text:
User:Mma666/Books/Nonlinear finite elements

Nonlinear Finite Elements

 * Nonlinear finite elements/Homework 1
 * Nonlinear finite elements/Homework 1/Solutions
 * Nonlinear finite elements/Homework 2
 * Nonlinear finite elements/Homework 2/Solutions
 * Nonlinear finite elements/Homework 3
 * Nonlinear finite elements/Homework 3/Solutions
 * Nonlinear finite elements/Homework 4
 * Nonlinear finite elements/Homework 4/Hints
 * Nonlinear finite elements/Homework 4/Solutions
 * Nonlinear finite elements/Homework 5
 * Nonlinear finite elements/Homework 5/Solutions
 * Nonlinear finite elements/Homework 6
 * Nonlinear finite elements/Homework 6/Hints
 * Nonlinear finite elements/Homework 6/Solutions


text:
Talk:TAO/Silver Knowledge

Concept and workplan for Modul Silver Knowledge
Topic: Silver Knowledge
 * short description of the problem and the aim
 * description of the objective targets
 * description of the approaches on several places
 * presentation of materials and methods
 * examples and testimonials

The introduction and the description of the learning activity Silver Knowledge will be presented as articel; the documantation of the exemplary approaches on several places as showcase will be illustrated also as articel with pictures and helpful regards. The main aspects of this showcase will be combined in a comparison table. The materials and methods will be described also in short articels and can be download. Testimonials, tips and symbols will visualise and exemplify the articels.

target group: who will be informed by this wikiversity and handbook articel Both of this target groups need not only informations about Silver Knowledge, important is, that they get an imagination about, how they can involve and integrate elderly people into the wikimedia as wikipedia author.
 * representative and multiplicator of senior- or education- institutions, which will take part on the project silver knowledge
 * Later on, this learning activity can also be used for representative and volunteers of other wikimedia chapters, which will transfer silver knowledge. Maybe the module must be adapted to their needs and specific situation.

collaboration on the module 'Silver Knowledge'
 * How will we develop this articel?
 * Who will be collaborator on this articel?
 * What are the next steeps?

We will work together with the wikipedia referents and responsible persons on the several places were senior knowledge take palce! Therefor we invite the referents of wikimedia into this wikiversity project to work together on the different parts of this module/articel. First we will translate this module in german, that the seniors and referents, who are not able to work in english can also involved. Until the end of February, we want to complete a draft, which will include all important elements and informations to retry the activities of silver knowledge on other places. On the first weekend in March we want to present this learning activity by the meeting of the wikipedia referents in Hannover to invite some of them to collaborate in this Community of Practice.

 Validation with the target groups

We will promote this learning activity for the representatives and multiplicators of such senior- or education- institutions, which will take part on the project silver knowledge. Here they should get all important informations about Silver Knowledge to arrange the activities, including all documents, checklists and other needed materials. We want to get an feedback on the discussion sites and we also ask them by a questionnaire.

 reflections on sustainability

This learning activity will be useful for the transfer of silver knowledge by the multiplication and internationalisation. It will be a part of the TAO-handbook and it could be a part of the developing consultant concept.

milestones and time line

Suggestions, questions and discussion
Do you have any suggestions, questions or discussionponits? We want to invite you to collaborate wit us in this module! --Marquard Markus 16:30, 28 January 2012 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:TAO/Silver Knowledge


text: User talk:Begoon

title: User talk:Begoon


text:
Topology/Lesson 5

Directed Sets
Directed sets are very useful in topology. We will explore a couple of their uses in this lesson.

Definition
A directed set is a set $$D$$ with a partial order denoted by $$\le$$ which satisfies the additional requirement that given $$a,b\in D$$ there is $$c\in D$$ such that $$a\le c$$ and $$b\le c$$.

Examples (directed set)

 * 1) Let $$X$$ be a set.  Then its power set $$2^X$$ is a directed set, ordered by set inclusion.  Indeed, if $$A,B\subset X$$ then $$A\subset A\cup B$$ and $$B\subset A\cup B$$.
 * 2) Suppose that $$X$$ is a topological space and $$x\in X$$.  Then the set $$D$$ of all neighborhoods of $$x$$ is a directed set, ordered by reverse set inclusion (that is, $$A\le B$$ if $$A\supset B$$).  The proof is left as an exercise.

Cofinal set
Let $$D$$ be a directed set. A subset $$D^\prime\subset D$$ is cofinal if for every $$d\in D$$ there is $$d^\prime\in D^\prime$$ such that $$d\le d^\prime$$.

Examples (cofinal set)

 * 1) Let $$X$$ be an infinite set.  Then, as above, its power set $$2^X$$ is a directed set.  The subset consisting of only infinite subsets of $$X$$ is a cofinal set.
 * 2) As in Example 2 above, let $$D$$ be the set of neighborhoods of the point $$x\in X$$.  Then the set of open neighborhoods of $$x$$ is a cofinal set.  If $$X$$ is Hausdorff and locally compact, then the set of compact neighborhoods of $$x$$ is also cofinal.

Nets
One of the main applications of directed sets is that of nets. A net is kind of like a sequence, but the indexing set is a directed set rather than an ordinal set (or, specifically the set $$\mathbb{N}$$). That is, a net in a space $$X$$ is a function $$f:D\to X$$, where $$D$$ is a directed set.

Subnet
Let $$f:D\to X$$ be a net. A subnet of $$f$$ is the restriction of $$f$$ to a subset $$D^\prime\subset D$$ that is also directed and is cofinal in $$D$$.

Nets are like sequences. Just as you can picture a sequence being a bunch of points in a space, and you usually think of that sequences limiting on some particular point, you can think of nets as a bunch of points in a space. And, just like sequences, nets are useful when they accumulate at a specific point (or multiple points).

Limits
Let $$f:D\to X$$ be a net. The net converges to a point $$x\in X$$ if for every neighborhood $$N\ni x$$, there is $$a\in D$$ such that $$f(\alpha)\in N$$ for all $$\alpha\ge a$$.

This definition looks surprisingly similar to the definition of the limit of a sequence, and it is very similar. However, note one significant difference. In $$D$$, not all points are assumed to be comparable (that is, there might be $$a,b\in D$$ for which neither $$a\le b$$ nor $$b\le a$$ is true). Therefore, the quantifier "for all $$\alpha\ge a$$" excludes any point in $$D$$ that is not comparable to $$a$$.

What's all the hype about? Why did topologists even invent the concept of a net? Consider the following results, prior to nets.

 * 1) If a set $$C$$ is compact, then every sequence in it has a convergent subsequence.
 * 2) If a function $$f:X\to Y$$ is continuous and $$x_n\to x$$ then $$f(x_n)\to f(x)$$.
 * 3) Let $$\{x_n\}$$ be a sequence in a set $$A\subset X$$.  If $$x_n\to x$$ in $$X$$ then $$x\in \bar A$$ (the closure of $$A$$).

Each of these is a very good result. However, for each one the converse is false. Consider the following examples.
 * 1) The space $$\omega_1$$ (the ordinal space, which consists of all finite/countable ordinals) is not compact but every sequence in it has a convergent subsequence (in particular, every monotone sequence in $$\omega_1$$ is convergent).
 * 2) Let $$f:[0,\omega_1]\to \{0,1\}$$ be defined by $$f(\alpha)=0$$ for $$\alpha<\omega_1>$$ and $$f(\omega_1)=1$$.  $$[0,\omega_1]$$ has the order topology, since it is an ordinal, and $$\{0,1\}$$ has the discrete topology.  Then $$f$$ is not continuous but every sequence in $$[0,\omega_1]$$ is preserved (that is, if $$x_n\to x$$ in $$[0,\omega_1]$$ then $$f(x_n)\to f(x)$$).
 * 3) The point $$\omega_1$$ in the space $$[0,\omega_1]$$ (as in the previous example) is in the closure of $$[0,\omega_1)$$ but is not the limit of any sequence in that set.

However, if we use nets instead of sequences, each of these results becomes a biconditional. The proof of each will be left as an exercise to the student. A suggestion for each would be to follow a proof of the case where only sequences are considered.

