Chasing%20Snakes%20in%20N-Space%20Cubes

1
Chasing Snakes in N-Space Cubes
N-space Snakes are special maximal length loops
through an N-space cube. Theyre full of
intriguing symmetries, puzzles and surprises.
Theyre simple structures that baffle us with
their complexities. Fascinating
creatures. Lets go find some Snakes.
2
Chasing Snakes in N-Space Cubes
In this session Well define what a Snake
is, Search for 3,4, and 5-space snakes by
hand, Identify snakes with binary names, Identify
snakes by their column changes, Find the unique
snakes up through 6-space, Look at a snakes
physique-l makeup, and ask some questions that
maybe you will answer.
3
Chasing Snakes in N-Space Cubes
Me.
So, what IS an N-space Snake?
4
A Snake is a closed path (loop) through an
N-space cube. But, the path must follow one
special rule. You must understand that rule in
order to create valid snakes.
000
100
010
001
110
101
011
111
The green lines form a valid 3-space snake of
length 6.
5
That special rule is No point on a snake (other
than the preceding and succeeding points on the
snake) can be within one line length of any
other point on the snake.
000
100
010
001
110
101
011
111
This is an invalid snake because point 011 is one
length away from 010, and both points are already
part of the snake.
6
Every point (b) on the snake has one point that
comes before it (a), and one that comes after it
(c). Points a and c are one length away from b.
a
000
b
100
010
001
c
110
101
011
111
No other point on the snake can be just one
length away from point b. If it is, the snake is
invalid. Thats the case here. Point 101 is
one length away from point 001.
7
A point is adjacent to another if it is one
line length away. The adjacents of a point are
those points that are one line length away.
a
000
b
100
010
001
c
d
110
101
011
111
The points a, c, and d are adjacent to point
b. a, c, and d are the adjacents of point b.
8
We will be looking for maximal length snakes
which I call Great Snakes. The snake shown here
is valid, but is not a Great Snake because it is
not the longest snake possible in 3-space.
a
000
b
100
010
001
c
d
110
101
011
111
This is a valid 3-space snake of length 4. It is
not a maximal length (Great) snake.
9
The longest snake possible in a 3-space cube is a
snake of length 6.
a
000
b
100
010
001
c
d
110
101
011
111
This is a valid 3-space Great Snake.
10
The longest snake in a 4-space cube is of length
8. You may wish to print this page and try to
find a 4-space Great Snake on your own.
11
This is an invalid 4-space snake. Do you see why?
0000
1000
0100
0010
0001
1100
1010
0110
1001
0011
0101
1110
1101
1011
0111
1111
12
It is invalid because points 0010 and 0110 (which
are already on the snake) are within one line
length of each other.
0000
1000
0001
0010
0100
1100
1010
1001
0011
0101
0110
1110
1101
1011
0111
1111
13
Do you see why this snake is invalid?
0000
1000
0001
0010
0100
1100
1010
1001
0011
0101
0110
1110
1101
1011
0111
1111
14
Actually, there are two problems here. The point
1010 is adjacent to both 0010 and 1011 which are
part of the snake.
0000
1000
0001
0010
0100
1100
1010
1001
0011
0101
0110
1110
1101
1011
0111
1111
15
Is this a valid 4-space snake? Is it a Great
Snake?
0000
1000
0001
0010
0100
1100
1010
1001
0011
0101
0110
1110
1101
1011
0111
1111
16
This snake is a valid 4-space Great Snake.
0000
1000
0001
0010
0100
1100
1010
1001
0011
0101
0110
1110
1101
1011
0111
1111
17
Heres another 4-space Great Snake. From now on,
when I say snake, I will usually be talking
about Great Snakes.
0000
1000
0001
0010
0100
1100
1010
1001
0011
0101
0110
1110
1101
1011
0111
1111
18
Chasing Snakes in N-Space Cubes
Find me...
Once you know the rules for finding a snake, it
is trivial to find a 3-space snake and easy to
find a 4-space snake. 5-space snakes take a
little more work, although most people can find
several without too much trouble. Give it a
try...
19
A 5-space cube Maximal length snake 14
20
Heres a 5-space Great Snake
10000
00100
00001
10100
10010
00011
11000
01100
01010
00101
01001
00111
11100
10110
11001
01011
10111
01111
11111
21
Chasing Snakes in N-Space Cubes
To become more familiar with our snakes, we have
to uniquely identify them . We have to name them.
My name is Joe Finklesnake III
22
One way to name a snake is to list the points
that make up the snake. They must be listed in
order otherwise they wont be a valid snake.
