Drorbn - User contributions [en]

10-1100/Assignment 1 Part 1 by Charles

2010-10-11T19:54:51Z

Tsangcharles:

'''Pyraminx''' by Charles Tsang

In this article, we will analyze Pyraminx, a variant of the Rubik's cube. We will restrict our attention to those permutations which can only be generated by turning the sides of the puzzle (not by “cheating” and reassembling the puzzle).

As an introduction, we shall describe the shape of the puzzle. It is shaped as a tetrahedron. i.e. It has 4 faces at which each face is a equilateral triangle with a different solid colour. And on each face, it is composed of 9 smaller equilateral triangles. The following are pictures of the puzzle and its decomposition into a net.

[[Image:10-1100-pyraminx.jpg]]
[[Image:10-1100-240px-Tetrahedron_fla1t.jpg]]

The faces are coloured green, blue, orange and yellow, and each smaller equilateral triangle is labelled with numbers.

Now we are ready to play!

To understand the puzzle, we need to observe what happens to the numbers when we turn each face.

When we turn the green face [“clockwise” with respect to the net], the number 1 is now in place of 5, and 5 goes to 25 etc... (At first this seems easy but because this is a tetrahedron, it is actually more complicated than it seems. So to aid this process, I actually created several nets on different sheets of papers and created a tetrahedron to keep track of the permutations.) We write this as a permutaiton : [25,14,13,2,1,11,12,23,24,35,36,26,15,4,3,9,10,18,19,20,21,22,27,16,5,7,8,28,29,30,31,32,33,34,17,6 ]
In this notation, the i'th term in the list correspond to the iverse image of number i under turning the green face.

Repeat the same procedure for blue, orange and yellow, we can see that they represent the permutations
[1,2,3,35,36,26,25,14,9,10,11,12,13,33,34,28,27,16,15,4,21,22,23,24,32,30,29,18,17,6,5,7,8,19,20,31]
[6,17,18,29,30,32,31,20,19,8,7,5,13,14,15,16,33,22,21,10,9,3,4,24,25,26,27,28,23,12,11,1,2,34,35,36] and
[31,20,3,4,5,6,7,8,9,29,30,32,21,10,15,16,17,18,19,27,28,34,33,22,11,1,2,13,14,25,26,36,35,24,23,12]
respectively.

Then after running the Schreier–Sims algorithm in Mathematica, we have found that there are 3,732,480 different configurations by turning each face of the puzzle. More specifically, if we restrict ourselves into only turning one side in the beginning, we get 3 different configurations, and if we restrict ourselves into turn two sides, we get 19440 configurations, which we can readily see from the output.

Here is the input/output log for those who are interested:

[[Image:10-1100-charlesinputoutput.jpg]]

I hope everyone has enjoyed it as much as I did analyzing it!

10-1100/Assignment 1 Part 1 by Charles

2010-10-11T19:41:05Z

Tsangcharles:

10-1100/Assignment 1 Part 1 by Charles

2010-10-11T19:37:10Z

Tsangcharles:

'''Pyraminx''' by Charles Tsang

In this article, we will analyze Pyraminx, a variant of the Rubik's cube. We will restrict our attention to those permutations which can only be generated by turning the sides of the puzzle (not by “cheating” and reassembling the puzzle).

As an introduction, we shall describe the shape of the puzzle. It is shaped as a tetrahedron. i.e. It has 4 faces at which each face is a equilateral triangle with a different solid colour. And on each face, it is composed of 9 smaller equilateral triangles. The following are pictures of the puzzle and its decomposition into a net.

[[Image:10-1100-pyraminx.jpg]]
[[Image:10-1100-240px-Tetrahedron_fla1t.jpg]]

The faces are coloured green, blue, orange and yellow, and each smaller equilateral triangle is labelled with numbers.

Now we are ready to play!

To understand the puzzle, we need to observe what happens to the numbers when we turn each face.

When we turn the green face(“clockwise” with respect to the net), the number 1 is now in place of 5, and 5 goes to 25 etc... (At first this seems easy but because this is a tetrahedron, it is actually more complicated than it seems. So to aid this process, I actually created several nets on different sheets of papers and created a tetrahedron to keep track of the permutations.) We write this as a permutaiton : [25,14,13,2,1,11,12,23,24,35,36,26,15,4,3,9,10,18,19,20,21,22,27,16,5,7,8,28,29,30,31,32,33,34,17,6 ]
In this notation, the i'th term in the list correspond to the iverse image of number i under turning the green face.

Repeat the same procedure for blue, orange and yellow, we can see that they represent the permutations
[1,2,3,35,36,26,25,14,9,10,11,12,13,33,34,28,27,16,15,4,21,22,23,24,32,30,29,18,17,6,5,7,8,19,20,31]
[6,17,18,29,30,32,31,20,19,8,7,5,13,14,15,16,33,22,21,10,9,3,4,24,25,26,27,28,23,12,11,1,2,34,35,36] and
[31,20,3,4,5,6,7,8,9,29,30,32,21,10,15,16,17,18,19,27,28,34,33,22,11,1,2,13,14,25,26,36,35,24,23,12]
respectively.