Exercises
Prove each of the following.
 * 1) A set $$C$$ is compact if and only if every net in $$C$$ has a convergent subnet.
 * 2) A function $$f:X\to Y$$ is continuous if and only if $$f(\nu(\alpha))\to f(x_0)$$ whenever $$\nu:D\to X$$ is a net converging to $$x_0\in X$$.
 * 3) Let $$A\subset X$$.  Then $$a\in \bar A$$ if and only if there is a net in $$A$$ that converges to $$a$$.

title: Topology/Lesson 5


text:
Healthy lifestyle/Cleanliness

Cleanliness means that there is no dirt, no dust, no stains, and no bad smells. The purposes of cleanliness is health, beauty, absence of offensive odor and to avoid the spreading of dirt and contaminants to oneself and others. In the case of glass objects such as windows or windshields, the purpose can also be transparency. Washing is one way of achieving cleanliness, usually with water and often some kind of soap or detergent.

Since the germ theory of disease appearance, it has also come to mean an absence of germs and other waste. In industry, certain processes need conditions of exceptional cleanliness which are achieved by working in cleanrooms. A very new idea is that ‘dirt’ may play a useful role in our immune systems.

Other websites

 * Boy Scouts. 2007. "Boy Scout's Oath, Law, Motto, and Slogan." Retrieved May 10, 2007, http://www.scouting.org/factsheets/02-503a.html
 * Cleaning Network. 2008. "Cleanliness Article" Retrieved March 10, 2008, http://www.cleaning-services.org/cleaningnews.html

title: Healthy lifestyle/Cleanliness


text:
User:CRIChouse

CRIChouse

title: User:CRIChouse


text:
Healthy lifestyle/Healthy diet

A healthy diet or balanced diet is a diet (what you eat) that contains the right amounts of all the food substances. It includes fruit, vegetables, grains, dairy products, and protein. It does not include too much or too little of any kind of food.

Other pages

 * Nutrition

title: Healthy lifestyle/Healthy diet


text:
Healthy lifestyle/Hygiene

Hygiene is the act of being clean. Washing the body to remove dirt and germs, brushing the teeth to keep them clean, shaving, using the toilet properly, and dressing correctly are some examples. Proper hygiene is often taught to children at a young age, and it becomes a habit. People who do not have good hygiene might smell bad, lose teeth, or become sick. .

title: Healthy lifestyle/Hygiene


text:
Healthy lifestyle/Jogging

Jogging is running at a slow speed. Many people go for a jog in order to keep fit and healthy. People go jogging for exercise, not for competitions. Jogging is a very cheap way of keeping fit, because all that is needed is a good pair of running shoes. One can jog anywhere, although it is good to stay away from main roads where the air is full of traffic fumes. It is also good to avoid too much jogging on hard pavements (sidewalks).

A softer surface such as grass is better for jogging because there is less hard jolting for the knees and hips.

Other websites

 * WorldwideRunning.com - A comprehensive resource for joggers and runners worldwide

title: Healthy lifestyle/Jogging


text:
Healthy lifestyle/Longevity

The word "longevity" means the quality of being long-lasting, especially of life. It is sometimes used as a synonym for life expectancy or to connote long life, especially when it concerns someone or something lasting longer than expected. .]] Longevity usually includes thinking about methods to extend life. Longevity has been a topic not only for the scientific community but also for writers of science fiction and utopian novels.

Longevity stories are cultural traditions and lore about exceptional human longevity, with or without eternal youth. These stories include sincere beliefs of claims of extreme age. Each category of belief is based on a different motivation for claiming exceptional age.

Stories of exaggerated longevity have been around since the earliest civilizations. The first longevity narratives were probably the patriarchal/matriarchal claims, which are often an attempt to link humans to the gods or to God due to fuller life. In some religious traditions there are claims that, if one follows a certain philosophy or practice, one can become immortal or at least live to an extreme age.

Predictions
The U.S. Census Bureau predicts that life expectancy in the United States will be in the mid-80s by 2050 (up from 77.85 in 2006) and will top out eventually in the low 90s, barring major scientific advances that can change the rate of human aging itself, as opposed to merely treating the effects of aging as is done today. The Census Bureau also predicted that the United States would have 5.3 million people aged over 100 in 2100. The United Nations has also made projections far out into the future, up to 2300, at which point it projects that life expectancies in most developed countries will be between 100 and 106 years and still rising, though more and more slowly than before. However, the UN has warned that these projections are uncertain, and caution that any change or advancement in medical technology could invalidate their projections.

Other websites

 * List of famous people, how old they died, year and month. For example: 40 years, 1 month, John Lennon

title: Healthy lifestyle/Longevity


text: User:Addihockey10 (automated)

title: User:Addihockey10 (automated)


