
I drew these diagrams in a technical illustration program.
I drew these diagrams in a technical illustration program.
smbelcas
2019-03-18 19:51:35
(Not in Asymptote.)
(Not in Asymptote.)
smbelcas
2019-03-18 19:51:39
So far we have only constructed the $n$-cube visually. Let us now situate it in space. Again, we'll go dimension by dimension.
So far we have only constructed the $n$-cube visually. Let us now situate it in space. Again, we'll go dimension by dimension.
smbelcas
2019-03-18 19:51:47
The most convenient place to put a single point (a $0$-cube) is at $0$.
The most convenient place to put a single point (a $0$-cube) is at $0$.
smbelcas
2019-03-18 19:51:52
What should the ends of our $1$-cube line segment be, in terms of coordinates?
What should the ends of our $1$-cube line segment be, in terms of coordinates?
villagevinegar
2019-03-18 19:52:43
0 and 1
0 and 1
Allen31415
2019-03-18 19:52:43
$(0), (1)$
$(0), (1)$
ad46578
2019-03-18 19:52:43
wait it could be anywhere 1 unit away from 0! cool!
wait it could be anywhere 1 unit away from 0! cool!
Damalone
2019-03-18 19:52:52
Or 0 and 1 on the number line
Or 0 and 1 on the number line
smbelcas
2019-03-18 19:52:54
Yes, we put them at $0$ and at $1$.
Yes, we put them at $0$ and at $1$.
smbelcas
2019-03-18 19:53:03
What about the corners of a square (a $2$-cube)?
What about the corners of a square (a $2$-cube)?
villagevinegar
2019-03-18 19:53:39
(0,0), (0,1), (1,0), (1,1)
(0,0), (0,1), (1,0), (1,1)
Mathlete99s
2019-03-18 19:53:39
(0,0) (1,0) (0,1) (1,1)
(0,0) (1,0) (0,1) (1,1)
MrMXS
2019-03-18 19:53:39
$(0,0),(1,0),(0,1),(1,1)$
$(0,0),(1,0),(0,1),(1,1)$
spoamath321
2019-03-18 19:53:39
(0,0)(1,0)(0,1)(1,1)
(0,0)(1,0)(0,1)(1,1)
Allen31415
2019-03-18 19:53:39
$(0,0), (0,1), (1,0), (1,1)$
$(0,0), (0,1), (1,0), (1,1)$
ChickenAgent2227-_-
2019-03-18 19:53:39
$(0,0);(0,1);(1,0);(1,1)$
$(0,0);(0,1);(1,0);(1,1)$
cad314
2019-03-18 19:53:39
(0,0) (0,1) (1,1) (1,0)
(0,0) (0,1) (1,1) (1,0)
mandrake41
2019-03-18 19:53:39
(0,0), (1,0), (1,1), (0,1)
(0,0), (1,0), (1,1), (0,1)
karthik_malasani
2019-03-18 19:53:39
(0,0),(0,1),(1,1),(1,0)
(0,0),(0,1),(1,1),(1,0)
SwimWithDolphin
2019-03-18 19:53:39
$(0, 0), (0, 1), (1, 0), (1, 1)$
$(0, 0), (0, 1), (1, 0), (1, 1)$
Damalone
2019-03-18 19:53:39
(0,0),(1,0),(0,1),(1,1)
(0,0),(1,0),(0,1),(1,1)
smbelcas
2019-03-18 19:53:39
For consistency, we place them at $(0,0), (0,1), (1,0)$, and $(1,1)$.
For consistency, we place them at $(0,0), (0,1), (1,0)$, and $(1,1)$.
smbelcas
2019-03-18 19:53:47
Now I'm just going to ask you a volley of questions: How many corners does a $3$-cube have? What about a $4$-cube? ...an $n$-cube? What are the coordinates of those corners? Is there an easy way to describe them?
Now I'm just going to ask you a volley of questions: How many corners does a $3$-cube have? What about a $4$-cube? ...an $n$-cube? What are the coordinates of those corners? Is there an easy way to describe them?
smbelcas
2019-03-18 19:56:28
I'm going to pass your responses through in batches by common theme.
I'm going to pass your responses through in batches by common theme.
kshah21
2019-03-18 19:57:04
8
8
da-rong_wae
2019-03-18 19:57:04
2^(n-1)
2^(n-1)
karthik_malasani
2019-03-18 19:57:04
3 cube 8 corners, 4 cube 16 corners, n cube 2^n corners
3 cube 8 corners, 4 cube 16 corners, n cube 2^n corners
motorfinn
2019-03-18 19:57:04
A 3-cube has the vertices of a cube, or 8.
A 3-cube has the vertices of a cube, or 8.
rubixsolver
2019-03-18 19:57:04
8 and 16, powers of 2
8 and 16, powers of 2
evankeri
2019-03-18 19:57:04
8 corners, but the real name is vertices
8 corners, but the real name is vertices
smbelcas
2019-03-18 19:57:05
A $3$-cube has $8$ corners. A $4$-cube has $16$ corners.
A $3$-cube has $8$ corners. A $4$-cube has $16$ corners.
villagevinegar
2019-03-18 19:57:43
8, 16...2^n
8, 16...2^n
SwimWithDolphin
2019-03-18 19:57:43
$2^n$ for number
$2^n$ for number
spoamath321
2019-03-18 19:57:43
2^n
2^n
da-rong_wae
2019-03-18 19:57:43
2^n
2^n
cad314
2019-03-18 19:57:43
3-cube: 8 verticies 4-cube: 18 verticies n-cube:$2^n$ verticies
3-cube: 8 verticies 4-cube: 18 verticies n-cube:$2^n$ verticies
motorfinn
2019-03-18 19:57:43
We can model the number of corners by 2^(n) where our figure is an n-cube.
We can model the number of corners by 2^(n) where our figure is an n-cube.
Mathlete99s
2019-03-18 19:57:43
it seems n cube 2^n corners
it seems n cube 2^n corners
ad46578
2019-03-18 19:57:43
a 3-cube has 8, a 4-cube has 16, and an n-cube has 2^n corners
a 3-cube has 8, a 4-cube has 16, and an n-cube has 2^n corners
mjz6202007
2019-03-18 19:57:43
a n-cube would have 2^n
a n-cube would have 2^n
DOGSTREET1
2019-03-18 19:57:43
an n cube as 2^n corners!
an n cube as 2^n corners!
smbelcas
2019-03-18 19:57:45
There are lots of conjectures that an $n$-cube has $2^n$ corners.
There are lots of conjectures that an $n$-cube has $2^n$ corners.
jerry_yang
2019-03-18 19:57:55
(0,0,0)(0,0,1)(0,1,0)(1,0,0)(0,1,1)(1,0,1)(1,1,0)(1,1,1)
(0,0,0)(0,0,1)(0,1,0)(1,0,0)(0,1,1)(1,0,1)(1,1,0)(1,1,1)
jumpmonkey
2019-03-18 19:57:55
(0,0,0), (0,0,1), (0,1,1), (0.1.0) , (1,1,0) , (1, 0,0)
(0,0,0), (0,0,1), (0,1,1), (0.1.0) , (1,1,0) , (1, 0,0)
avisioner
2019-03-18 19:57:55
ok
3d is 0,0,0;0,0,1;0,1,0;0,1,1;1,0,0;1,0,1;1,1,0 and 1,1,1 in other words binary
ok
3d is 0,0,0;0,0,1;0,1,0;0,1,1;1,0,0;1,0,1;1,1,0 and 1,1,1 in other words binary
kshah21
2019-03-18 19:58:06
oops, (0,0,0) (0,0,1) (0,1,0) (1,0,0) and so on
oops, (0,0,0) (0,0,1) (0,1,0) (1,0,0) and so on
avisioner
2019-03-18 19:58:13
4d is
0,0,0,0;0,0,0,1;0,0,1,0;0,0,1,1;0,1,0,0;0,1,0,1;0,1,1,0;0,1,1,1;1,0,0,0;1,0,0,1;1,0,1,0;1,0,1,1;1,1,0,0;1,1,0,1;1,1,1,0;1,1,1,1
4d is
0,0,0,0;0,0,0,1;0,0,1,0;0,0,1,1;0,1,0,0;0,1,0,1;0,1,1,0;0,1,1,1;1,0,0,0;1,0,0,1;1,0,1,0;1,0,1,1;1,1,0,0;1,1,0,1;1,1,1,0;1,1,1,1
smbelcas
2019-03-18 19:58:15
A 3-cube has corners at $(0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1)$, and $(1,1,1)$.
A 3-cube has corners at $(0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1)$, and $(1,1,1)$.
Allen31415
2019-03-18 19:58:47
There are $2^n$ vertices on an $n-$cube, the vertices are all possible configurations of 0 and 1 as coordinates.
There are $2^n$ vertices on an $n-$cube, the vertices are all possible configurations of 0 and 1 as coordinates.