0000 0001 0011 0111 1111 1110 1100 1000
23
But since there is no head or tail to the snake,
you can start anywhere on the snake, and list the
points as you follow the path back to your
starting point.
0000 0001 0011 0111 1111 1110 1100 1000
1111 0111 0011 0001 0000 1000 1100 1110
24
Although the two lists are different, they are
really the same snake. They just start at
different points and go in opposite directions.
Start
0000 0001 0011 0111 1111 1110 1100 1000
1111 0111 0011 0001 0000 1000 1100 1110
Start
25
So a single snake can have many different binary
names. Since these particular lists appear to
rotate vertically, they are called vertical
rotations of each other.
0000 0001 0011 0111 1111 1110 1100 1000
0011 0111 1111 1110 1100 1000 0000 0001
0111 1111 1110 1100 1000 0000 0001 0011
1111 1110 1100 1000 0000 0001 0011 0111
1110 1100 1000 0000 0001 0011 0111 1111
1100 1000 0000 0001 0011 0111 1111 1110
1000 0000 0001 0011 0111 1111 1110 1100
0001 0011 0111 1111 1110 1100 1000 0000
26
Pretend that our 4-cube is a round transparent
Christmas tree ornament suspended by a red ribbon
from the 0000 point.
There are other rotations too.
0000 0001 0011 0111 1111 1110 1100 1000
27
If we slowly twirl the ornament, some of the
points would appear to change places with other
points on the same level and the snake would
appear to move around the ornament.
0000 0001 0011 0111 1111 1110 1100 1000
28
If you twirled just the snake, and not the
ornament, you could make an intuitive leap and
call the resulting snakes horizontal rotations
of each other.
0000 0001 1001 1101 1111 1110 0110 0010
0000
1000
0001
0010
0100
1100
1010
1001
0011
0101
0110
1110
1101
1011
0111
1111
29
Rotate the 4 Column to the right hand side.
Rotated Snake
Old Snake
Columns
4321
4
3
2
1
4
3
2
1
3214
0000 0001 0011 0111 1111 1110 1100 1000
0 0 0 0 1 1 1 1
0 0 0 1 1 1 1 0
0 0 1 1 1 1 0 0
0 1 1 1 1 0 0 0
0 0 0 0 1 1 1 1
0 0 0 1 1 1 1 0
0 0 1 1 1 1 0 0
0 1 1 1 1 0 0 0
0000 0010 0110 1110 1111 1101 1001 0001
The horizontally rotated list of points looks
very different, so you might think that you have
a new, different snake. But, its really the
same old snake rotated.
30
Horizontally inter-mixed Columns
New Snake
Old Snake
Columns
4321
4
3
2
1
4
3
2
1
2431
0000 0001 0011 0111 1111 1110 1100 1000
0 0 0 0 1 1 1 1
0 0 0 1 1 1 1 0
0 0 1 1 1 1 0 0
0 1 1 1 1 0 0 0
0 0 0 0 1 1 1 1
0 0 0 1 1 1 1 0
0 0 1 1 1 1 0 0
0 1 1 1 1 0 0 0
0000 0001 1001 1011 1111 1110 0110 0100
In fact, if you exchange any column of a given
snake with any other column of the same snake,
you have an intermixed rotation of the snake, and
it is really the same snake as before even
though the list of points is very different.
31
Chasing Snakes in N-Space Cubes
You can call me Joe
There are other intriguing ways to name our
snakes.
My name is Joe Finklesnake III
32
This picture shows colored linesets as well as
points of a 4-space cube.
0000
1
2
3
4
2
31
4
1
3
421
42
3
3
42
421
3
1
31
2
4
4
3
2
1
1111
33
0000 0001 001101111111111011001000
0000
1
2
3
4
1000
0100
0010
0001
Instead of using the points to name the snake, we
can use the column number between each of the
snakes 8 points. This snakes name would then
be 1 2 3 4 1 2 3 4
2
31
4
1
3
421
42
3
1100
1010
0110
1001
0011
0101
3
42
421
3
1
31
2
4
1110
1101
1011
0111
4
3
2
1
1111
34
Snake named by its points
Snake named by column changes
Columns
4321
4
3
2
1
0000 0001 0011 0111 1111 1110 1100 1000 0000
0 0 0 0 1 1 1 1 0
0 0 0 1 1 1 1 0 0
0 0 1 1 1 1 0 0 0
0 1 1 1 1 0 0 0 0
1 2 3 4 1 2 3 4
It turns out that the column-change naming
convention is a more effective, efficient,
easy method of naming snakes. And it highlights
something we might not have seen otherwise.
35
This snake appears to be made from two
identical halves.