text:
Rubik's Cube/Customize Your Moves

 +-+ +-+
 * HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE                              |
 * (Preliminary April 20, 2007 version)            |
 * by Mr. Ray Calvin Baker                         |
 * FREE Educational Material                       |
 * Chapter Six - - - - - - - - - - Customize Your Moves -- Commutation                        |
 * There is a clever trick which can often be used in manipulating Rubik's Cube (and many     |
 * other puzzles, too). It's called "commutation". When you know how to do something when the |
 * cubies are at the "correct" locations, but some cubies are not at the correct locations,   |
 * try this three step process. First, move the cubies to their "correct" location. (You will |
 * need to remember exactly what moves you made to do this for step three, later. Take notes  |
 * if you must!) Second, do the thing you know how to do. Third, undo whatever you did in the |
 * first step.                                                                                |
 * An example: we know how to convert                                                         |
 * 0 - + + 0   0 0 0 0 0                                                                 |
 * 0 - - + 0 to 0 0 0 0 0                                                                |
 * ^           ^         (all cubies in correct orientation).                            |
 * (Note: I put in the "^" symbol just to emphasize the orientation.)                     |
 * We just use our 0 + - - 0                                                                  |
 * 0 + + - 0 operator, Fv R^ Fv R^ Fv R^.                                     |
 * So, how do we solve                                                                        |
 * + 0 - + +                                                                             |
 * + 0 - - + ?                                                                           |
 * First step: rotate the entire cube to this position:                                       |
 * 0 - + + 0                                                                             |
 * 0 - - + 0                                                                             |
 * Second step: Do Fv R^ Fv R^ Fv R^. This converts                                           |
 * 0 - + + 0         0 0 0 0 0                                                           |
 * 0 - - + 0   to    0 0 0 0 0                                                           |
 * Third step: undo whatever we did in the first step. In this case, we rotate the entire Cube |
 * back to its original orientation, from 0 0 0 0 0   0 0 0 0 0                               |
 * ^        to   ^       .                             |
 * Before you say, "This is strange stuff -- it has no relevance for me!", let me point out   |
 * some uses you have already made using this principle, from previous chapters.              |
 * You have some understanding of how to rotate the entire Cube to any of its 24 possible     |
 * orientations. You have used some of these moves to place several corner cubies in their    |
 * correct locations. In effect, you have learned one series of moves, but you now know 24    |
 * ways to use that series of moves.                                                          |
 * Here is a portion of one of the diagrams from Chapter Four, "Ignoring Details -- Moving    |
 * Corner Cubies". See if you can find some of the ways this principle has been used (even    |
 * though I didn't tell you that you were using it).                                          |
 * (L) klt (K)  : (L) klt (K) : (L) klt (K) : (L) klt (K)    : (L) klt (K)   : (L) klt (K)    |
 * flt T krt   :  flt T frt  :  frt T krt  :  frt T flt     :  krt T flt    :  krt T frt     |
 * F frt  R    :  F  krt  R  :  F  flt  R  :  F  krt  R     :  F  frt  R    :  F  flt  R     |
 * :            :             :  (3-cycle)     :  (diagonal    :  (3-cycle)     |
 * :            :             :                :   corners)    :                |
 * WOW! All 8   : 3R^         : 3Fv         : K^ 3R^ 3T^     : This is done  : K^ 3R^ 3T^     |
 * corner cubies : DO THE SWAP : DO THE SWAP : Diagram 4-2:  : exactly like  : Diagram 4-2:   |
 * are properly : diagram 4-5 : diagram 4-5 : Fv R^ F^ Rv T2 : diagrams 4-8A : Fv R^ F^ Rv T2 |
 * located!     : 3Rv         : 3Fv         : 3Tv 3Rv Kv     : through 4-8J. : 3Tv 3Rv Kv     |
 * :            :             : Then look at   :               :                |
 * :            :             : diagram 4-11F. :               :                |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-11A.       : 4-11B.      : 4-11C.      : 4-11D.         : 4-11E.        : 4-11F.         |
 * Look at diagram 4-11B. Step one of the useful principle is, "3R^". Step two is, "DO THE    |
 * SWAP". Step three is, "3Rv", which "undoes" step one.                                      |
 * Look at diagram 4-11F. Step one is, "K^ 3R^ 3T^". Step two is, "Fv R^ F^ Rv T2". Step three |
 * is, "3Tv 3Rv Kv", which "undoes" step one.                                                 |
 * Do you remember the clumsy way to exchange two diagonal corner cubies? Here is a much      |
 * easier way! Starting with diagram 4-11E, but using commutation, step one is shown in       |
 * diagram 6-1.                                                                               |
 * (L) klt (K)      (L)_ ? _(K)      (L)_ ? _(K)       (L)_ ? _(K)  Step one:                 |
 * krt T flt        klt T _ ?        ? _ T _flt        klt T _ ?   we have put the           |
 * F frt  R   3R^   |  flt  |  3T^   |  klt  |   Tv    |  flt  |   corner cubies to be       |
 * diagonal  -->   | F | R |  -->   | F | R |   -->   | F | R |   SWAPped into "correct"    |
 * corners)        krt | _ ?        ? _ | _frt        ? _ | _frt  FRT and BFR positions.    |
 * frt             krt               krt                                |
 * (Expanding the diagram                                                        |
 * a little bit)                                                                |
 * DIAGRAM 6-1. Example Step One                                                              |
 * (L)_ ? _(K) Step two:    (L)_ ? _(K)                                                       |
 * klt T _ ?  DO THE SWAP   klt T _ ?                                                        |
 * | flt  |                 |  krt  |                                                        |
 * | F | R |                | F | R |                                                        |
 * ? _ | _frt               ? _ | _frt                                                       |
 * krt                      flt                                                           |
 * DIAGRAM 6-2. Example Step Two                                                              |
 * Step three:   (L)_ ? _(K)       (L)_ ? _(K)       (L)_ ? _(K)       (L) klt (K)     |
 * "Undo" what    klt T _ ?         ? _ T _krt        klt T _ ?         flt T krt      |
 * you did in     |  krt  |   T^    |  klt  |   3Tv   |  krt  |   3Rv   |  frt  |      |
 * step one.      | F | R |   -->   | F | R |         | F | R |   -->   | F | R |      |
 * ? _ | _frt       ? _ | _frt        flt | _ ?         ? _ | _ ?      |
 * flt              flt               frt                ?          |
 * DIAGRAM 6-3. Example Step Three                                                     |
 * Wasn't that much faster and easier than the clumsy way shown in Chapter Four, ""?          |
 * Here is a quick review and summary of how to "undo" one move. (You learned most of this    |
 * when the moves were introduced.).                                                          |
 * B^ undoes Bv.  B2 undoes  B2.  Bv undoes  B^.                                        |
 * F^ undoes Fv.  F2 undoes  F2.  Fv undoes  F^.                                        |
 * K^ undoes Kv.  K2 undoes  K2.  Kv undoes  K^.                                        |
 * L^ undoes Lv.  L2 undoes  L2.  Lv undoes  L^.                                        |
 * B^ undoes Bv.  B2 undoes  B2.  Bv undoes  B^.                                        |
 * B^ undoes Bv.  B2 undoes  B2.  Bv undoes  B^.                                        |
 * 2B^ undoes 2Bv. 2B2 undoes 2B2. 2Bv undoes 2B^.                                       |
 * 2F^ undoes 2Fv. 2F2 undoes 2F2. 2Fv undoes 2F^.                                       |
 * 2K^ undoes 2Kv. 2K2 undoes 2K2. 2Kv undoes 2K^.                                       |
 * 2L^ undoes 2Lv. 2L2 undoes 2L2. 2Lv undoes 2L^.                                       |
 * 2B^ undoes 2Bv. 2B2 undoes 2B2. 2Bv undoes 2B^.                                       |
 * 2B^ undoes 2Bv. 2B2 undoes 2B2. 2Bv undoes 2B^.                                       |
 * 3B^ undoes 3Bv. 3B2 undoes 3B2. 3Bv undoes 3B^.                                       |
 * 3F^ undoes 3Fv. 3F2 undoes 3F2. 3Fv undoes 3F^.                                       |
 * 3K^ undoes 3Kv. 3K2 undoes 3K2. 3Kv undoes 3K^.                                       |
 * 3L^ undoes 3Lv. 3L2 undoes 3L2. 3Lv undoes 3L^.                                       |
 * 3B^ undoes 3Bv. 3B2 undoes 3B2. 3Bv undoes 3B^.                                       |
 * 3B^ undoes 3Bv. 3B2 undoes 3B2. 3Bv undoes 3B^.                                       |
 * DIAGRAM 6-4. Summary: How to Undo One Move                                            |
 * How does one undo a sequence of moves, such as "F^ Rv T^"? Start by considering the last   |
 * move made in "step 1" of the "customizing" process ("T^"), then undo it ("Tv"). Then,      |
 * consider the next- to-last move ("Rv"); undo it ("R^"). Continue this process. Eventually, |
 * you will consider the first move made in "step 1" ("F^"); undo that step ("Fv"). You're    |
 * done undoing the sequence of moves! To summarize the example, undo the sequence "F^ Rv T^" |
 * by doing "Tv R^ Fv".                                                                       |
 * B^ undoes Bv.  B2 undoes  B2.  Bv undoes  B^.                                        |
 * F^ undoes Fv.  F2 undoes  F2.  Fv undoes  F^.                                        |
 * K^ undoes Kv.  K2 undoes  K2.  Kv undoes  K^.                                        |
 * L^ undoes Lv.  L2 undoes  L2.  Lv undoes  L^.                                        |
 * B^ undoes Bv.  B2 undoes  B2.  Bv undoes  B^.                                        |
 * B^ undoes Bv.  B2 undoes  B2.  Bv undoes  B^.                                        |
 * 2B^ undoes 2Bv. 2B2 undoes 2B2. 2Bv undoes 2B^.                                       |
 * 2F^ undoes 2Fv. 2F2 undoes 2F2. 2Fv undoes 2F^.                                       |
 * 2K^ undoes 2Kv. 2K2 undoes 2K2. 2Kv undoes 2K^.                                       |
 * 2L^ undoes 2Lv. 2L2 undoes 2L2. 2Lv undoes 2L^.                                       |
 * 2B^ undoes 2Bv. 2B2 undoes 2B2. 2Bv undoes 2B^.                                       |
 * 2B^ undoes 2Bv. 2B2 undoes 2B2. 2Bv undoes 2B^.                                       |
 * 3B^ undoes 3Bv. 3B2 undoes 3B2. 3Bv undoes 3B^.                                       |
 * 3F^ undoes 3Fv. 3F2 undoes 3F2. 3Fv undoes 3F^.                                       |
 * 3K^ undoes 3Kv. 3K2 undoes 3K2. 3Kv undoes 3K^.                                       |
 * 3L^ undoes 3Lv. 3L2 undoes 3L2. 3Lv undoes 3L^.                                       |
 * 3B^ undoes 3Bv. 3B2 undoes 3B2. 3Bv undoes 3B^.                                       |
 * 3B^ undoes 3Bv. 3B2 undoes 3B2. 3Bv undoes 3B^.                                       |
 * DIAGRAM 6-4. Summary: How to Undo One Move                                            |
 * How does one undo a sequence of moves, such as "F^ Rv T^"? Start by considering the last   |
 * move made in "step 1" of the "customizing" process ("T^"), then undo it ("Tv"). Then,      |
 * consider the next- to-last move ("Rv"); undo it ("R^"). Continue this process. Eventually, |
 * you will consider the first move made in "step 1" ("F^"); undo that step ("Fv"). You're    |
 * done undoing the sequence of moves! To summarize the example, undo the sequence "F^ Rv T^" |
 * by doing "Tv R^ Fv".                                                                       |
 * move made in "step 1" of the "customizing" process ("T^"), then undo it ("Tv"). Then,      |
 * consider the next- to-last move ("Rv"); undo it ("R^"). Continue this process. Eventually, |
 * you will consider the first move made in "step 1" ("F^"); undo that step ("Fv"). You're    |
 * done undoing the sequence of moves! To summarize the example, undo the sequence "F^ Rv T^" |
 * by doing "Tv R^ Fv".                                                                       |