ChickenAgent2227-_-
2019-03-18 19:58:47
a $n$-cube has $2^n$ vertices. They are all in the form $(x,y,z,...)$ where there are $n$ coordinates and each one can be 0 or 1
a $n$-cube has $2^n$ vertices. They are all in the form $(x,y,z,...)$ where there are $n$ coordinates and each one can be 0 or 1
mandrake41
2019-03-18 19:58:47
an n-cube has 2^n corners, and they go from(0,0,...0(n zeroes)) to (1,1,...1(n ones))
an n-cube has 2^n corners, and they go from(0,0,...0(n zeroes)) to (1,1,...1(n ones))
Damalone
2019-03-18 19:58:47
2^n corners. The corners are of the form (a_1,a_2,...a_n) where a_i can be 0 or 1
2^n corners. The corners are of the form (a_1,a_2,...a_n) where a_i can be 0 or 1
MrMXS
2019-03-18 19:58:47
yeah basically the $2^{n}$ points represented by $(c_{1},c_{2},\dots c_{n})$ where each $c_{i}=0$ or $1$
yeah basically the $2^{n}$ points represented by $(c_{1},c_{2},\dots c_{n})$ where each $c_{i}=0$ or $1$
villagevinegar
2019-03-18 19:58:47
for an n-cube, there are 2^n corners. Each vertex is described by n numbers -- all zeros and ones: for example for n=6, one of the vertices might be (p1,p2,p3,p4,p5,p6) where p_i=0 or 1
for an n-cube, there are 2^n corners. Each vertex is described by n numbers -- all zeros and ones: for example for n=6, one of the vertices might be (p1,p2,p3,p4,p5,p6) where p_i=0 or 1
cad314
2019-03-18 19:58:47
n-cube: all coordinates of (n-1)-cube with both a 0 added and a 1 added
n-cube: all coordinates of (n-1)-cube with both a 0 added and a 1 added
smbelcas
2019-03-18 19:58:49
People think that the corners of an $n$-cube can be described as all $n$-tuples with entries that are $0$ or $1$. But this is a conjecture...
People think that the corners of an $n$-cube can be described as all $n$-tuples with entries that are $0$ or $1$. But this is a conjecture...
smbelcas
2019-03-18 19:58:51
...and so here are more questions. Can you prove that an $n$-cube has $2^n$ corners? Are you sure that every corner of an $n$-cube should have coordinates with entries that are $0$ or $1$; why? Does every $n$-tuple with $0$ and/or $1$ entries represent a corner of an $n$-cube?
...and so here are more questions. Can you prove that an $n$-cube has $2^n$ corners? Are you sure that every corner of an $n$-cube should have coordinates with entries that are $0$ or $1$; why? Does every $n$-tuple with $0$ and/or $1$ entries represent a corner of an $n$-cube?
smbelcas
2019-03-18 20:00:09
These questions are much tougher, but here is where we are really getting to understand the $n$-cube. Let's answer them one at a time.
These questions are much tougher, but here is where we are really getting to understand the $n$-cube. Let's answer them one at a time.
smbelcas
2019-03-18 20:00:11
Prove that an $n$-cube has $2^n$ corners.
Prove that an $n$-cube has $2^n$ corners.
MrMXS
2019-03-18 20:01:00
it has $2^{n}$ corners by induction since at each step we "copy" our previous cube, going from $v$ vertices to $2v$ vertices
it has $2^{n}$ corners by induction since at each step we "copy" our previous cube, going from $v$ vertices to $2v$ vertices
ChickenAgent2227-_-
2019-03-18 20:01:00
Yes, by induction. Every time you sweep an n-1 cube it you make a copy of it and have double the vertices in an n cube
Yes, by induction. Every time you sweep an n-1 cube it you make a copy of it and have double the vertices in an n cube
DOGSTREET1
2019-03-18 20:01:00
yes, because you double the number of corners every time you moosh the cube
yes, because you double the number of corners every time you moosh the cube
mandrake41
2019-03-18 20:01:00
Each time you moosh an n cube to an n+1 cube, you double the number of corners. One for the starting position and one for the ending position.
Each time you moosh an n cube to an n+1 cube, you double the number of corners. One for the starting position and one for the ending position.
spoamath321
2019-03-18 20:01:00
Yes, a 0-cube (not really a cube) is 2^0, a 1-cube is 2^1 points
Yes, a 0-cube (not really a cube) is 2^0, a 1-cube is 2^1 points
rubixsolver
2019-03-18 20:01:00
Every time we moosh an n-cube, we double the number of corners it has. This results in (current number of corners * 2) which is doubling it.
Every time we moosh an n-cube, we double the number of corners it has. This results in (current number of corners * 2) which is doubling it.
cad314
2019-03-18 20:01:06
verticies always double everytime we add a dimension because we are just copying the (n-1)-cube and connecting all the verticies
verticies always double everytime we add a dimension because we are just copying the (n-1)-cube and connecting all the verticies
smbelcas
2019-03-18 20:01:07
When we make an $n$-cube from an $(n-1)$-cube, we moosh that $(n-1)$-cube by one unit. There are the "starting" corners and the "ending" corners, so there are twice as many corners in an $n$-cube as in an $(n-1)$-cube.
When we make an $n$-cube from an $(n-1)$-cube, we moosh that $(n-1)$-cube by one unit. There are the "starting" corners and the "ending" corners, so there are twice as many corners in an $n$-cube as in an $(n-1)$-cube.
smbelcas
2019-03-18 20:01:20
That's not enough to say that the number of corners is $2^n$, though. We also have to remember that we can manually count to see that the number of corners of the $\{0$-cube, $1$-cube, $2$-cube, $3$-cube$\}$ is $\{1, 2, 4, 8\}$ so if we continue to double we'll always get $2^n$.
That's not enough to say that the number of corners is $2^n$, though. We also have to remember that we can manually count to see that the number of corners of the $\{0$-cube, $1$-cube, $2$-cube, $3$-cube$\}$ is $\{1, 2, 4, 8\}$ so if we continue to double we'll always get $2^n$.
smbelcas
2019-03-18 20:01:36
Are you sure that every corner of an $n$-cube should have coordinates with entries that are $0$ or $1$; why?
Are you sure that every corner of an $n$-cube should have coordinates with entries that are $0$ or $1$; why?
ad46578
2019-03-18 20:02:41
if it has a corner on the origin, then yes because the corners must always be 1 unit away from the origin
if it has a corner on the origin, then yes because the corners must always be 1 unit away from the origin
spoamath321
2019-03-18 20:02:41
It starts with one point and the unit increases by one
It starts with one point and the unit increases by one
cad314
2019-03-18 20:02:41
because all the side lengths are always 1 (why they are cubes)
because all the side lengths are always 1 (why they are cubes)
mandrake41
2019-03-18 20:02:41
since each time you moosh in a perpendicular direction, you will only ever reach a maximum of 1 for any corner; corners that are already at 1 for one axis will never go past 1 on that dimension
since each time you moosh in a perpendicular direction, you will only ever reach a maximum of 1 for any corner; corners that are already at 1 for one axis will never go past 1 on that dimension
Damalone
2019-03-18 20:02:41
Because mooshing is a type of Cartesian product as you said
Because mooshing is a type of Cartesian product as you said
ChickenAgent2227-_-
2019-03-18 20:02:41
yes, you just moosh it 1 unit in the next dimension, so if each previous cube has coordinates with entries that are all 0 or 1, then each new cube should have those, but with 0 or 1 added on at the end
yes, you just moosh it 1 unit in the next dimension, so if each previous cube has coordinates with entries that are all 0 or 1, then each new cube should have those, but with 0 or 1 added on at the end
smbelcas
2019-03-18 20:02:43
The corners of an $n$-cube must have coordinate entries that are $0$ or $1$ if we situate our original point at $0$, because every corner is either a "starting" corner ($0$ entry) or an "ending" corner ($1$ entry) in the last coordinate, and has the $0$ and/or $1$ entries from lower-dimensional cubes in the first $n-1$ coordinates.
The corners of an $n$-cube must have coordinate entries that are $0$ or $1$ if we situate our original point at $0$, because every corner is either a "starting" corner ($0$ entry) or an "ending" corner ($1$ entry) in the last coordinate, and has the $0$ and/or $1$ entries from lower-dimensional cubes in the first $n-1$ coordinates.
smbelcas
2019-03-18 20:02:56
Does every $n$-tuple with $0$ and/or $1$ entries represent a corner of an $n$-cube?
Does every $n$-tuple with $0$ and/or $1$ entries represent a corner of an $n$-cube?
Allen31415
2019-03-18 20:04:16
Every direction has an axis, each axis can have two possible coordinates
Every direction has an axis, each axis can have two possible coordinates
mandrake41
2019-03-18 20:04:16
As you increase the dimensions of a cube, the number of corners double, as do the number of combinations of 1s and 0s. Since each corner ends up in a different location, this means that each combination of 1s and 0s are used.
As you increase the dimensions of a cube, the number of corners double, as do the number of combinations of 1s and 0s. Since each corner ends up in a different location, this means that each combination of 1s and 0s are used.