Then after running the Schreier–Sims algorithm in Mathematica, we have found that there are 3,732,480 different configurations by turning each face of the puzzle. More specifically, if we restrict ourselves into only turning one side in the beginning, we get 3 different configurations, and if we restrict ourselves into turn two sides, we get 19440 configurations, which we can readily see from the output.

Here is the input/output log for those who are interested:

[[Image:10-1100-charlesinputoutput.jpg]]

I hope everyone has enjoyed it as much as I did analyzing it!

File:10-1100-charlesinputoutput.jpg

2010-10-11T19:37:04Z

Tsangcharles:

File:10-1100-pyraminx.jpg

2010-10-11T19:36:08Z

Tsangcharles:

File:10-1100-240px-Tetrahedron fla1t.jpg

2010-10-11T19:34:59Z

Tsangcharles:

10-1100/Assignment 1 Part 1 by Charles

2010-10-11T19:09:30Z

Tsangcharles:

'''Pyraminx''' by Charles Tsang 995305753

In this article, we will analyze Pyraminx, a variant of the Rubik's cube. We will restrict our attention to those permutations which can only be generated by turning the sides of the puzzle (not by “cheating” and reassembling the puzzle).

As an introduction, we shall describe the shape of the puzzle. It is shaped as a tetrahedron. i.e. It has 4 faces at which each face is a equilateral triangle with a different solid colour. And on each face, it is composed of 9 smaller equilateral triangles. The following are pictures of the puzzle and its decomposition into a net.

[[Image:Pyraminx.jpg]]
[[Image:240px-Tetrahedron_fla1t.jpg]]

The faces are coloured green, blue, orange and yellow, and each smaller equilateral triangle is labelled with numbers.

Now we are ready to play!

To understand the puzzle, we need to observe what happens to the numbers when we turn each face.

When we turn the green face(“clockwise” with respect to the net), the number 1 is now in place of 5, and 5 goes to 25 etc... (At first this seems easy but because this is a tetrahedron, it is actually more complicated than it seems. So to aid this process, I actually created several nets on different sheets of papers and created a tetrahedron to keep track of the permutations.) We write this as a permutaiton : [25,14,13,2,1,11,12,23,24,35,36,26,15,4,3,9,10,18,19,20,21,22,27,16,5,7,8,28,29,30,31,32,33,34,17,6 ]
In this notation, the i'th term in the list correspond to the iverse image of number i under turning the green face.

Repeat the same procedure for blue, orange and yellow, we can see that they represent the permutations
[1,2,3,35,36,26,25,14,9,10,11,12,13,33,34,28,27,16,15,4,21,22,23,24,32,30,29,18,17,6,5,7,8,19,20,31]
[6,17,18,29,30,32,31,20,19,8,7,5,13,14,15,16,33,22,21,10,9,3,4,24,25,26,27,28,23,12,11,1,2,34,35,36] and
[31,20,3,4,5,6,7,8,9,29,30,32,21,10,15,16,17,18,19,27,28,34,33,22,11,1,2,13,14,25,26,36,35,24,23,12]
respectively.

Then after running the Schreier–Sims algorithm in Mathematica, we have found that there are 3,732,480 different configurations by turning each face of the puzzle. More specifically, if we restrict ourselves into only turning one side in the beginning, we get 3 different configurations, and if we restrict ourselves into turn two sides, we get 19440 configurations, which we can readily see from the output.

Here is the input/output log for those who are interested:

[[Image:Charlesinputoutput.jpg]]

I hope everyone has enjoyed it as much as I did analyzing it!

10-1100/Assignment 1 Part 1 by Charles

2010-10-11T19:04:16Z

Tsangcharles:

'''Pyraminx''' by Charles Tsang 995305753

In this article, we will analyze Pyraminx, a variant of the Rubik's cube. We will restrict our attention to those permutations which can only be generated by turning the sides of the puzzle (not by “cheating” and reassembling the puzzle).

As an introduction, we shall describe the shape of the puzzle. It is shaped as a tetrahedron. i.e. It has 4 faces at which each face is a equilateral triangle with a different solid colour. And on each face, it is composed of 9 smaller equilateral triangles. The following are pictures of the puzzle and its decomposition into a net.

[[Image:Pyraminx.jpg]]
[[Image:240px-Tetrahedron_fla1t.jpg]]

The faces are coloured green, blue, orange and yellow, and each smaller equilateral triangle is labelled with numbers.

Now we are ready to play!

To understand the puzzle, we need to observe what happens to the numbers when we turn each face.

When we turn green(“clockwise” with respect to the net), the number 1 is now in place of 5, and 5 goes to 25 etc... (At first this seems easy but because this is a tetrahedron, it is actually more complicated than it seems. So to aid this process, I actually created several nets on different sheets of papers and created a tetrahedron to keep track of the permutations.) We write this as a permutaiton : [25,14,13,2,1,11,12,23,24,35,36,26,15,4,3,9,10,18,19,20,21,22,27,16,5,7,8,28,29,30,31,32,33,34,17,6 ]
In this notation, the i'th term in the list correspond to the iverse image of number i under turning the green face.