title: Rubik's Cube/Customize Your Moves


text:
Rubik's Cube/Finishing the Orientation of Corner Cubies

 +-+ +-+
 * HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE                              |
 * (Preliminary April 20, 2007 version)            |
 * by Mr. Ray Calvin Baker                         |
 * FREE Educational Materials                      |
 * Chapter Seven - - - - - - - - - Finishing the Orientation of Corner Cubies                 |
 * In Chapter Five, we found some new patterns for rotating corner cubies. Each pattern is a  |
 * (coded) solution for \i some\i0 Rubik's Cube problem.                                      |
 * Can you find these patterns?                                                               |
 * + 0 + 0 +    - + - + -     0 + 0 + 0     0 - 0 - 0     0 0 0 0 0     Some sample      |
 * - 0 - 0 -    + + + 0 +     0 + 0 0 0     0 0 0 - 0     0 + 0 - 0     problems.        |
 * [F] [K]      [F] [K]       [F] [K]       [F] [K]       [F] [K]                      |
 * DIAGRAM 7-1. Typical Coded Rotation Problems                                          |
 * These last two patterns, or variations of them, will solve ALL problems af rotated corner  |
 * cubies (unless your cube has been physically damaged.) Yes, there are simpler ways -- can  |
 * you find some?                                                                             |
 * We now have tools which will enable us to find a way to rotate just TWO corner cubes (in   |
 * opposite directions). Let's work out a detailed plan to do this.                           |
 * Starting with a Cube with all eight corner cubies in their proper locations is essential   |
 * for completing this chapter. The orientations of these eight corner cubies has not yet     |
 * determined -- it can be completely arbitrary, within the constraints imposed by the        |
 * essential geometry of the Cube itself. (If seven corner cubies are properly located and    |
 * oriented, then the eighth corner cubie is also properly located and oriented -- unless your |
 * Cube has been physically damaged or is "out of orbit".)                                    |                                           |                                                                                |
 * Since we are still doing "paper and pencil" work, we can start with an arbitrary config-   |
 * uration of properly located corner cubies. Let's start with                                |
 * 0 0 0 0 0                                                                            |
 * 0 0 0 0 0                                                                            |
 * [F] [K].                                                                           |
 * From our work in chapter five, we know that this sequence of moves changes the orientation |
 * of six corner cubies (but it doesn't change their locations).                              |
 * 0 0 0 0 0    Fv R^ Fv R^ Fv R^     0 + - - 0                                         |
 * 0 0 0 0 0    -> -> -> -> -> ->     0 + + - 0                                         |
 * [F] [K]                            [F] [K]                                         |
 * DIAGRAM 7-2. A Sequence of Moves Which Changes Orientations od Corner Cubies         |
 * If we do that again, we have this.                                                         |
 * 0 + + - 0    Fv R^ Fv R^ Fv R^     0 - + + 0                                         |
 * 0 + - - 0    -> -> -> -> -> ->     0 - - + 0                                         |
 * [F] [K]                            [F] [K]                                         |
 * DIAGRAM 7-3. Repeating the Sequence of Moves                                         |
 * We expect that, if we do it a third time, everything will cancel out, and we will be right |
 * back at the starting configuration -- not very useful! But what if we could fix things so  |
 * that not everything cancels out? Let's try an experiment.                                  |
 * 0 - + + 0   Fv    0 - - + 0    Fv R^ Fv R^ Fv R^    0 0 + 0 0    F^    0 + - 0 0       |
 * 0 - - + 0   ->    0 - + + 0    -> -> -> -> -> ->    0 0 - 0 0    ->    0 0 0 0 0       |
 * [F] [K]           [F] [K]                           [F] [K]            [F] [K]       |
 * DIAGRAM 7-4. A Sucessful Experiment                                                    |
 * This can be extremely useful! Just be careful to have your Cube precisely oriented before  |
 * you start the sequence of moves. The corner cubie at FRONT LEFT TOP will be rotated        |
 * counterclockwise, and the corner cubie at FRONT RIGHT TOP will be rotated clockwise. (Hint: |
 * there may be a useful way to turn your Cube 180 degrees if you need to do this.)           |
 * How do you solve this problem?                                                             |
 * Step              Step two:                            Step                  |
 * one:              (You know how!)                      three:                |
 * 0 0 + 0 0   L^    0 - + 0 0      Fv R^ Fv R^ Fv R^      0 0 0 0 0   Lv    0 0 0 0 0      |
 * 0 - 0 0 0  --->   0 0 0 0 0      Fv R^ Fv R^ Fv R^      0 0 0 0 0  --->   0 0 0 0 0      |
 * [F] \{K\}           [F] [K]   Fv Fv R^ Fv R^ Fv R^ F^     [F] [K]           [F] [K]    |
 * A Problem!                                                                  Solved!      |
 * DIAGRAM 7-5. Solving a Typical Problem                                                   |
 * NOTE: Step one, a one-layer turn, moves four corner cubies away from their proper          |
 * locations. Therefore, it is very important not to forget to do step three, another one-    |
 * layer move, in order to return those four corner cubies to their proper places. There are a |
 * lot of details not shown in the above simplified diagrams!                                 |
 * We can also demonstrate that knowing how to rotate two corner cubies in opposite directions |
 * will allow us to solve the problem of three corner cubies all rotated in the same          |
 * direction. See if you can follow this sequence of moves.                                   |
 * 0 0 0 0 0 ROTATE  0 - + 0 0  ROTATE  0 + - 0 0  3Tv  + - 0 0 0  ROTATE   + + + 0 0        |
 * 0 0 0 0 0   TWO   0 0 0 0 0    TWO   0 0 0 0 0  -->  0 0 0 0 0    TWO    0 0 0 0 0        |
 * [F] [K] CORNERS   [F] [K]  CORNERS   [F] [K]         [F] [K]  CORNERS    [F] [K]        |
 * DIAGRAM 7-6. Solving Three Corners Rotated in Same Direction                              |
 * At this time, you should be able to pick up your Cube and rotate all eight corner cubies   |
 * into their proper orientation. You should be able to accomplish Goal Two. Basically, there |
 * are three ways to do this.                                                                 |
 * The first way is, "everything all at once". You have tools to make lots of diagrams of     |
 * patterns of rotation for the corner cubies. You should be able to make a diagram of your   |
 * partially unscrambled Cube. You should be able to determine what pattern will rotate the   |
 * corner cubies to their proper orientation.                                                 |
 * Here's an example of what I mean by this. Suppose your Cube has this pattern:              |
 * 0 0 + + +  What pattern will     0 0 - - -         For each "0" in the problem,          |
 * + - 0 - +  solve this problem?   - + 0 + -            write "0" in the solution.         |
 * A possible                       The pattern       For each "+" in the problem,          |
 * problem.                         that solves it.      write "-" in the solution.         |
 * For each "-" in the problem,         |
 * write "+" in the solution.        |
 * DIAGRAM 7-7. Find a Solution For a Rotation Problem                                      |
 * Now, all you need to do is find this solution in your notebook (You did make a notebook,   |
 * didn't you?) and apply it to your Cube (You did record the moves to make your patterns,    |
 * didn't you?). The advantage of "everything all at once" is that it gives you an immediate  |
 * sequence of moves to complete fixing all eight corner cubies. The disadvantages are (1) you |
 * must have lots of good, accurate notes -- there are (3 to the 8th power) / 3 = 2,187       |
 * possible patterns, and (2) I can't remember 2,187 moves, or even what the next             |
 * disadvantage was.                                                                          |
 * The second method is, "one or two cubies at a time". This is easy to memorize, but slow -- |
 * you may have to use this method up to seven times. Diagrams 7-2, 7-3, and 7-4 show how to  |
 * rotate two corner cubies in opposite directions. So, find two corner cubies which are not  |
 * oriented properly, move them into position (step 1 of customizing), rotate the two corners, |
 * and undo step 1 (step 3 of customizing). One more, or possibly two more, corner cubies     |
 * should now be correctly oriented. Keep doing this until all eight corners are correctly    |
 * oriented.                                                                                |
 * The third way is to use a combination of methods -- find a rotation pattern that matches   |
 * several corners of your required solution, apply it, then clean up any remaining problems  |
 * using the "one or two cubies at a time" method.                                            |
 * The second and third methods are something fairly easy to remember and use. If you can     |
 * finish this phase of the solution by yourself, that's great! If you need the help of a     |
 * recipe to do this, here it comes.                                                          |
 * But first, I need to show you another diagram. You may use this diagram to find the        |
 * "customization" moves you need.                                                            |
 * Case 1: ?? #1 #2 ?? ?? No customization is needed.                                    |
 * [F]  [K]                                                                  |
 * Case 2: ?? #1 ?? #2 ?? Customize by doing "R^".                                       |
 * [F]  [K]                                                                  |
 * Case 3: #2 #1 ?? ?? #2 Customize by doing "3T^".                                      |
 * [F]  [K]                                                                  |
 * Case 4: ?? #1 ?? ?? ?? Customize by doing "K^ 3T^".                                   |
 * #2 ?? ?? ?? #2                                                                |
 * [F]  [K]                                                                  |
 * Case 5: ?? #1 ?? ?? ?? Customize by doing "3Fv".                                      |
 * ?? #2 ?? ?? ??                                                                |
 * [F]  [K]                                                                  |
 * Case 6: ?? #1 ?? ?? ?? Customize by doing "Rv".                                       |