ChickenAgent2227-_-
2019-03-18 20:04:16
yes, there are 2^n such n-tuples and each corresponds to a corner, which there are 2^n of
yes, there are 2^n such n-tuples and each corresponds to a corner, which there are 2^n of
Allen31415
2019-03-18 20:04:16
Yes, it is a 1-1 correspondence between the coordinates and vertices on an $n$-cube.
Yes, it is a 1-1 correspondence between the coordinates and vertices on an $n$-cube.
smbelcas
2019-03-18 20:04:20
One way we can say that every $n$-tuple with $0$ and/or $1$ entries represents a corner of an $n$-cube is by using our previous two arguments: There are $2^n$ $n$-tuples with $0$ and/or $1$ entries, and every corner must be among them. However, there are also $2^n$ corners, so each of those $n$-tuples represents a corner!
One way we can say that every $n$-tuple with $0$ and/or $1$ entries represents a corner of an $n$-cube is by using our previous two arguments: There are $2^n$ $n$-tuples with $0$ and/or $1$ entries, and every corner must be among them. However, there are also $2^n$ corners, so each of those $n$-tuples represents a corner!
MrMXS
2019-03-18 20:04:28
yeah cause for each dimension and for each vertex that vertex was either translated or not translated in that dimension, which correspond to a $1$ and a $0$, respectively (and we represent this by having each coordinate correspond to a dimension/translation)
yeah cause for each dimension and for each vertex that vertex was either translated or not translated in that dimension, which correspond to a $1$ and a $0$, respectively (and we represent this by having each coordinate correspond to a dimension/translation)
da-rong_wae
2019-03-18 20:04:32
Suppose one isnt, then we have 2^n corners but fewer n-tuples
Suppose one isnt, then we have 2^n corners but fewer n-tuples
smbelcas
2019-03-18 20:04:48
Now let's count parts of $n$-cubes: Please help to fill in the table:
Now let's count parts of $n$-cubes: Please help to fill in the table:
smbelcas
2019-03-18 20:04:52
$$\begin{array}{c||c|c|c|c|c|c|c}
{\rm dim} &0&1&2&3&4&5&\dots\\ \hline
\hline
{\rm points} & 1 &2&&& &&\dots\\
\hline
{\rm lines} &0 &1 & & & & &\dots\\
\hline
{\rm squares} &0 &0 & & & & &\dots\\
\hline
{\rm 3-cubes} &0 &\hspace{1cm} & & & & &\dots\\
\hline
{\rm 4-cubes}&\hspace{1cm} &0 &\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\dots\\
\hline
\end{array}$$
$$\begin{array}{c||c|c|c|c|c|c|c}
{\rm dim} &0&1&2&3&4&5&\dots\\ \hline
\hline
{\rm points} & 1 &2&&& &&\dots\\
\hline
{\rm lines} &0 &1 & & & & &\dots\\
\hline
{\rm squares} &0 &0 & & & & &\dots\\
\hline
{\rm 3-cubes} &0 &\hspace{1cm} & & & & &\dots\\
\hline
{\rm 4-cubes}&\hspace{1cm} &0 &\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\dots\\
\hline
\end{array}$$
smbelcas
2019-03-18 20:04:56
So, for example, you can say "points in $2$-cube is $4$" to fill in the next entry of the "points" row.
So, for example, you can say "points in $2$-cube is $4$" to fill in the next entry of the "points" row.
smbelcas
2019-03-18 20:06:39
I'll pass the answers through, column by column.
I'll pass the answers through, column by column.
smbelcas
2019-03-18 20:06:48
For $2$-cubes:
For $2$-cubes:
mjz6202007
2019-03-18 20:06:50
squares in 2-cube is 1
squares in 2-cube is 1
ChickenAgent2227-_-
2019-03-18 20:06:50
points in $2$-cube is $4$
points in $2$-cube is $4$
cad314
2019-03-18 20:06:50
lines in 2-cube is 4
lines in 2-cube is 4
cad314
2019-03-18 20:06:50
squares in 2-cube is 1
squares in 2-cube is 1
avisioner
2019-03-18 20:06:50
lines in 2 cube is 4
lines in 2 cube is 4
DOGSTREET1
2019-03-18 20:06:50
lines in 2 cube is 4
lines in 2 cube is 4
DOGSTREET1
2019-03-18 20:06:50
squares in 2 cube is 1
squares in 2 cube is 1
mjz6202007
2019-03-18 20:06:50
lines in 2-cube is 4
lines in 2-cube is 4
karthik_malasani
2019-03-18 20:07:11
2 dimensions: points = 4, lines = 4, squares = 1, 3-cubes = 0, 4-cubes = 0
2 dimensions: points = 4, lines = 4, squares = 1, 3-cubes = 0, 4-cubes = 0
smbelcas
2019-03-18 20:07:44
For $3$-cubes:
For $3$-cubes:
mandrake41
2019-03-18 20:07:46
lines in 3-cube is 12
lines in 3-cube is 12
mandrake41
2019-03-18 20:07:46
squares in 3-cube is 6
squares in 3-cube is 6
Mathlete161
2019-03-18 20:07:46
squares in 3 cubes is 6
squares in 3 cubes is 6
mjz6202007
2019-03-18 20:07:46
points in 3-cube is 8
points in 3-cube is 8
avisioner
2019-03-18 20:07:46
points in 3-cube is 8
points in 3-cube is 8
mjz6202007
2019-03-18 20:07:46
lines in 3-cube is 12
lines in 3-cube is 12
avisioner
2019-03-18 20:07:46
lines in 3 cube is 12
lines in 3 cube is 12
DOGSTREET1
2019-03-18 20:07:46
points in 3 cube is 8
points in 3 cube is 8
cad314
2019-03-18 20:07:46
points in 3-cube is 8
points in 3-cube is 8
DOGSTREET1
2019-03-18 20:07:46
lines for 3 cube is 12
lines for 3 cube is 12
DOGSTREET1
2019-03-18 20:07:46
squares in 3 cube 6
squares in 3 cube 6
DOGSTREET1
2019-03-18 20:07:46
3 cubes in 3 cube is 1
3 cubes in 3 cube is 1
mandrake41
2019-03-18 20:07:46
3-cubes in 3-cube is 1
3-cubes in 3-cube is 1
karthik_malasani
2019-03-18 20:08:31
3- dimensions: points = 8, lines = 12, squares = 6, 3-cubes = 1, 4-cubes = 0
3- dimensions: points = 8, lines = 12, squares = 6, 3-cubes = 1, 4-cubes = 0
smbelcas
2019-03-18 20:09:25
For $4$-cubes:
For $4$-cubes:
avisioner
2019-03-18 20:09:27
points in 4 cube is 16
points in 4 cube is 16
karthik_malasani
2019-03-18 20:09:27
4- dimensions: points = 2^4 = 16, lines = 4*2^3 = 32, squares = 12?, 3-cubes = 2, 4-cubes = 1
4- dimensions: points = 2^4 = 16, lines = 4*2^3 = 32, squares = 12?, 3-cubes = 2, 4-cubes = 1
DOGSTREET1
2019-03-18 20:09:27
points in 4 cube is 16
points in 4 cube is 16
ChickenAgent2227-_-
2019-03-18 20:09:27
4-cubes in 4-cube is 1
4-cubes in 4-cube is 1
Allen31415
2019-03-18 20:09:27
$3$-cubes in 4-cube is 8
$3$-cubes in 4-cube is 8
StickyWashington
2019-03-18 20:09:27
points in 4-cube is 16
points in 4-cube is 16
DOGSTREET1
2019-03-18 20:09:27
4 cube in 4 cube is 1
4 cube in 4 cube is 1
Allen31415
2019-03-18 20:09:27
Lines in 4-cube is 32
Lines in 4-cube is 32
Allen31415
2019-03-18 20:09:27
points in 4-cube is 16
points in 4-cube is 16
ChickenAgent2227-_-
2019-03-18 20:09:27
3-cubes in 4-cube is 8?
3-cubes in 4-cube is 8?
smbelcas
2019-03-18 20:10:29
For $5$-cubes:
For $5$-cubes:
mandrake41
2019-03-18 20:10:31
points in 5-cube is 32
points in 5-cube is 32
ad46578
2019-03-18 20:10:31
points in 5- cube is 32
points in 5- cube is 32
avisioner
2019-03-18 20:10:31
points in 5 cube is 32
points in 5 cube is 32
als123
2019-03-18 20:10:31
points in 5-cube is 32
points in 5-cube is 32
DOGSTREET1
2019-03-18 20:10:31
points in 5 cube is 32
points in 5 cube is 32
avisioner
2019-03-18 20:11:04
lines in a 5d cube is 80
lines in a 5d cube is 80
Allen31415
2019-03-18 20:11:04
lines in 5 cube is 80
lines in 5 cube is 80
mandrake41
2019-03-18 20:11:04
5-cubes in 5-cube is 1
5-cubes in 5-cube is 1
avisioner
2019-03-18 20:11:04
5d cube has 80 lines
5d cube has 80 lines
Allen31415
2019-03-18 20:11:04
3-cubes in 5-cube is 40
3-cubes in 5-cube is 40
smbelcas
2019-03-18 20:11:07
Wait a minute. Where are those numbers coming from? Did you draw a $5$-cube earlier, and have been counting from it?