Snake named by its points
Snake named by column changes
4321
0000 0001 0011 0111 1111 1110 1100 1000 0000
1 2 3 4 1 2 3 4
1 2 3 4 and 1 2 3 4
The column-change naming convention reveals
structures within the snake that we did not
expect to find.
36
Now, we can name this 5-space snake two different
ways.
00000
Binary snake name
Column-change name
00000 0001000110 01110 11110 11010 11011 10011
10001 10101 11101 01101 01001 01000
2 3 4 5 3 1 4 2 3 4 5 3 1 4
10000
01000
00100
00010
00001
10100
10010
00110
01001
00011
11000
01100
01010
10001
00101
11010
01110
10101
10011
00111
11100
10110
11001
01101
01011
11110
11101
11011
10111
01111
11111
37
A 5-space Cube
Symmetry, symmetry, everywhere and what a lot to
think.
A 4-space cube
00000
10000
01000
00100
00010
00001
10100
10010
00110
01001
00011
11000
01100
01010
10001
00101
11010
01110
10101
10011
00111
11100
10110
11001
01101
01011
11110
11101
11011
10111
01111
2345314 2345314
A 4-space cube
11111
38
5-space cube
This gives us a clue as to how we might construct
N-space snakes from (N-1)-space snakes.
A 4-space cube
00000
10000
01000
00100
00010
00001
10100
10010
00110
01001
00011
11000
01100
01010
10001
00101
11010
01110
10101
10011
00111
11100
10110
11001
01101
01011
11110
11101
11011
10111
01111
23453142345314
A 4-space cube
11111
39
Chasing Snakes in N-Space Cubes
Just how big do these snakes get?
40
We dont know how big they are above 7-space.
This Big
0-space 0 1-space 1 2-space 4 3-space 6
4-space 8 5-space 14 6-space 26 7-space 48
41
Chasing Snakes in N-Space Cubes
Now, it might be informative to catalog all of
the snakes in an N-space cube to see how each of
them is constructed. That could give us a clue
as to how to construct snakes in higher N-space
cubes. However, a lot of the snakes are just
transformations of each other. The N-cubes
appear to be infested with snakes!
42
Chasing Snakes in N-Space Cubes
If we throw out all of the duplicate snakes, how
many are left? How many UNIQUE snakes are there
in each N-cube?
43
Chasing Snakes in N-Space Cubes
First, you have to find them all. How do you do
that? One way is to write a computer program
that exhaustively searches for them. I wrote
one and named it TailWagger
44
You could find all of the snakes in an N-space
cube if you tried all of the possible paths.
This is called the BFI or Brute Force and
Ignorance method.
0000
1000
0100
0010
0001
1100
1010
0110
1001
0011
0101
1110
1101
1011
0111
1111
45
TailWagger starts at point 0000. It chooses one
of four possible points. It then has three more
choices, chooses one and checks to see if the
snake has violated any rules.
0000
1000
0100
0010
0001
1100
1010
0110
1001
0011
0101
1110
1101
1011
0111
1111
46
If TailWagger chooses a point that violates a
rule, it backtracks and tries one of the other
points.
0010 would have to link with 0000 but the snake
is still too small.
0000
1000
0100
0010
0001
1100
1010
0110
1001
0011
0101
1110
1101
1011
0111
1111
47
If no rules have been violated, it continues
choosing new points. If all three choices violate
a rule, it backtracks to the previous point and
chooses another point there.
0000
1000
0100
0010
0001
1100
1010
0110
1001
0011
0101
1110
1101
1011
0111
1111
48
When it finds a valid snake it prints it out.
Then it backtracks (as if it had found an error)
and chooses other points that havent been tried.
0000
1000
0100
0010
0001
1100
1010
0110
1001
0011
0101
1110
1101
1011
0111
1111
49
Eventually, it backtracks all the way to the
third node where the program stops. Do you see
why it isnt necessary to backtrack to the first
point to try all of the possibilities there?
0000
1000
0100
0010
0001
1100
1010
0110
1001
0011
0101
1110
1101
1011
0111
1111
50
Once TailWagger found all of the snakes (up
through 6-space) all of the duplicate snakes had
to be thrown out in order to determine the number
of unique snakes and their composition.
X
X
X
The matter required a bit of careful thought.
51
Are these two snakes the same?
1 2 3 4 1 2 4 3
1 2 4 3 1 2 3 4
They are if the second snake is a vertical,
horizontal, or intermixed rotation of the
first snake.
52
Yes, the second snake is a rotation of the first.