 * ?? ?? #2 ?? ??                                                                |
 * [F]  [K]                                                                  |
 * Case 7: ?? #1 ?? ?? ?? Customize by doing "R2".                                       |
 * ?? ?? ?? #2 ??                                                                |
 * [F]  [K]                                                                  |
 * DIAGRAM 7-8. Find Second Corner Cubie to Rotate, Then Apply These Moves               |
 * While looking at diagram 7-8, you may have thought that there are often several different  |
 * ways to "customize". You are correct. Pick one way -- whatever you feel comfortable with -- |
 * and stick with it. Just don't change your mind part way through the three-step process.    |
 * Paragraph A:                                                                               |
 * Find a corner cubie which is properly located, but not properly oriented. Call this cubie  |
 * "#1". Rotate the entire Cube until this cubie is located at the FRONT LEFT TOP corner. Find |
 * another corner cubie which is also not properly oriented. There are seven possibilities, as |
 * shown in diagram 7-8. Follow the directions to "customize" your moves, then perform the    |
 * sequence,                                                                                |
 * Fv R^ Fv R^ Fv R^                                                    |
 * Fv R^ Fv R^ Fv R^                                                    |
 * Fv Fv R^ Fv R^ Fv R^ F^.                                                |
 * Paragraph B:                                                                               |
 * It is possible that no improvement resulted from performing paragraph A, because both      |
 * corner cubies were rotated in the wrong direction. If the corner cubie in the FRONT LEFT   |
 * TOP position is still not correcxtly oriented, simply repeat this sequence again,          |
 * Fv R^ Fv R^ Fv R^                                                    |
 * Fv R^ Fv R^ Fv R^                                                    |
 * Fv Fv R^ Fv R^ Fv R^ F^.                                                |
 * _ * _                                       _ * _                 |
 * _ * _  _ * _                                _ * _   _ * _             |
 * _ * _  _ * _   _ * _                        _ * _   _ * _   _ * _         |
 * * _ T _ * _ T _ * _  _ *                    * _ l _ * _ T _ * _   _ *       |
 * (L)...|  * _   _ * _   _ *   |              (f)...|   * _   _ * _   _ *   |       |
 * | F |  * _#2 _ *   |   |                    | t |   * _#2 _ *   |   |       |
 * The      * _ |   |   *   |   | _ *          The       * _ |   |   *   |   | _ *       |
 * FRONT    |   * _ |#2 |#2 | _ *   |          FRONT     |   * _ |#2 |#2 | _ *   |       |
 * LEFT     |   |   * _ | _ *   |   |          LEFT      |   |   * _ | _ *   |   |       |
 * TOP      * _ | F |   *   | R | _ *          TOP       * _ | F |   *   | R | _ *       |
 * cubie    |   * _ |   |   | _ *   |          cubie     |   * _ |   |   | _ *   |       |
 * is       |   |   * _ | _ *   |   |          needs     |   |   * _ | _ *   |   |       |
 * correctly * _ |  |   *   |   | _ *          more      * _ |   |   *   |   | _ *       |
 * oriented;    * _ |   |   | _ *              work;         * _ |   |   | _ *           |
 * go on to         * _ | _ *                  repeat the        * _ | _ *               |
 * paragraph C.         *                      sequence of moves.    *                   |
 * DIAGRAM 7-9A.                               DIAGRAM 7-9B.                             |
 * DIAGRAM 7-9. Is the FRONT LEFT TOP Corner Cubie Properly Oriented?                    |
 * Paragraph C:                                                                               |
 * Undo the customization you used in paragraph A. (You do remember what you did, don't you?) |
 * Paragraph D:                                                                               |
 * If all eight corner cubies are properly positioned and properly oriented, you are done!    |
 * Otherwise, repeat this process (starting at paragraph A) until all eight corner cubies are |
 * properly oriented.                                                                         |
 * 1) How much progress have we made at the end of chapter seven? There are now only              #
 * 2) ( ( 12 factorial) * (2 to the 12th power) / 4 ) = 479,001,600 * 4,096 / 4 =                 #
 * 3) 490,497,638,400 ways to arrange the cubies of your Cube.                                    #
 * By the way, you should now be able to see a pretty "X" pattern of matching cubies on all   |
 * six sides of your Cube.                                                                    |
 * |TOP| ? |TOP|                                                  |
 * | ? |TOP| ? |                                                  |
 * |TOP| ? |TOP|                                                  |
 * | L | ? | L | F | ? | F | R | ? | R | K | ? | K |                          |
 * | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? |                          |
 * | L | ? | L | F | ? | F | R | ? | R | K | ? | K |                          |
 * | B | ? | B |                                                  |
 * | ? | B | ? |                                                  |
 * | B | ? | B |                                                  |
 * DIAGRAM 7-10. X Marks Our Progress                                         |
 * Fv R^ Fv R^ Fv R^                                                    |
 * Fv Fv R^ Fv R^ Fv R^ F^.                                                |
 * Paragraph B:                                                                               |
 * It is possible that no improvement resulted from performing paragraph A, because both      |
 * corner cubies were rotated in the wrong direction. If the corner cubie in the FRONT LEFT   |
 * TOP position is still not correcxtly oriented, simply repeat this sequence again,          |
 * Fv R^ Fv R^ Fv R^                                                    |
 * Fv R^ Fv R^ Fv R^                                                    |
 * Fv Fv R^ Fv R^ Fv R^ F^.                                                |
 * _ * _                                       _ * _                 |
 * _ * _  _ * _                                _ * _   _ * _             |
 * _ * _  _ * _   _ * _                        _ * _   _ * _   _ * _         |
 * * _ T _ * _ T _ * _  _ *                    * _ l _ * _ T _ * _   _ *       |
 * (L)...|  * _   _ * _   _ *   |              (f)...|   * _   _ * _   _ *   |       |
 * | F |  * _#2 _ *   |   |                    | t |   * _#2 _ *   |   |       |
 * The      * _ |   |   *   |   | _ *          The       * _ |   |   *   |   | _ *       |
 * FRONT    |   * _ |#2 |#2 | _ *   |          FRONT     |   * _ |#2 |#2 | _ *   |       |
 * LEFT     |   |   * _ | _ *   |   |          LEFT      |   |   * _ | _ *   |   |       |
 * TOP      * _ | F |   *   | R | _ *          TOP       * _ | F |   *   | R | _ *       |
 * cubie    |   * _ |   |   | _ *   |          cubie     |   * _ |   |   | _ *   |       |
 * is       |   |   * _ | _ *   |   |          needs     |   |   * _ | _ *   |   |       |
 * correctly * _ |  |   *   |   | _ *          more      * _ |   |   *   |   | _ *       |
 * oriented;    * _ |   |   | _ *              work;         * _ |   |   | _ *           |
 * go on to         * _ | _ *                  repeat the        * _ | _ *               |
 * paragraph C.         *                      sequence of moves.    *                   |
 * DIAGRAM 7-9A.                               DIAGRAM 7-9B.                             |
 * DIAGRAM 7-9. Is the FRONT LEFT TOP Corner Cubie Properly Oriented?                    |
 * Paragraph C:                                                                               |
 * Undo the customization you used in paragraph A. (You do remember what you did, don't you?) |
 * Paragraph D:                                                                               |
 * If all eight corner cubies are properly positioned and properly oriented, you are done!    |
 * Otherwise, repeat this process (starting at paragraph A) until all eight corner cubies are |
 * properly oriented.                                                                         |
 * 1) How much progress have we made at the end of chapter seven? There are now only              #
 * 2) ( ( 12 factorial) * (2 to the 12th power) / 4 ) = 479,001,600 * 4,096 / 4 =                 #
 * 3) 490,497,638,400 ways to arrange the cubies of your Cube.                                    #
 * By the way, you should now be able to see a pretty "X" pattern of matching cubies on all   |
 * six sides of your Cube.                                                                    |
 * |TOP| ? |TOP|                                                  |
 * | ? |TOP| ? |                                                  |
 * |TOP| ? |TOP|                                                  |
 * | L | ? | L | F | ? | F | R | ? | R | K | ? | K |                          |
 * | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? |                          |
 * | L | ? | L | F | ? | F | R | ? | R | K | ? | K |                          |
 * | B | ? | B |                                                  |
 * | ? | B | ? |                                                  |
 * | B | ? | B |                                                  |
 * DIAGRAM 7-10. X Marks Our Progress                                         |
 * |TOP| ? |TOP|                                                  |
 * | ? |TOP| ? |                                                  |
 * |TOP| ? |TOP|                                                  |
 * | L | ? | L | F | ? | F | R | ? | R | K | ? | K |                          |
 * | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? |                          |
 * | L | ? | L | F | ? | F | R | ? | R | K | ? | K |                          |
 * | B | ? | B |                                                  |
 * | ? | B | ? |                                                  |
 * | B | ? | B |                                                  |
 * DIAGRAM 7-10. X Marks Our Progress                                         |
 * | L | ? | L | F | ? | F | R | ? | R | K | ? | K |                          |
 * | B | ? | B |                                                  |
 * | ? | B | ? |                                                  |
 * | B | ? | B |                                                  |
 * DIAGRAM 7-10. X Marks Our Progress                                         |
 * | B | ? | B |                                                  |
 * DIAGRAM 7-10. X Marks Our Progress                                         |
 * DIAGRAM 7-10. X Marks Our Progress                                         |
 * DIAGRAM 7-10. X Marks Our Progress                                         |
 * DIAGRAM 7-10. X Marks Our Progress                                         |