Wait a minute. Where are those numbers coming from? Did you draw a $5$-cube earlier, and have been counting from it?
DOGSTREET1
2019-03-18 20:11:30
nope
nope
avisioner
2019-03-18 20:11:30
no!
no!
mandrake41
2019-03-18 20:11:30
...maybe
...maybe
smbelcas
2019-03-18 20:11:32
I'm just not sure of all those numbers. Well, here is the updated table:
I'm just not sure of all those numbers. Well, here is the updated table:
smbelcas
2019-03-18 20:11:33
$$\begin{array}{c||c|c|c|c|c|c|c}
{\rm dim} &0&1&2&3&4&5&\dots\\ \hline
\hline
{\rm points} & 1 &2&4&8&16&32&\dots\\
\hline
{\rm lines} &0 &1 &4 &12 &32 & &\dots\\
\hline
{\rm squares} &0 &0 &1 &6 &24 & &\dots\\
\hline
{\rm 3-cubes} &0 &0 &0 &1 &8 & &\dots\\
\hline
{\rm 4-cubes}&0 &0 &0 &0 &1 & &\dots\\ \hline
&\hspace{1cm} &\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\hspace{1cm}& \\
\end{array}$$
$$\begin{array}{c||c|c|c|c|c|c|c}
{\rm dim} &0&1&2&3&4&5&\dots\\ \hline
\hline
{\rm points} & 1 &2&4&8&16&32&\dots\\
\hline
{\rm lines} &0 &1 &4 &12 &32 & &\dots\\
\hline
{\rm squares} &0 &0 &1 &6 &24 & &\dots\\
\hline
{\rm 3-cubes} &0 &0 &0 &1 &8 & &\dots\\
\hline
{\rm 4-cubes}&0 &0 &0 &0 &1 & &\dots\\ \hline
&\hspace{1cm} &\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\hspace{1cm}&\hspace{1cm}& \\
\end{array}$$
smbelcas
2019-03-18 20:11:40
The $5$-cube numbers are mostly left out because I'm just not convinced yet.
The $5$-cube numbers are mostly left out because I'm just not convinced yet.
smbelcas
2019-03-18 20:11:46
Let me give you some notation: Let $C_n$ denote the $n$-cube.
Let me give you some notation: Let $C_n$ denote the $n$-cube.
smbelcas
2019-03-18 20:11:52
And let $f_k(C_n) = $ the number of $k$-cubes in $C_n$. (The $f$ stands for $f$aces.)
And let $f_k(C_n) = $ the number of $k$-cubes in $C_n$. (The $f$ stands for $f$aces.)
smbelcas
2019-03-18 20:12:08
We already proved that $f_0(C_n) = 2^n$. Do you have any conjectures based on the data we have in the table?
We already proved that $f_0(C_n) = 2^n$. Do you have any conjectures based on the data we have in the table?
Allen31415
2019-03-18 20:14:50
$f_1(C_n)=n \cdot 2^{n-1}$
$f_1(C_n)=n \cdot 2^{n-1}$
MrMXS
2019-03-18 20:14:50
$f_{1}(C_{n})=2f_{1}(C_{n-1})+2^{n-1}$
$f_{1}(C_{n})=2f_{1}(C_{n-1})+2^{n-1}$
ChickenAgent2227-_-
2019-03-18 20:14:50
$f_1(C_n) = n\cdot 2^{n-1}$
$f_1(C_n) = n\cdot 2^{n-1}$
mandrake41
2019-03-18 20:14:50
f1(cn) = n * 2^n
f1(cn) = n * 2^n
da-rong_wae
2019-03-18 20:14:50
F0(c_(n+1))=2f0(c_n)
F0(c_(n+1))=2f0(c_n)
ChickenAgent2227-_-
2019-03-18 20:14:50
$f_2(C_n) = n(n-1)\cdot 2^{n-3}$
$f_2(C_n) = n(n-1)\cdot 2^{n-3}$
jerry_yang
2019-03-18 20:14:50
f_1(C_n)=n*2^(n-1)
f_1(C_n)=n*2^(n-1)
ChickenAgent2227-_-
2019-03-18 20:14:50
$f_3(C_n) = n(n-1)(n-2)\cdot 2^{n-3}$
$f_3(C_n) = n(n-1)(n-2)\cdot 2^{n-3}$
mandrake41
2019-03-18 20:14:50
f1(cn) = n * 2^(n-1)
f1(cn) = n * 2^(n-1)
ChickenAgent2227-_-
2019-03-18 20:14:50
sum of columns is 3^n!
sum of columns is 3^n!
DoingWhatCounts
2019-03-18 20:14:50
coefficients in expansion of (1 + 2)^n
coefficients in expansion of (1 + 2)^n
smbelcas
2019-03-18 20:15:58
These are interesting.
These are interesting.
smbelcas
2019-03-18 20:16:16
Does anyone have a conjecture for what $f_k(C_n)$ is?
Does anyone have a conjecture for what $f_k(C_n)$ is?
smbelcas
2019-03-18 20:16:26
Either in terms of smaller cubes or faces, or directly?
Either in terms of smaller cubes or faces, or directly?
da-rong_wae
2019-03-18 20:18:04
N!/k!*2^n-k
N!/k!*2^n-k
MrMXS
2019-03-18 20:18:04
$f_{k}(C_{n})=\cfrac{n!}{(n-k)!}2^{n-k}$?? Kind of a guess
$f_{k}(C_{n})=\cfrac{n!}{(n-k)!}2^{n-k}$?? Kind of a guess
da-rong_wae
2019-03-18 20:18:04
M!/(n-k)!*2^n-k
M!/(n-k)!*2^n-k
smbelcas
2019-03-18 20:18:09
These are all close.
These are all close.
smbelcas
2019-03-18 20:18:31
Any ideas for a recurrence?
Any ideas for a recurrence?
smbelcas
2019-03-18 20:19:15
Here are cleaned-up versions of what some of you have been saying:
Here are cleaned-up versions of what some of you have been saying:
smbelcas
2019-03-18 20:19:19
Conjecture 1: $f_k(C_n) = 2f_k(C_{n-1})+f_{k-1}(C_{n-1})$.
Conjecture 1: $f_k(C_n) = 2f_k(C_{n-1})+f_{k-1}(C_{n-1})$.
smbelcas
2019-03-18 20:19:24
Conjecture 2: $f_k(C_n) = {n\choose k}2^{n-k}$. (This one only makes sense if you know binomial coefficients already.)
Conjecture 2: $f_k(C_n) = {n\choose k}2^{n-k}$. (This one only makes sense if you know binomial coefficients already.)
smbelcas
2019-03-18 20:19:38
Conjecture 3: The sum of the $n$-th column is $3^n$.
Conjecture 3: The sum of the $n$-th column is $3^n$.
smbelcas
2019-03-18 20:19:44
Can you prove any of these?
Can you prove any of these?
smbelcas
2019-03-18 20:20:54
Conjecture 1 follows from our construction of the $n$-cube. How does that work, exactly?
Conjecture 1 follows from our construction of the $n$-cube. How does that work, exactly?
smbelcas
2019-03-18 20:21:28
Let's go through the details. What happens to a corner point when we moosh?
Let's go through the details. What happens to a corner point when we moosh?
Allen31415
2019-03-18 20:22:07
It copies itself, creating an edge.
It copies itself, creating an edge.
smbelcas
2019-03-18 20:22:10
It turns into a line segment. Basically it has the "starting" (or $0$) point, and the "ending" (or $1$) point, and the trail left is a line segment.
It turns into a line segment. Basically it has the "starting" (or $0$) point, and the "ending" (or $1$) point, and the trail left is a line segment.
mandrake41
2019-03-18 20:22:17
copies, creating a line
copies, creating a line
smbelcas
2019-03-18 20:22:20
What happens to a line segment when we moosh?
What happens to a line segment when we moosh?
ChickenAgent2227-_-
2019-03-18 20:22:51
creates a face
creates a face
mandrake41
2019-03-18 20:22:51
copies, creating a 2-cube
copies, creating a 2-cube
Allen31415
2019-03-18 20:22:51
It copies itself, creating a square.
It copies itself, creating a square.
gozomete
2019-03-18 20:22:51
creats a square
creats a square
mjz6202007
2019-03-18 20:22:51
It copies itself and moves perpendicular to the original shape, leaving the trail it took to get to the new shape.
It copies itself and moves perpendicular to the original shape, leaving the trail it took to get to the new shape.
MrMXS
2019-03-18 20:22:51
gives us a square
gives us a square
smbelcas
2019-03-18 20:22:54
It turns into a square. Basically it has the "starting" (or $0$) segment, and the "ending" (or $1$) segment, and the trail left is a square.
It turns into a square. Basically it has the "starting" (or $0$) segment, and the "ending" (or $1$) segment, and the trail left is a square.
smbelcas
2019-03-18 20:22:56
What happens to a square when we moosh?
