An ambitious plan

Laterally Speaking · 2008-02-12 06:30:52

I think the lowest known lower bound for the solution is 6, so it could be said that the premise of G is rather flawed, but it is an upper bound to the problem.

However, I don't quite think you've got the entire concept; then again, nobody really does.

John E. Franklin · 2008-02-12 07:42:41

I'm surprised you doubt me.
I am correct you know.

John E. Franklin · 2008-02-12 13:34:34

@ganesh, I gotta thank you man, for
getting me to think about these hypercubes.
I have thought about them for many years
since my final project on logic circuit minimization.
The hypercube is analogous to the Karnaugh
map, and hence, that is why I have thought
about it so much. Great Fun!!

John E. Franklin · 2008-02-13 13:38:28

So anyway, what makes this easy for me to see is
to imagine the larger and larger hypercubes, not
in higher dimensions, but simply in one linear dimension,
like a number line. Just put the new half of the
hypercube to the right on the number line and
imagine connecting together the identical points
from each half together. So for example,
1,2,3,4 get connected to 5,6,7,8, respectively, I just imagine
these connected lines as 4 rainbows that connect
to the number that is 4 higher. It's all just
kindergarten stuff, really. I'll explain more later.
~ means connected by a line.
1-D consists of 2 points: 1~2
2-D consists of 4 points: keep 1-D points, clones(by adding 2 which is sizeof(1-D)) it to 3~4 and adds lines 1~3, 2~4
3-D consists of 8 points: keeps 2-D, clones(by adding 4, which is sizeof(2-D)) it to 5~6 7~8 5~7 6~8 and adds lines 1~5, 2~6, 3~7, 4~8.
4-D consists of 16 points: keeps 3-D, clones it to (by adding 8, which
is sizeof(3-D)) 9~10 11~12 9~11 10~12 13~14 15~16 13~15 14~16
9~13 10~14 11~15 12~16 and add lines
between cloned and new halves with 1~9 2~10 3~11
4~12 5~13 6~14 7~15 8~16.

Now the fun comes when you start trying out examples
of creating the squares or 4 points in a plane. After
trying examples, I think you will eventually see the
trick to this, which leads me to say that the additional
connecting lines added at each higher dimension can be
any color at all, as long as the cloned part is the negative
color to the less dimension. Have fun working on this.
Start on paper, and try to do it in your head without
paper, because that helps, once you understand what's
going on.
Anyway, there are probably other ways to color these
lines between points as well, but that is not part of my
method. My method is just a subset of the ways to
color the hypercubes that ensure no square where the
square's 4 lines are the same color.
Have fun!!! It's all about playing like a child on paper
and then in your head perhaps. I figured it out in the
shower.

Last edited by John E. Franklin (2008-02-23 09:15:55)

John E. Franklin · 2008-02-15 06:57:27

One thing that you have to come to believe before this
makes sense is that when you are making squares,
say you have a 75-dimension hypercube. Now say
for your first edge, you traverse the 67th dimension
in the the 67th's positive direction, then at the other
side you can go traverse say the 56th dimension in the
positive direction. Now to complete this square
perimeter traversal, you are forced into taking the 67th
dimension in the negative direction, and then the
56th dimension in the negative direction. Coming to
believe this notion is at the root of the simplicity of
the problem. Proving is harder than believing, however.

John E. Franklin · 2008-02-23 09:34:44

Additionally, it might be fun to examine this structure
with binary numbers instead of the natural counting
numbers.
Here I repeat from before in green:

~ means connected by a line.
1-D consists of 2 points: 1~2
2-D consists of 4 points: keep 1-D points, clones(by adding 2 which is sizeof(1-D)) it to 3~4 and adds lines 1~3, 2~4
3-D consists of 8 points: keeps 2-D, clones(by adding 4, which is sizeof(2-D)) it to 5~6 7~8 5~7 6~8 and adds lines 1~5, 2~6, 3~7, 4~8.
4-D consists of 16 points: keeps 3-D, clones it to (by adding 8, which
is sizeof(3-D)) 9~10 11~12 9~11 10~12 13~14 15~16 13~15 14~16
9~13 10~14 11~15 12~16 and add lines
between cloned and new halves with 1~9 2~10 3~11
4~12 5~13 6~14 7~15 8~16.

...
But now instead let's subtract 1 from all the
numbers above and convert it to binary.
This will create a nice situtation where connecting
lines between vertices connect two points
whose numbers only differ in one digit. chiffre

So now in binary below:
~ means connected by a line.
1-D consists of 2 points: 0~1
2-D consists of 4 points: keep 1-D points, clones(by adding 2 which is sizeof(1-D)) it to 10~11 and adds lines 0~10, 1~11
3-D consists of 8 points: keeps 2-D, clones(by adding 4, which is sizeof(2-D)) it to 100~101 110~111 100~110 101~111 and adds lines
0~100, 1~101, 10~110, 11~111.
4-D consists of 16 points: keeps 3-D, clones it to (by adding 8, which
is sizeof(3-D)) 1000~1001 1010~1011 1000~1010 1001~1011 1100~1101 1110~1111 1100~1110 1101~1111
1000~1100 1001~1101 1010~1110 1011~1111 and add lines
between cloned and new halves with 0~1000 1~1001 10~1010
11~1011 100~1100 101~1101 110~1110 111~1111.

Last edited by John E. Franklin (2008-02-23 09:42:08)

Math Is Fun Forum

#26 2008-02-12 06:30:52

Re: An ambitious plan

#27 2008-02-12 07:42:41

Re: An ambitious plan

#28 2008-02-12 13:34:34

Re: An ambitious plan

#29 2008-02-13 13:38:28

Re: An ambitious plan

#30 2008-02-15 06:57:27

Re: An ambitious plan

#31 2008-02-23 09:34:44

Re: An ambitious plan

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