Repeat the same procedure for blue, orange and yellow, we can see that they represent the permutations
[1,2,3,35,36,26,25,14,9,10,11,12,13,33,34,28,27,16,15,4,21,22,23,24,32,30,29,18,17,6,5,7,8,19,20,31]
[6,17,18,29,30,32,31,20,19,8,7,5,13,14,15,16,33,22,21,10,9,3,4,24,25,26,27,28,23,12,11,1,2,34,35,36] and
[31,20,3,4,5,6,7,8,9,29,30,32,21,10,15,16,17,18,19,27,28,34,33,22,11,1,2,13,14,25,26,36,35,24,23,12]
respectively.

Then after running the Schreier–Sims algorithm in Mathematica, we have found that there are 3,732,480 different configurations by turning each face of the puzzle. More specifically, if we restrict ourselves into only turning one side in the beginning, we get 3 different configurations, and if we restrict ourselves into turn two sides, we get 19440 configurations, which we can readily see from the output.

Here is the input/output log for those who are interested:

[[Image:Charlesinputoutput.jpg]]

I hope everyone has enjoyed it as much as I did analyzing it!

10-1100/Assignment 1 Part 1 by Charles

2010-10-11T18:58:57Z

Tsangcharles: Charles' assignment 1 part 1

'''Pyraminx''' by Charles Tsang 995305753

In this article, we will analyze Pyraminx, which is variant of the Rubik's cube. We will restrict our attention to those permutations which can only be generated by turning the sides of the puzzle (not by “cheating” and reassembling the puzzle).

As an introduction, we shall describe the shape of the puzzle. It is shaped as a tetrahedron. i.e. It has 4 faces at which each face is a equilateral triangle with a different solid colour. And on each face, it is composed of 9 smaller equilateral triangles. The following are pictures of the puzzle and its decomposition into a net.

[[Image:Pyraminx.jpg]]
[[Image:240px-Tetrahedron_fla1t.jpg]]

The faces are coloured green, blue, orange and yellow, and each smaller equilateral triangle is labelled with numbers.

Now we are ready to play!

To understand the puzzle, we need to observe what happens to the numbers when we turn each face.

When we turn green(“clockwise” with respect to the net), the number 1 is now in place of 5, and 5 goes to 25 etc... (At first this seems easy but because this is a tetrahedron, it is actually more complicated than it seems. So to aid this process, I actually created several nets on different sheets of papers and created a tetrahedron to keep track of the permutations.) We write this as a permutaiton : [25,14,13,2,1,11,12,23,24,35,36,26,15,4,3,9,10,18,19,20,21,22,27,16,5,7,8,28,29,30,31,32,33,34,17,6 ]
In this notation, the i'th term in the list correspond to the iverse image of number i under turning the green face.

Repeat the same procedure for blue, orange and yellow, we can see that they represent the permutations
[1,2,3,35,36,26,25,14,9,10,11,12,13,33,34,28,27,16,15,4,21,22,23,24,32,30,29,18,17,6,5,7,8,19,20,31]
[6,17,18,29,30,32,31,20,19,8,7,5,13,14,15,16,33,22,21,10,9,3,4,24,25,26,27,28,23,12,11,1,2,34,35,36] and
[31,20,3,4,5,6,7,8,9,29,30,32,21,10,15,16,17,18,19,27,28,34,33,22,11,1,2,13,14,25,26,36,35,24,23,12]
respectively.

Then after running the Schreier–Sims algorithm in Mathematica, we have found that there are 3,732,480 different configurations by turning each face of the puzzle. More specifically, if we restrict ourselves into only turning one side in the beginning, we get 3 different configurations, and if we restrict ourselves into turn two sides, we get 19440 configurations, which we can readily see from the output.

Here is the input/output log for those who are interested:

[[Image:Charlesinputoutput.jpg]]

I hope everyone has enjoyed it as much as I did analyzing it!

10-1100/Class Photo

2010-09-29T14:14:41Z

Tsangcharles: /* Who We Are... */

Our class on September 28, 2010:

[[Image:10-1100-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{10-1100/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=0
|-
!First name
!Last name
!UserID
!Email
!In the photo
!Comments
{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math.toronto.edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}
{{Photo Entry|last=Dema|first=Ilir|userid=idema|email=ilir.dema@ utoronto.ca|location=The only one with gray hair|comments=First name is ILIR - just want to avoid confusion by similar-looking capital I and lower case l}}
{{Photo Entry|last=Eagle|first=Chris|userid=cjeagle|email=cjeagle@ math.utoronto.ca|location=Towards the right, in the back row. You can see only the top of my head, as I'm directly behind someone.|comments=}}
{{Photo Entry|last=MeneZes|first=Allan|userid=Amenezes007|email=amenezes007@ sympatico.ca|location=The only brown dude in the middle of the photo|comments=Knowledge is Power, Ignorance is Dangerous, War is Peace}}
{{Photo Entry|last=Tsang|first=Charles|userid=Tsangcharles|email=ck.tsang@ utoronto.ca|location=The Asian dude to the right of the word Sn under the definition of equivalence of towers|comments=add me on PSN if you play PS3, my PSN id is tsangcharles}}

|}