What happens to a square when we moosh?
cad314
2019-03-18 20:23:12
and a plane becomes a cube
and a plane becomes a cube
Allen31415
2019-03-18 20:23:12
It copies itself, creating a cube
It copies itself, creating a cube
mandrake41
2019-03-18 20:23:12
creates a cube
creates a cube
ChickenAgent2227-_-
2019-03-18 20:23:12
It copies itself and makes a cube
It copies itself and makes a cube
cad314
2019-03-18 20:23:12
a square becomes a cube
a square becomes a cube
phanithans1
2019-03-18 20:23:12
into a cube
into a cube
smbelcas
2019-03-18 20:23:13
It turns into a $3$-cube. Basically it has the "starting" (or $0$) square, and the "ending" (or $1$) square, and the trail left is a $3$-cube.
It turns into a $3$-cube. Basically it has the "starting" (or $0$) square, and the "ending" (or $1$) square, and the trail left is a $3$-cube.
smbelcas
2019-03-18 20:23:16
So, more generally, what happens to a $k$-cube when we moosh?
So, more generally, what happens to a $k$-cube when we moosh?
MrMXS
2019-03-18 20:23:39
a $k$-dimensional object gives us a $(k+1)$-dimensional object
a $k$-dimensional object gives us a $(k+1)$-dimensional object
Allen31415
2019-03-18 20:23:39
It copies itself, creating a $k+1$-cube.
It copies itself, creating a $k+1$-cube.
mandrake41
2019-03-18 20:23:39
k+1 cube
k+1 cube
ChickenAgent2227-_-
2019-03-18 20:23:39
It copies itself and makes a k+1 cube
It copies itself and makes a k+1 cube
da-rong_wae
2019-03-18 20:23:39
Becomes k+1 cube
Becomes k+1 cube
spoamath321
2019-03-18 20:23:39
it becomes a k+1 cube
it becomes a k+1 cube
gozomete
2019-03-18 20:23:39
k+1-cube
k+1-cube
smbelcas
2019-03-18 20:23:41
It turns into a $(k+1)$-cube. Basically it has the "starting" (or $0$) $k$-cube, and the "ending" (or $1$) $k$-cube, and the trail left is a $(k+1)$-cube.
It turns into a $(k+1)$-cube. Basically it has the "starting" (or $0$) $k$-cube, and the "ending" (or $1$) $k$-cube, and the trail left is a $(k+1)$-cube.
smbelcas
2019-03-18 20:23:44
How does that help us prove Conjecture 1?
How does that help us prove Conjecture 1?
smbelcas
2019-03-18 20:24:23
The formula says the number of $k$-cubes in an $n$-cube is the same as $2($the number of $k$-cubes in an $(n-1)$-cube$) + ($the number of $(k-1)$-cubes in an $(n-1)$-cube$)$.
The formula says the number of $k$-cubes in an $n$-cube is the same as $2($the number of $k$-cubes in an $(n-1)$-cube$) + ($the number of $(k-1)$-cubes in an $(n-1)$-cube$)$.
da-rong_wae
2019-03-18 20:24:42
We count twice everything in previous step, cuz we are copying the whole Cn, then account for the "mooshed" stuff
We count twice everything in previous step, cuz we are copying the whole Cn, then account for the "mooshed" stuff
smbelcas
2019-03-18 20:25:00
The $2($the number of $k$-cubes in an $(n-1)$-cube$)$ counts the "starting" and "ending" $k$-cubes from a moosh.
The $2($the number of $k$-cubes in an $(n-1)$-cube$)$ counts the "starting" and "ending" $k$-cubes from a moosh.
mandrake41
2019-03-18 20:25:17
everything is doubled, plus the k-1 cubes are mooshed into k cubes
everything is doubled, plus the k-1 cubes are mooshed into k cubes
MrMXS
2019-03-18 20:25:17
the number of $k$-dimensional objects is twice the number of $k$-dimensional objects in our previous cube (because it's copied) plus all of the $(k-1)$ dimensional stuff which becomes $k$-dimensional stuff
the number of $k$-dimensional objects is twice the number of $k$-dimensional objects in our previous cube (because it's copied) plus all of the $(k-1)$ dimensional stuff which becomes $k$-dimensional stuff
ChickenAgent2227-_-
2019-03-18 20:25:17
Each k-dimesnional face of a n-cube can be a new one made from mooshing, or an old one that alredy existed. The new ones are made from each $k-1$ D face of the old cube. The old ones are the k-D faces from the old cube, copies to have 2 times as much
Each k-dimesnional face of a n-cube can be a new one made from mooshing, or an old one that alredy existed. The new ones are made from each $k-1$ D face of the old cube. The old ones are the k-D faces from the old cube, copies to have 2 times as much
smbelcas
2019-03-18 20:25:22
The $($the number of $(k-1)$-cubes in an $(n-1)$-cube$)$ counts the new $k$-cubes from the moosh, which all came from the trails of $(k-1)$-cubes.
The $($the number of $(k-1)$-cubes in an $(n-1)$-cube$)$ counts the new $k$-cubes from the moosh, which all came from the trails of $(k-1)$-cubes.
cad314
2019-03-18 20:25:26
the first term is that every one of the objects gets duplicated from the (n-1)-cube, the second term is that everything that is (k-1)-cube becomes 1 k-cube
the first term is that every one of the objects gets duplicated from the (n-1)-cube, the second term is that everything that is (k-1)-cube becomes 1 k-cube
smbelcas
2019-03-18 20:25:38
Conjecture 2 is tougher, so we won't discuss it tonight.
Conjecture 2 is tougher, so we won't discuss it tonight.
smbelcas
2019-03-18 20:26:04
But we can use Conjecture 1 to prove Conjecture 3. Do you see how?
But we can use Conjecture 1 to prove Conjecture 3. Do you see how?
ChickenAgent2227-_-
2019-03-18 20:26:46
induction and adding everything up?
induction and adding everything up?
Allen31415
2019-03-18 20:26:46
induction?
induction?
sansae
2019-03-18 20:26:46
Just a induction!
Just a induction!
MrMXS
2019-03-18 20:26:46
induction?
induction?
da-rong_wae
2019-03-18 20:26:46
Telescopic sums?
Telescopic sums?
smbelcas
2019-03-18 20:26:57
Before working on Conjecture 3, let's restate it in mathematical notation: $\sum_{k=0}^n f_k(C_n)= 3^n.$
Before working on Conjecture 3, let's restate it in mathematical notation: $\sum_{k=0}^n f_k(C_n)= 3^n.$
smbelcas
2019-03-18 20:27:29
We'll use Conjecture 1: $\sum_{k=0}^n f_k(C_n) = \sum_{k=0}^n 2f_k(C_{n-1})+f_{k-1}(C_{n-1})$.
We'll use Conjecture 1: $\sum_{k=0}^n f_k(C_n) = \sum_{k=0}^n 2f_k(C_{n-1})+f_{k-1}(C_{n-1})$.
smbelcas
2019-03-18 20:27:38
What can we do?
What can we do?
ChickenAgent2227-_-
2019-03-18 20:28:00
Split into two sums
Split into two sums
smbelcas
2019-03-18 20:28:01
If we distribute, we have $\sum_{k=0}^n 2f_k(C_{n-1})+ \sum_{k=0}^nf_{k-1}(C_{n-1})$.
If we distribute, we have $\sum_{k=0}^n 2f_k(C_{n-1})+ \sum_{k=0}^nf_{k-1}(C_{n-1})$.
smbelcas
2019-03-18 20:28:24
So that's $2$ times the total sum for an $(n-1)$-cube, plus another sum that represents all that stuff from the mooshing. What is that sum from the mooshing?
So that's $2$ times the total sum for an $(n-1)$-cube, plus another sum that represents all that stuff from the mooshing. What is that sum from the mooshing?
ChickenAgent2227-_-
2019-03-18 20:28:35
reindex
reindex
smbelcas
2019-03-18 20:28:43
$\sum_{k=0}^nf_{k-1}(C_{n-1}) = \sum_{k=0}^nf_{k}(C_{n-1}) $ because the first term on the left-hand side is just $0$, and the last term on the right-hand side is also $0$. Why?
$\sum_{k=0}^nf_{k-1}(C_{n-1}) = \sum_{k=0}^nf_{k}(C_{n-1}) $ because the first term on the left-hand side is just $0$, and the last term on the right-hand side is also $0$. Why?
smbelcas
2019-03-18 20:29:30
There are no $(-1)$-cubes! And there are no $n$-cubes in an $(n-1)$-cube.
There are no $(-1)$-cubes! And there are no $n$-cubes in an $(n-1)$-cube.
smbelcas
2019-03-18 20:29:35
So now our total sum is $\sum_{k=0}^n2f_k(C_{n-1})+f_k(C_{n-1}) = 3 \sum_{k=0}^nf_k(C_{n-1})$.
So now our total sum is $\sum_{k=0}^n2f_k(C_{n-1})+f_k(C_{n-1}) = 3 \sum_{k=0}^nf_k(C_{n-1})$.
smbelcas
2019-03-18 20:29:51
Just one more step---how do we finish the proof?