text:
Rubik's Cube/Moving Edge Cubies

 +-+ +-+
 * HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE                              |
 * (Preliminary April 20, 2007 version)               |
 * by Mr. Ray Calvin Baker                         |
 * FREE Educational Material                       |
 * Chapter Eight - - - - - - - - - Moving Edge Cubies                                         |
 * Goal Three is to locate each of the twelve edge cubies at the proper edge of the Cube. This |
 * is going to take some different moves that leave the locations and orientations of the     |
 * corner cubies unchanged. We will be using some two-layer turns that move some of the       |
 * centers of several faces as well. Consequently, we will need some new landmarks.           |
 * This is not as difficult as it sounds, because, at this stage, all of the corner cubies    |
 * should be properly located and oriented. Here is what is called the "down-slice" move.     |
 * (It's called a "down-slice" because the three cubies on the FRONT move DOWN.)              |
 * _ * _    (1)                 _ * _     (7)                  _ * _     (7)      |
 * _ * _ T _ * _ /              _ * _ T _ * _ /                _ * _ T _ * _ /        |
 * _ * _ ? _ * _ 2 _ * _        _ * _ ? _ * _ 8 _ * _          _ * _ ? _ * _ 8 _ * _      |
 * * _ T _ * _ T _ * _ T _ *    * _ T _ * _ k _ * _ k _ *      * _ T _ * _ k _ * _ T _ *    |
 * |  * _ 3 _ * _ ? _ *   |     |   * _ 1 _ * _ ? _ *   |      |   * _ 1 _ * _ ? _ *   |    |
 * | F |  * _ T _ *   | R |     | F |   * _ k _ *   | R |      | F |   * _ T _ *   | R |    |
 * * _ | 4 |  *   | ? | _ *     * _ | 2 |   *   | ? | _ *      * _ | 2 |   *   | ? | _ *    |
 * |  * _ | F | R | _ *   |     |   * _ | t | R | _ *   |      |   * _ | F | R | _ *   |    |
 * | ? |  * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |      | ? |   * _ | _ *   | ? |    |
 * * _ | F |  *   | R | _ * 2R^ * _ | t |   *   | R | _ *  Rv  * _ | t |   *   | R | _ *    |
 * |  * _ | ? | ? | _ *   | --> |   * _ | ? | ? | _ *   | -->  |   * _ | ? | ? | _ *   |    |
 * | F |  * _ | _ *   | R |     | F |   * _ | _ *   | R |      | F |   * _ | _ *   | R |    |
 * * _ | 5 |  *   | ? | _ *     * _ | 3 |   *   | ? | _ *      * _ | 3 |   *   | ? | _ *    |
 * * _ | F | R | _ *            * _ | t | R | _ *              * _ | F | R | _ *        |
 * / * _ | _ *                  / * _ | _ *                    / * _ | _ *            |
 * (6)    *                     (4)     *                      (4)     *                |
 * DIAGRAM 8-1. The Down-Slice Move                                                         |
 * Notice the new landmarks in this process -- the LEFT and RIGHT sides, and the BOTTOM,      |
 * FRONT, BACK, and TOP edges of the LEFT layer have all stayed neatly arranged. The change to |
 * the BOTTOM, FRONT, BACK, and TOP was only temporary. (I used "t" and "k" to indicate those |
 * faces that started on TOP or on the BACK; notice how they ended up exactly where they      |
 * started.)                                                                                |
 * Next, I will show you how a simple move (T2) and the "up-slice" can be used together to    |
 * create a very useful three way shuffle of edge cubies.                                     |
 * _ * _    (7)                 _ * _     (2)                                     |
 * _ * _ T _ * _ /              _ * _ T _ * _ /                                       |
 * _ * _ ? _ * _ 8 _ * _        _ * _ ? _ * _ 1 _ * _                                     |
 * * _ T _ * _ k _ * _ T _ *    * _ T _ * _ k _ * _ T _ *                                   |
 * |  * _ 1 _ * _ ? _ *   |     |   * _ 8 _ * _ ? _ *   |                                   |
 * | F |  * _ T _ *   | R |     | k |   * _ T _ *   | l |                                   |
 * * _ | 2 |  *   | ? | _ *     * _ | 7 |   *   | ? | _ *                                   |
 * |  * _ | F | R | _ *   |     |   * _ | k | l | _ *   |                                   |
 * | ? |  * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |                                   |
 * * _ | t |  *   | R | _ * T2  * _ | t |   *   | R | _ *                                   |
 * |  * _ | ? | ? | _ *   | --> |   * _ | ? | ? | _ *   |                                   |
 * | F |  * _ | _ *   | R |     | F |   * _ | _ *   | R |                                   |
 * * _ | 3 |  *   | ? | _ *     * _ | 3 |   *   | ? | _ *                                   |
 * * _ | F | R | _ *            * _ | F | R | _ *                                       |
 * / * _ | _ *                  / * _ | _ *                                           |
 * (4)    *                     (4)     *                                               |
 * DIAGRAM 8-2. Turn the TOP Layer Twice                                                    |
 * _ * _    (2)                 _ * _     (8)                  _ * _     (8)      |
 * _ * _ T _ * _ /              _ * _ T _ * _ /                _ * _ T _ * _ /        |
 * _ * _ ? _ * _ 1 _ * _        _ * _ ? _ * _ 7 _ * _          _ * _ ? _ * _ 7 _ * _      |
 * * _ T _ * _ k _ * _ T _ *    * _ T _ * _ T _ * _ k _ *      * _ T _ * _ T _ * _ T _ *    |
 * |  * _ 8 _ * _ ? _ *   |     |   * _ 3 _ * _ ? _ *   |      |   * _ 3 _ * _ ? _ *   |    |
 * | k |  * _ T _ *   | l |     | k |   * _ f _ *   | l |      | k |   * _ T _ *   | l |    |
 * * _ | 7 |  *   | ? | _ *     * _ | 4 |   *   | ? | _ *      * _ | 4 |   *   | ? | _ *    |
 * |  * _ | k | l | _ *   |     |   * _ | b | R | _ *   |      |   * _ | k | l | _ *   |    |
 * | ? |  * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |      | ? |   * _ | _ *   | ? |    |
 * * _ | t |  *   | R | _ * 2Rv * _ | F |   *   | R | _ *  R^  * _ | F |   *   | R | _ *    |
 * |  * _ | ? | ? | _ *   | --> |   * _ | ? | ? | _ *   | -->  |   * _ | ? | ? | _ *   |    |
 * | F |  * _ | _ *   | R |     | F |   * _ | _ *   | l |      | F |   * _ | _ *   | R |    |
 * * _ | 3 |  *   | ? | _ *     * _ | 5 |   *   | ? | _ *      * _ | 5 |   *   | ? | _ *    |
 * * _ | F | R | _ *            * _ | b | R | _ *              * _ | F | R | _ *        |
 * / * _ | _ *                  / * _ | _ *                   /  * _ | _ *            |
 * (4)    *                     (6)     *                     (6)      *                |
 * DIAGRAM 8-3. The Up-Slice Move                                                            |
 * _ * _    (8)                 _ * _     (4)                  _ * _     (1)      |
 * _ * _ T _ * _ /              _ * _ T _ * _ /                _ * _ T _ * _ /        |
 * _ * _ ? _ * _ 7 _ * _        _ * _ ? _ * _ 3 _ * _          _ * _ ? _ * _ 2 _ * _      |
 * * _ T _ * _ T _ * _ T _ *    * _ T _ * _ T _ * _ T _ *      * _ T _ * _ T _ * _ T _ *    |
 * |  * _ 3 _ * _ ? _ *   |     |   * _ 7 _ * _ ? _ *   |      |   * _ 3 _ * _ ? _ *   |    |
 * | k |  * _ T _ *   | l |     | F |   * _ T _ *   | R |      | F |   * _ T _ *   | R |    |
 * * _ | 4 |  *   | ? | _ *     * _ | 8 |   *   | ? | _ *      * _ | 4 |   *   | ? | _ *    |
 * |  * _ | k | l | _ *   |     |   * _ | F | R | _ *   |      |   * _ | F | R | _ *   |    |
 * | ? |  * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |      | ? |   * _ | _ *   | ? |    |
 * * _ | F |  *   | R | _ * T2  * _ | F |   *   | R | _ *      * _ | F |   *   | R | _ *    |
 * |  * _ | ? | ? | _ *   | --> |   * _ | ? | ? | _ *   |      |   * _ | ? | ? | _ *   |    |
 * | F |  * _ | _ *   | R |     | F |   * _ | _ *   | R |      | F |   * _ | _ *   | R |    |
 * * _ | 5 |  *   | ? | _ *     * _ | 5 |   *   | ? | _ *      * _ | 5 |   *   | ? | _ *    |
 * * _ | F | R | _ *            * _ | F | R | _ *              * _ | F | R | _ *        |
 * / * _ | _ *                  / * _ | _ *                    / * _ | _ *            |