1 2 3 4 1 2 4 3 1 2 3 4 1 2 4 3
1 2 4 3 1 2 3 4
Here, we duplicated the first snake (red numbers)
and shifted the second snake to the right. The
numbers match. The snakes are the same.
53
Are these two snakes the same?
1 2 3 4 1 2 4 3
3 2 1 3 4 2 1 4
They are if the second snake is a vertical,
horizontal, or intermixed rotation of the
first snake.
54
Yes, the second snake is a rotation of the first.
1 2 3 4 1 2 4 3 1 2 3 4 1 2 4 3
4 1 2 4 3 1 2 3
Here, we duplicated the first snake (red
numbers), turned the second snake around
(32134214 to 41243123) and shifted the second
snake to the right. The numbers match. The
snakes are the same.
55
Are these two snakes the same?
1 2 3 4 1 2 4 3
4 1 3 4 2 1 3 2
They are if the second snake is a vertical,
horizontal, or intermixed rotation of the
first snake.
56
Yes, the second snake is a shifted, inter-mixed
rotation of the first.
1 2 3 4 1 2 4 3 1 2 3 4 1 2 4 3 first snake
4 1 3 4 2 1 3 2 second snake
4 1 2 4 3 1 2 3 second snake with 3s and 2s
swapped
4 1 2 4 3 1 2 3 second snake shifted right
1 2 3 4 1 2 4 3 1 2 3 4 1 2 4 3 first snake
In the second snake we changed every 2 to a 3 and
every 3 to a 2. Then we shifted it to the
right. The numbers match. The snakes are the same.
57
Chasing Snakes in N-Space Cubes
Im from the class of 65
I promised you a third way to name snakes.
My name is Joe Finklesnake III
58
Snakes can be partially described by using the
following trick.
2345314234531423453142345314 .....1......1......1.
.....1. 2......2......2......2...... .3..3...3..3.
..3..3...3..3.. ..4...4..4...4..4...4..4...4 ...5.
.....5......5......5...
1 7 7 1 occurs every 7th number 2 7 7 2 occurs
every 7th number 3 3 4 3 4 3 occurs every 3rd,
4th, 3rd, 4th number 4 4 3 4 3 4 occurs every
4th, 3rd, 4th, 3rd number 5 7 7 5 occurs every
7th number
59
Because transformations or rotations of snakes
are equivalent, the following two snakes are in
the same class. The are the same snake.

Snake 1 2 3 4 5 3 1 4 2 3 4 5 3 1 4 Snake 2 2 5 4
3 5 1 4 2 5 4 3 5 1 4
Snake 1 1 7 7 2 7 7 3 3 4 3 4 4 4 3 4 3 5 7 7
Snake 2 1 7 7 2 7 7 3 7 7 4 4 3 4 3 5 3 4 3 4
60
In order to unmask the unique snakes, every
snake in an N-space cube must be compared to
every other snake in the N-space cube to see
whether they are forward, backward
(vertical) and / or intermixed rotations
of each other. Will the Real
Unique Snakes Please Step
forward ?
61
These are unique snakes for N lt 7.
3-space 1 2 3 1 2 3 4-space 1 2 3 4 1 2 3
4 1 2 3 4 1 2 4 3 1 2 3 4 2 1 4 3
5-space 1 2 3 4 5 2 1 4 2 3 4 5 2 4 1 2
3 4 5 2 3 1 2 4 3 2 5 3 1 2 3 4 5 2 4 1 2 3
4 5 2 4 6-space 1 2 3 4 5 6 1 2 5 4 1 5 6 1 2 3
6 5 4 1 2 5 6 1 5 4 1 2 3 4 5 6 1 2 5 4 2 3 4 1
2 5 4 3 6 1 2 3 4 2 5 4 1 2 3 4 5 6 3 4 2 3 5 4
1 5 3 6 2 5 6 4 3 5 6 2 5 3 1 2 3 4 5 6 3 4 2 3
5 4 3 1 2 3 4 5 6 3 4 2 3 5 4 3 1 2 3 4 5 6 3 4
2 3 5 4 3 1 2 4 3 5 6 3 4 2 3 5 4 3
62
How long did it take to find every snake in
7-space? About 30 years. Why so long?
3-space 36 7.2x102 729 4-space 48
6.5x104 65536 5-space 514 6.1109
6103515625 6-space 626 1.71020
170581728179578208256 7-space 748
3.6x1040 3670336821729412544123021103203366018
8801
63
So, weve come to the end with lots of
questions. How long are the snakes in any N-space
cube? What are the unique snakes in an N-space
cube? What governs the construction of
snakes? Are there equations that describe all
of these things?
We dont know yet.