Just one more step---how do we finish the proof?
Allen31415
2019-03-18 20:30:19
Base Case when $n=0$
Base Case when $n=0$
MrMXS
2019-03-18 20:30:19
look at the sum in the first column
look at the sum in the first column
smbelcas
2019-03-18 20:30:21
Because our first few sums are powers of $3$, that pattern continues.
Because our first few sums are powers of $3$, that pattern continues.
smbelcas
2019-03-18 20:30:51
There is also a neat proof based on Conjecture 2---you can think about that later.
There is also a neat proof based on Conjecture 2---you can think about that later.
smbelcas
2019-03-18 20:30:55
Okay, let's switch gears and talk about {MathILy, MathILy-Er} so that there's time for you to ask lots of questions. (I hope you enjoyed the math!)
Okay, let's switch gears and talk about {MathILy, MathILy-Er} so that there's time for you to ask lots of questions. (I hope you enjoyed the math!)
smbelcas
2019-03-18 20:31:07
{MathILy, MathILy-Er} are intensive residential summer programs for mathematically excellent secondary students.
{MathILy, MathILy-Er} are intensive residential summer programs for mathematically excellent secondary students.
smbelcas
2019-03-18 20:31:14
As we say on the website (http://www.mathily.org), {MathILy, MathILy-Er} focus on participants exploring and creating mathematics. Instructors provide the framework and you get to make (and prove!) the conjectures. You will encounter new ideas, improve your problem-solving skills, learn lots and lots of advanced mathematics, and hone your overall thinking skills. You'll meet others like you. (Yes, really. We promise.) Most of all, you will find serious mathematics infused with levity.
{MathILy, MathILy-Er} are five weeks of maximized mathematical marvelousness.
As we say on the website (http://www.mathily.org), {MathILy, MathILy-Er} focus on participants exploring and creating mathematics. Instructors provide the framework and you get to make (and prove!) the conjectures. You will encounter new ideas, improve your problem-solving skills, learn lots and lots of advanced mathematics, and hone your overall thinking skills. You'll meet others like you. (Yes, really. We promise.) Most of all, you will find serious mathematics infused with levity.
{MathILy, MathILy-Er} are five weeks of maximized mathematical marvelousness.
smbelcas
2019-03-18 20:31:30
The programs share an application process---you apply to both programs at once. You take an Exam Assessing Readiness and fill out some information on the Short Form and Not-as-Short Form. Based on these things (and comments from a recommender) the {MathILy, MathILy-Er} Directors decide whether you are qualified, and if so, for which program. MathILy-Er is designed for students who are a little bit earlier in their mathematical development than MathILy students.
The programs share an application process---you apply to both programs at once. You take an Exam Assessing Readiness and fill out some information on the Short Form and Not-as-Short Form. Based on these things (and comments from a recommender) the {MathILy, MathILy-Er} Directors decide whether you are qualified, and if so, for which program. MathILy-Er is designed for students who are a little bit earlier in their mathematical development than MathILy students.
smbelcas
2019-03-18 20:31:46
Please ask questions!
Please ask questions!
Potato12
2019-03-18 20:31:50
what are the combined admission rates of mathily and mathily EAR? If we don't solve all the problems do we still have a chance of getting in?
what are the combined admission rates of mathily and mathily EAR? If we don't solve all the problems do we still have a chance of getting in?
smbelcas
2019-03-18 20:32:09
I don't know what you mean by "combined admission rates." But no one solves all the problems correctly.
I don't know what you mean by "combined admission rates." But no one solves all the problems correctly.
spoamath321
2019-03-18 20:32:17
How long is it?
How long is it?
smbelcas
2019-03-18 20:32:20
5 weeks!
5 weeks!
ibtstrash_
2019-03-18 20:32:23
What ages are these programs for?
What ages are these programs for?
smbelcas
2019-03-18 20:32:40
We focus on 14--17, but we take applications from younger and older students.
We focus on 14--17, but we take applications from younger and older students.
smbelcas
2019-03-18 20:33:02
That said, we haven't admitted anyone younger than 13 (no one has done well enough on the EAR) or older than 18.
That said, we haven't admitted anyone younger than 13 (no one has done well enough on the EAR) or older than 18.
Potato12
2019-03-18 20:33:07
Like out of all the students who apply, what are the percentage of those accepted?
Like out of all the students who apply, what are the percentage of those accepted?
smbelcas
2019-03-18 20:33:17
Last year it was something like 20%.
Last year it was something like 20%.
smbelcas
2019-03-18 20:33:21
I think this year it will be lower.
I think this year it will be lower.
Allen31415
2019-03-18 20:33:31
Have you been in MathILY, and is it fun?
Have you been in MathILY, and is it fun?
smbelcas
2019-03-18 20:33:39
Yes, always, and yes, always.
Yes, always, and yes, always.
Potato12
2019-03-18 20:33:43
Are the admissions rolling?
Are the admissions rolling?
smbelcas
2019-03-18 20:33:46
Yup.
Yup.
cad314
2019-03-18 20:33:50
What typical high school courses are helpful as prerequisites for {MathILy, MathILy-Er} respectively?
What typical high school courses are helpful as prerequisites for {MathILy, MathILy-Er} respectively?
mjz6202007
2019-03-18 20:33:50
Around what grade level material are most of the problems centered around?
Around what grade level material are most of the problems centered around?
BeastAtMath12
2019-03-18 20:33:50
Are there any middle schoolers there?
Are there any middle schoolers there?
smbelcas
2019-03-18 20:34:07
We don't have prerequisites, or grade-level material. There are sometimes middle-schoolers.
We don't have prerequisites, or grade-level material. There are sometimes middle-schoolers.
SandyK
2019-03-18 20:34:17
@Allen I've gone to MathILy-Er twice. Oh yeah. Best experience of my life.
@Allen I've gone to MathILy-Er twice. Oh yeah. Best experience of my life.
Allen31415
2019-03-18 20:34:32
Are the questions in the program roughly the same level as the conjectures we just have proven?
Are the questions in the program roughly the same level as the conjectures we just have proven?
smbelcas
2019-03-18 20:34:40
OOh! great question!
OOh! great question!
smbelcas
2019-03-18 20:34:48
The problems in the program are much harder.
The problems in the program are much harder.
Potato12
2019-03-18 20:35:00
What is the percentage of applicants accepted to just Mathily?
What is the percentage of applicants accepted to just Mathily?
smbelcas
2019-03-18 20:35:17
Something like 20%, and then there's another selection process for MathILy-Er that's roughly the same.
Something like 20%, and then there's another selection process for MathILy-Er that's roughly the same.
ww1234
2019-03-18 20:35:21
Is the flavor of the camp more "fun" type that touch different variety of topics in discrete math? or is the camp more systematic and goes in depth like AoPS classes?
Is the flavor of the camp more "fun" type that touch different variety of topics in discrete math? or is the camp more systematic and goes in depth like AoPS classes?
smbelcas
2019-03-18 20:35:25
All of the above!
All of the above!
DOGSTREET1
2019-03-18 20:35:32
Do you get to choose what you learn?
Do you get to choose what you learn?
smbelcas
2019-03-18 20:35:41
During some parts of the program, yes; during other parts,no.
During some parts of the program, yes; during other parts,no.
phanithans1
2019-03-18 20:35:46
why will the percent of applicants passing be lower this year?
why will the percent of applicants passing be lower this year?
smbelcas
2019-03-18 20:35:57
More applications, and fewer of those are qualified.
More applications, and fewer of those are qualified.
mgrimalo
2019-03-18 20:36:01
What is the dorm situation at the college? (Room accompaniment, curfew, rule strictness, etc.)
What is the dorm situation at the college? (Room accompaniment, curfew, rule strictness, etc.)
smbelcas
2019-03-18 20:36:16
I don't know how to answer that. But it's kind of like college.
I don't know how to answer that. But it's kind of like college.
ww1234
2019-03-18 20:36:22
Are there lots of exercises and problems in addition to classroom introduction/discussion?
Are there lots of exercises and problems in addition to classroom introduction/discussion?
smbelcas
2019-03-18 20:36:39
This... we don't have that kind of classroom.
This... we don't have that kind of classroom.
smbelcas
2019-03-18 20:36:44
It's all mixed together.
It's all mixed together.
smbelcas
2019-03-18 20:36:50
We do problems in class.
We do problems in class.
ww1234
2019-03-18 20:36:52
What is the percentage of girls in the past?
What is the percentage of girls in the past?
smbelcas
2019-03-18 20:37:17
That's varied from year to year. Often it's near 1/3. I hope it will be more this year.
That's varied from year to year. Often it's near 1/3. I hope it will be more this year.
Potato12
2019-03-18 20:37:29
What is the cost of the program? And is there a significant amount of financial aid provided?
What is the cost of the program? And is there a significant amount of financial aid provided?
smbelcas
2019-03-18 20:37:56
$4600 and all financial aid is need-based. We expect that we'll be able to meet all demonstrated need, as we have in the past several years.