 * (6)    *                     (6)     *                      (6)     *                |
 * Compare this...            ... with the starting position. |
 * DIAGRAM 8-4. Exchanging Three Edge Cubies                                                 |
 * After doing T2 a second time, all corner cubies have been restored to their places, with   |
 * the correct orientation. The edge cubie marked "5" and "6" is also back where it started.  |
 * But the edge cubie marked "1" and "2" has gone to the BOTTOM BACK location. The edge cubie |
 * marked "7" and "8" has gone to the FRONT TOP location, and the cubie marked "3" and "4" has |
 * gone to the BACK TOP location. We have a very useful 3-cycle!                              |
 * In fact, this three-way shuffle is so simple, easy, and effective that you should memorize |
 * it (forwards and backwards, if you would like!).                                           |
 * You may use customization to shuffle any three edge cubies, but be careful -- many a Cube  |
 * has gotten re-scrambled because someone forgot (or messed up) step 3 of the customization  |
 * process. You can proceed one edge cubie at a time, until only three edge cubies are out of |
 * place. (One final three-way shuffle will put these three cubies into their proper places.) |
 * I suggest that you line up (1) one edge cubie (#1) which is out of place, (2) the edge     |
 * cubie which belongs in the space occupied by edge cubie #1, and (3) any other edge cubie   |
 * which is out of place (#3). Then the three way shuffle (if done in the proper direction)   |
 * will restore one edge cubie (#2) to its proper location. Do not forget step 3 of the       |
 * customizing process!                                                                       |
 * _ * _    (1)                                                 |
 * _ * _ T _ * _ /...........Locate the edge cubie which belongs    |
 * _ * _ ? _ * _ 2 _ * _        in the BOTTOM BACK location "#2"        |
 * * _ T _ * _ T _ * _ T _ *      here at the BACK TOP edge.              |
 * Locate any other ........._ 3 _ * _ ? _ *  |                                               |
 * out-of-place edge  | F |   * _ T _ *   | R |                                               |
 * cubie "#3" here    * _ | 4 |   *   | ? | _ *                                               |
 * at the FRONT TOP   |   * _ | F | R | _ *   |                                               |
 * edge.              | ? |   * _ | _ *   | ? |                                               |
 * * _ | F |  *   | R | _ * ..... Locate an out-of-place edge cubie "#1"  |
 * |  * _ | ? | ? | _ *   |       at the BOTTOM BACK edge.                |
 * | F |  * _ | _ *   | R |       (it's hidden in back.)                  |
 * The edge cubie     * _ | 5 |   *   | ? | _ *                                               |
 * at BOTTOM FRONT..........._ | F | R | _ *          This is another view of the starting    |
 * gets returned to         / * _ | _ *               position for shuffling three edge       |
 * this position.         (6)     *                   cubies.                                 |
 * DIAGRAM 8-5. Recommended Way to Position Edge Cubies                   |
 * Let's try to develop another type of diagram to showw all twelve edge cubies. This may make |
 * "customization" easier to visualize, as we put all twelve edge cubies into their proper    |
 * places. I start with diagram 5-6, but, instead of emphasizing the corner cubies, I         |
 * concentrate on the twelve edge cubies. The result is diagram 8-6. Leaving out a few more   |
 * lines results in diagram 8-7.                                                              |
 * *         This is a repeat         *                                                      |
 * |\        of diagram 5-6F         /|                                                      |
 * | *       from Chapter Five.     * |                                                      |
 * *d|\                            /|d*                                                      |
 * * |\| \_                   _/  |/| *                                                      |
 * |\| *    * _           _ *     * |/|                  Exploded TOP Side                   |
 * | *?|\ A |   * _   _ *   |  3 /|?* |                                                      |
 * *?|\| * _ | ? |  -   | ? | _ * |/|?*    vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv   |
 * |\| *a|  * _ |   B   | _ *   |c* |/|   | d | ? |   A   | ? |   B   | | |   3   | ? | d |  |
 * | *L|\| 1 |  * _   _ *   | C |/|K* |   +---+---+---+---+---+---+---+---+---+---+---+---+  |
 * *\|?* _ | ? |  *   | ? | _ * |/| * |   | ? | a | 1 | ? | 2 | b | ? | C | c | ? |   |   |  |
 * | \| * |  * _ | 2 | b | _ *  |?* |/|   +---+---+---+---+---+---+---+---+---+---+---+---+  |
 * | *?|\| ? |  * _ | _ *   | ? |/|?* |   | ? | L | ? | ? FRONT ? | ? RIGHT ? | ? | K | ? |  |
 * * |\| * _ FRONT  *   RIGHT _ * |/| *   +---+---+---+---+---+---+---+---+---+---+---+---+  |
 * \| *E|  * _ | ? | ? | _ *   |7* |/    |   | ? | E | e | ? | 6 | f | ? | g | 7 | ? |   |  |
 * *?|\| e |  * _ | _ *   | g |/|?*     +---+---+---+---+---+---+---+---+---+---+---+---+  |
 * \| * _ | ? |  *   | ? | _ * |/      |   | ? |   5   | ? |   F   | ? |   G   | ? |   |  |
 * *    * _ | 6 | f | _ *     *        ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^   |
 * \ 5 |   * _ | _ *   |  G /                                                           |
 * \_ | ? |  *   | ? | _/                         Exploded BOTTOM Side                |
 * * _ |  F   | _ *                                                                |
 * * _  _ *                 This time, we will ignore the corner cubies...     |
 * We Are Unwrapping the Cube (Again!)                                                       |
 * DIAGRAM 8-6A.                          DIAGRAM 8-6B.                                      |
 * (exploded TOP)                                      |
 * | lt |        | ft |         | rt |         | kt |                    |
 * |   | lt |    |    | ft |    |    | rt |    |    | kt |    |               |
 * | kl | kl |LEFT| fl | fl |FRONT fr | fr |RIGHT kr | kr |BACK| kl | kl |         |
 * /    |    | bl |    |    | bf |    |    | br |    |    | bk |    |     \         |
 * Overlap     +  --++--  +  --++--  +  --++--  +  --++--  +      Note     |
 * | bl |        | bf |         | br |         | bk |          the       |
 * ++        ++         ++         ++         overlap.   |
 * (exploded BOTTOM)                                    |
 * DIAGRAM 8-6C. ... and emphasize the edge cubies.                                |
 * DIAGRAM 8-6. A New Type of Diagram Emphasizes the Edge Cubies                             |
 * Eventually, you will line up the last three corner cubies, shuffle them into place, and    |
 * undo the moves you made to line up those three edge cubies for the shuffle. That should    |
 * complete the accomplishment of goal three for you! (Yes, it is possible that all twelve    |
 * edge cubies were already in their correct locations. You don't need to use Chapter Eight at |
 * all when that happens!)                                                                    |
 * For the sake of completeness, here is a discussion of a way to position the three cubies   |
 * (#1,#2, and #3) as indicated in diagram 8-5. I am going to assume that you can identify one |
 * edge cubie (#1) which is not in the correct location, then rotate the entire Cube until #1 |
 * is positioned at the BOTTOM BACK edge of the Cube. You may want to keep careful notes as   |
 * you proceed. You will need to remember how to undo both parts of this two-part             |
 * customization process.                                                                     |