$4600 and all financial aid is need-based. We expect that we'll be able to meet all demonstrated need, as we have in the past several years.
ibtstrash_
2019-03-18 20:37:59
What level of math is taught in the programs?
What level of math is taught in the programs?
smbelcas
2019-03-18 20:38:02
college.
college.
smbelcas
2019-03-18 20:38:06
And sometimes graduate.
And sometimes graduate.
ww1234
2019-03-18 20:38:09
Does the dorm has AC? Are there different cafeteria open during the camp and different variety of food available?
Does the dorm has AC? Are there different cafeteria open during the camp and different variety of food available?
smbelcas
2019-03-18 20:38:28
At MathILy, there are room ACs and at MathILy-Er the weather is cool so no AC is needed.
At MathILy, there are room ACs and at MathILy-Er the weather is cool so no AC is needed.
smbelcas
2019-03-18 20:38:45
There is exactly one dining hall for each campus and it has lots of different interesting food.
There is exactly one dining hall for each campus and it has lots of different interesting food.
Allen31415
2019-03-18 20:38:50
Where and when can I take this EAR exam?
Where and when can I take this EAR exam?
del-math
2019-03-18 20:38:50
Is the exam the same every year, or does it change?
Is the exam the same every year, or does it change?
smbelcas
2019-03-18 20:39:04
You can take the EAR, new every year, by submitting a Short Form on the website.
You can take the EAR, new every year, by submitting a Short Form on the website.
BeastAtMath12
2019-03-18 20:39:12
Is the application process done for this year?
Is the application process done for this year?
smbelcas
2019-03-18 20:39:16
HA HA HA HA HA no.
HA HA HA HA HA no.
mandrake41
2019-03-18 20:39:19
How long will applications be open for
How long will applications be open for
smbelcas
2019-03-18 20:39:24
At least until April 23rd.
At least until April 23rd.
smbelcas
2019-03-18 20:39:29
And then after, if there are spaces left.
And then after, if there are spaces left.
Damalone
2019-03-18 20:39:38
How does this compare to Mathcamp?
How does this compare to Mathcamp?
smbelcas
2019-03-18 20:39:48
Good question. It's philosophically very different.
Good question. It's philosophically very different.
smbelcas
2019-03-18 20:40:00
Mathcamp has everything optional, and focuses on forming community.
Mathcamp has everything optional, and focuses on forming community.
smbelcas
2019-03-18 20:40:13
MathILy and MathILy-Er have everything required, and focus on learning a lot.
MathILy and MathILy-Er have everything required, and focus on learning a lot.
smbelcas
2019-03-18 20:40:20
We are sillier than Mathcamp is.
We are sillier than Mathcamp is.
ww1234
2019-03-18 20:40:27
How long will we hear back after submit the application?
How long will we hear back after submit the application?
smbelcas
2019-03-18 20:40:39
Usually about a week, but sometimes two or more during heavy admissions times.
Usually about a week, but sometimes two or more during heavy admissions times.
del-math
2019-03-18 20:40:46
Is there an age limit? I know you said 13-ish to 18-ish, but is there an age where you draw the line?
Is there an age limit? I know you said 13-ish to 18-ish, but is there an age where you draw the line?
smbelcas
2019-03-18 20:41:04
We tend not to draw lines except on chalkboards.
We tend not to draw lines except on chalkboards.
Potato12
2019-03-18 20:41:08
Where are the locations of the programs?
Where are the locations of the programs?
smbelcas
2019-03-18 20:41:12
MathILy: Bryn Mawr.
MathILy: Bryn Mawr.
smbelcas
2019-03-18 20:41:18
MathILy-Er: Bowdoin.
MathILy-Er: Bowdoin.
Allen31415
2019-03-18 20:41:25
If I got a 6 on the AIME, is it possible for me to make the program?
If I got a 6 on the AIME, is it possible for me to make the program?
smbelcas
2019-03-18 20:41:35
How would I know? I know nothing about contests or scoring them.
How would I know? I know nothing about contests or scoring them.
ww1234
2019-03-18 20:41:41
How different is MathIly compared with Promys?
How different is MathIly compared with Promys?
smbelcas
2019-03-18 20:41:46
Extremely different.
Extremely different.
smbelcas
2019-03-18 20:42:00
At PROMYS the focus is on number theory, and at MathILy the focus is on discrete math.
At PROMYS the focus is on number theory, and at MathILy the focus is on discrete math.
smbelcas
2019-03-18 20:42:30
At PROMYS there are problem sets done independently. At MathILy we do pretty much everything communally.
At PROMYS there are problem sets done independently. At MathILy we do pretty much everything communally.
ChickenAgent2227-_-
2019-03-18 20:42:34
do you teach at MathIly?
do you teach at MathIly?
smbelcas
2019-03-18 20:42:44
Yes, ChickenAgent. I've taught there every summer and I run the program.
Yes, ChickenAgent. I've taught there every summer and I run the program.
ww1234
2019-03-18 20:42:49
But depth are similar other than the topics?
But depth are similar other than the topics?
smbelcas
2019-03-18 20:42:52
Yes.
Yes.
mgrimalo
2019-03-18 20:42:56
Are there ever one-on-one sessions with the instructors? Maybe something like office hours?
Are there ever one-on-one sessions with the instructors? Maybe something like office hours?
smbelcas
2019-03-18 20:43:00
Yes, and no.
Yes, and no.
smbelcas
2019-03-18 20:43:11
That is, there are constantly one-on-one sessions, but not office hours.
That is, there are constantly one-on-one sessions, but not office hours.
smbelcas
2019-03-18 20:43:43
We talk to students approximately all the time. Well, actually, we listen to them. And then ask them questions after they've also talked to each other.
We talk to students approximately all the time. Well, actually, we listen to them. And then ask them questions after they've also talked to each other.
Potato12
2019-03-18 20:43:47
Should I wait to submit my application after I receive my AIME score or will it not make a big difference?
Should I wait to submit my application after I receive my AIME score or will it not make a big difference?
smbelcas
2019-03-18 20:44:02
If you're even asking that question, I think you maybe haven't read my earlier responses...
If you're even asking that question, I think you maybe haven't read my earlier responses...
aopsUserNY
2019-03-18 20:44:13
What is the daily schedule like?
What is the daily schedule like?
smbelcas
2019-03-18 20:44:22
Morning class: 4 hours.
Morning class: 4 hours.
smbelcas
2019-03-18 20:44:25
Evening class: 3 hours.
Evening class: 3 hours.
smbelcas
2019-03-18 20:44:31
Daily Gather in the late afternoon.
Daily Gather in the late afternoon.
smbelcas
2019-03-18 20:44:35
And meals. Three of them.
And meals. Three of them.
smbelcas
2019-03-18 20:44:41
Also optional bedtime stories.
Also optional bedtime stories.
gozomete
2019-03-18 20:44:47
what type of math do you do at mathily-er?
what type of math do you do at mathily-er?
smbelcas
2019-03-18 20:45:08
Very similar to what is done at MathILy, and yet totally different. That is, both programs do discrete math but do different parts of it.
Very similar to what is done at MathILy, and yet totally different. That is, both programs do discrete math but do different parts of it.
ChickenAgent2227-_-
2019-03-18 20:45:12
Do earlier applicants have higher priority?
Do earlier applicants have higher priority?
smbelcas
2019-03-18 20:45:40
We have rolling admissions, so not really. The only way in which earlier applicants have priority is if we were to run out of spots, which we never do before our "deadline."
We have rolling admissions, so not really. The only way in which earlier applicants have priority is if we were to run out of spots, which we never do before our "deadline."
spoamath321
2019-03-18 20:45:55
Do you stay there for 5 weeks?
Do you stay there for 5 weeks?
smbelcas
2019-03-18 20:46:03
I do. I live in the dorms with the students, as do all instructors.
I do. I live in the dorms with the students, as do all instructors.
Allen31415
2019-03-18 20:46:12
What are the dates of the program?
What are the dates of the program?
smbelcas
2019-03-18 20:46:33
Uhhh.... I think June 30 to August 3? Check the website (I would have to check in order to be sure).
Uhhh.... I think June 30 to August 3? Check the website (I would have to check in order to be sure).
ChickenAgent2227-_-
2019-03-18 20:46:36
how many days a week are there classes?
how many days a week are there classes?
smbelcas
2019-03-18 20:46:44
5.5 days per week.
5.5 days per week.
smbelcas
2019-03-18 20:47:15
We have morning/evening class and Daily Gather Monday--Friday, and morning class on Saturday, and then Life Seminar on Saturday afternoon.
We have morning/evening class and Daily Gather Monday--Friday, and morning class on Saturday, and then Life Seminar on Saturday afternoon.
cad314
2019-03-18 20:47:19
what do students do on the days without classes?
what do students do on the days without classes?
smbelcas
2019-03-18 20:47:44
Depends on the day. We have big activities on some of the days. On others people play games and stuff.
Depends on the day. We have big activities on some of the days. On others people play games and stuff.
smbelcas
2019-03-18 20:47:52
But people also play games and stuff on class days.
But people also play games and stuff on class days.
smbelcas
2019-03-18 20:47:53
So.
So.
StickyWashington
2019-03-18 20:47:56
I play saxophone. Would I have time and a place to practice every day?
I play saxophone. Would I have time and a place to practice every day?
smbelcas
2019-03-18 20:48:26
Time? Yes. Place? Depends on how private you want it to be. We usually have a practice room, and people sign it out for various times.
Time? Yes. Place? Depends on how private you want it to be. We usually have a practice room, and people sign it out for various times.
Waterfall1234
2019-03-18 20:48:29
What is the difference between Mathily and Mathily-er
What is the difference between Mathily and Mathily-er
smbelcas
2019-03-18 20:48:51
MathILy-Er is for students who are slightly younger in their mathematical development.
MathILy-Er is for students who are slightly younger in their mathematical development.
aopsUserNY
2019-03-18 20:48:58
Is there access to an athletic center?
Is there access to an athletic center?
smbelcas
2019-03-18 20:49:16
Yes, of sorts. Some of the parts can only be accessed with a staff member around.
Yes, of sorts. Some of the parts can only be accessed with a staff member around.
Allen31415
2019-03-18 20:49:20
Is there an electronic policy?
Is there an electronic policy?
smbelcas
2019-03-18 20:49:31
What is an electronic policy? One that we post on the internet?
What is an electronic policy? One that we post on the internet?
Damalone
2019-03-18 20:49:38
what are some advantages of this over other math camps?
what are some advantages of this over other math camps?
smbelcas
2019-03-18 20:49:45
That depends on who you are.
That depends on who you are.
smbelcas
2019-03-18 20:50:10
The different math programs fit different people well.
The different math programs fit different people well.
smbelcas
2019-03-18 20:50:39
We do inquiry-based learning at MathILy and MathILy-Er, so students who like to invent math and discuss their ideas are well suited to our approach.
We do inquiry-based learning at MathILy and MathILy-Er, so students who like to invent math and discuss their ideas are well suited to our approach.
smbelcas
2019-03-18 20:50:45
Students who are gooftastic also tend to fit in.
Students who are gooftastic also tend to fit in.
Potato12
2019-03-18 20:51:08
Will there be shuttles to take people from the airport to the site?
Will there be shuttles to take people from the airport to the site?
smbelcas
2019-03-18 20:51:17
No, just cars and trains and vans.
No, just cars and trains and vans.
smbelcas
2019-03-18 20:51:23
We don't have access to the space program.
We don't have access to the space program.
ww1234
2019-03-18 20:52:49
Will the program help to pickup kids flying unaccompanied minor?
Will the program help to pickup kids flying unaccompanied minor?
smbelcas
2019-03-18 20:52:55
As much as we can, yes!
As much as we can, yes!
smbelcas
2019-03-18 20:53:14
Usually if someone needs to fly unaccompanied minor, our Minion works with the family to arrange things in advance.
Usually if someone needs to fly unaccompanied minor, our Minion works with the family to arrange things in advance.
PracticingMath
2019-03-18 20:53:23
Do they provide resources, for example food?
Do they provide resources, for example food?
smbelcas
2019-03-18 20:53:33
Does *who* provide resources?
Does *who* provide resources?
Potato12
2019-03-18 20:54:04
If I submit within the next three days, will I get my decision back within a week or is this a busy time?
If I submit within the next three days, will I get my decision back within a week or is this a busy time?
smbelcas
2019-03-18 20:54:15
That depends on how easy the decision is
That depends on how easy the decision is
phanithans1
2019-03-18 20:54:26
what about unacomppannied major
what about unacomppannied major
smbelcas
2019-03-18 20:54:41
That's a pretty good question. I think there is a piano in our dorm, or maybe two?
That's a pretty good question. I think there is a piano in our dorm, or maybe two?
smbelcas
2019-03-18 20:54:48
So no need for unaccompanied majors.
So no need for unaccompanied majors.
casi
2019-03-18 20:55:18
What's the minimum age?
What's the minimum age?
smbelcas
2019-03-18 20:55:34
We don't do minimums, or deadlines, or (for the most part) rules.
We don't do minimums, or deadlines, or (for the most part) rules.
smbelcas
2019-03-18 20:55:42
We just do stuff that makes sense.
We just do stuff that makes sense.
smbelcas
2019-03-18 20:56:09
I think I might have missed some questions earlier.
I think I might have missed some questions earlier.
smbelcas
2019-03-18 20:56:21
If I didn't answer something you tried to ask, please try again now!
If I didn't answer something you tried to ask, please try again now!
cooljoseph
2019-03-18 20:56:43
How hard are the classes?
How hard are the classes?
smbelcas
2019-03-18 20:56:54
Super-hard! Also not that hard.
Super-hard! Also not that hard.
bkim0325
2019-03-18 20:56:57
will we be doing original research?
will we be doing original research?
smbelcas
2019-03-18 20:57:18
No. That's for research programs. I don't think we've ever had a student who arrived qualified to do research.
No. That's for research programs. I don't think we've ever had a student who arrived qualified to do research.
pad
2019-03-18 20:57:25
where can I find the ear for mathily?
where can I find the ear for mathily?
smbelcas
2019-03-18 20:57:28
You can't!
You can't!
cooljoseph
2019-03-18 20:57:33
Hard is relative. What do you mean by hard? Is it harder than AIME level questions?
Hard is relative. What do you mean by hard? Is it harder than AIME level questions?
smbelcas
2019-03-18 20:57:43
I know nothing about math contests.
I know nothing about math contests.
Potato12
2019-03-18 20:57:52
Since it's 20% for mathily and 20% for mathilyear, does that mean there is about a 40% admittance rate to either program?
Since it's 20% for mathily and 20% for mathilyear, does that mean there is about a 40% admittance rate to either program?
smbelcas
2019-03-18 20:58:01
There are a lot of ways of measuring.
There are a lot of ways of measuring.
smbelcas
2019-03-18 20:58:21
Because the missions of the programs are slightly different, it's not quite additive.
Because the missions of the programs are slightly different, it's not quite additive.
Damalone
2019-03-18 20:58:36
is this a good way to prepare myself for a research program?
is this a good way to prepare myself for a research program?
smbelcas
2019-03-18 20:58:39
Yes!
Yes!
smbelcas
2019-03-18 20:58:55
It's an excellent way, because we structure our classrooms to be research-like experiences.
It's an excellent way, because we structure our classrooms to be research-like experiences.
Potato12
2019-03-18 20:58:59
If we get rejected but there are spots open at the end or if someone drops out, can we still join?
If we get rejected but there are spots open at the end or if someone drops out, can we still join?
smbelcas
2019-03-18 20:59:18
Nope. If you're not admitted, then it's generally because you're not qualified this year.
Nope. If you're not admitted, then it's generally because you're not qualified this year.
cad314
2019-03-18 20:59:41
is there a waitlist?
is there a waitlist?
smbelcas
2019-03-18 20:59:46
If we fill up near the "deadline", we will make a waitlist.
If we fill up near the "deadline", we will make a waitlist.
smbelcas
2019-03-18 21:00:10
So far we've managed to avoid an actual waitlist by asking people to make decisions more quickly.
So far we've managed to avoid an actual waitlist by asking people to make decisions more quickly.
Potato12
2019-03-18 21:00:13
So there's no advantage towards applying early?
So there's no advantage towards applying early?
smbelcas
2019-03-18 21:00:18
You get a decision earlier!
You get a decision earlier!
Potato12
2019-03-18 21:00:43
I mean like application-wise, is there a higher chance we will be admitted?
I mean like application-wise, is there a higher chance we will be admitted?
smbelcas
2019-03-18 21:00:46
Nope.
Nope.
smbelcas
2019-03-18 21:01:34
Okay! Any last questions?
Okay! Any last questions?
Waterfall1234
2019-03-18 21:02:31
Would the program be too hard for people who aren't very experienced in proofs?
Would the program be too hard for people who aren't very experienced in proofs?
smbelcas
2019-03-18 21:02:58
That depends on the person. Some people have no experience in proofs and do really well. Others can't handle it. That's why we have the EAR!
That depends on the person. Some people have no experience in proofs and do really well. Others can't handle it. That's why we have the EAR!
gozomete
2019-03-18 21:03:01
Is Mathil-er online?
Is Mathil-er online?
smbelcas
2019-03-18 21:03:05
Nope, it's in person.
Nope, it's in person.
smbelcas
2019-03-18 21:04:20
Okay, I'm not hearing any more questions, so I think that's about it!
Okay, I'm not hearing any more questions, so I think that's about it!
smbelcas
2019-03-18 21:04:30
Thanks for coming to the {MathILy, MathILy-Er} Math Jam.
Thanks for coming to the {MathILy, MathILy-Er} Math Jam.
StickyWashington
2019-03-18 21:04:33
Thanks for the class!
Thanks for the class!
smbelcas
2019-03-18 21:04:37
You are super welcome!
You are super welcome!
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