 * After we have rotated the Cube to get the #1 edge cubie into the BOTTOM BACK      |
 * position, there are eleven possible places to find edge cubie #2.                 |
 * Part 1 Case 1. ++    ++    ++    ++                                        |
 * | #2 |   |    |    |    |    |    |<--This is where we want #2.            |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Move:                |
 * ++++++++++ "Tv".                            |
 * |   |    |    |    |    |    | #1 |                                        |
 * Part 1 Case 2. ++    ++    ++    ++                                        |
 * |   |    | #2 |    |    |    |    |<--This is where we want #2.            |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Move:                |
 * ++++++++++ "T2".                            |
 * |   |    |    |    |    |    | #1 |                                        |
 * Part 1 Case 3. ++    ++    ++    ++                                        |
 * |   |    |    |    | #2 |    |    |<--This is where we want #2.            |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Move:                |
 * ++++++++++ "T^".                            |
 * |   |    |    |    |    |    | #1 |                                        |
 * Part 1 Case 4. ++    ++    ++    ++                                        |
 * |   |    |    |    |    |    | #2 |<--This is where we want #2.            |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ none needed for part one.        |
 * |   |    |    |    |    |    | #1 |                                        |
 * Part 1 Case 5. ++    ++    ++    ++                                        |
 * |   |    |    |    |    |    |    |<--This is where we want #2.            |
 * | #2 | L |    | F  |    | R  |    | K  | #2 |  Customizing Moves:               |
 * ++++++++++ "Lv Tv".                         |
 * |   |    |    |    |    |    | #1 |                                        |
 * Part 1 Case 6. ++    ++    ++    ++                                        |
 * |   |    |    |    |    |    |    |<--This is where we want #2.            |
 * |   | L  | #2 | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "L^ Tv".                         |
 * |   |    |    |    |    |    | #1 |                                        |
 * Part 1 Case 7. ++    ++    ++    ++                                        |
 * |   |    |    |    |    |    |    |<--This is where we want #2.            |
 * |   | L  |    | F  | #2 | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "F^ T2".                         |
 * |   |    |    |    |    |    | #1 |                                        |
 * Part 1 Case 8. ++    ++    ++    ++                                        |
 * |   |    |    |    |    |    |    |<--This is where we want #2.            |
 * |   | L  |    | F  |    | R  | #2 | K  |    |  Customizing Moves:               |
 * ++++++++++ "R^ T^".                         |
 * |   |    |    |    |    |    | #1 |                                        |
 * Part 1 Case 9. ++    ++    ++    ++                                        |
 * |   |    |    |    |    |    |    |<--This is where we want #2.            |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "L2 Tv".                         |
 * | #2 |   |    |    |    |    | #1 |                                        |
 * Part 1 Case 10. ++   ++    ++    ++                                        |
 * |   |    |    |    |    |    |    |<--This is where we want #2.            |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "F2 T2".                         |
 * |   |    | #2 |    |    |    | #1 |                                        |
 * Part 1 Case 11. ++   ++    ++    ++                                        |
 * |   |    |    |    |    |    |    |<--This is where we want #2.            |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "R2 T^".                         |
 * |   |    |    |    | #2 |    | #1 |                                        |
 * DIAGRAM 8-7. Positioning Cubie #2 Without Disturbing #1                                    |
 * Now we need to position any other out-of-place edge cubie (#3), without disturbing either  |
 * edge cubie #1 or edge cubie #2. This is the second part of the two-part customization      |
 * process. You should be able to find an out-of-place edge cubie (#3) in one of the ten      |
 * locations shown in diagram 8-8.                                                            |
 * After we have rotated the Cube to get the #1 edge cubie into the BOTTOM BACK      |
 * position, and we have done Part 1 customizing moves to get the #2 edge cubie      |
 * into the BACK TOP position, there are ten possible places to find edge cubie #3.  |
 * __ This is where we want #3.                               |
 * Part 2 Case 1.               /                                                             |
 * | #3 |   |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "Lv Fv".                         |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 2.               /                                                             |
 * |   |    | #3 |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ none needed for part two.        |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 3.               /                                                             |
 * |   |    |    |    | #3 |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "R^ F^".                         |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 4.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * | #3 | L |    | F  |    | R  |    | K  | #3 |  Customizing Moves:               |
 * ++++++++++ "L2 Fv".                         |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 5.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  | #3 | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "Fv".                            |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 6.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  | #3 | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "F^".                            |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 7.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  | #3 | K  |    |  Customizing Moves:               |
 * ++++++++++ "R2 F^".                         |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 8.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "L^ Fv".                         |
 * | #3 |   |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 9.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "F2".                            |
 * |   |    | #3 |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 10.              /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "Rv F^".                         |
 * |   |    |    |    | #3 |    | #1 |                                        |
 * DIAGRAM 8-8. Positioning Cubie #3 Without Disturbing #2 Or #1                              |
 * When edge cubies #1, #2, and #3 are in position, do the three-way shuffle of edge cubies,  |
 * just as we developed it earlier in this chapter. Then undo the customization process you   |
 * used to get the three edge cubies into position. Be sure to undo part two (the part which  |
 * positioned edge cubie #3) first, then undo part one (the part which positioned edge        |
 * cubie #2) second.                                                                          |
 * Let's just check to see if you really understand how to "undo" the "customization" process |
 * whenn there are two parts to it, such as we have in this chapter. Here is an example.      |
 * Suppose, after manipulating the entire Cube to get edge cubie #1 into position, we find    |
 * that "Part 1 Case 11" is our situation, according to diagram 8-7. We apply the customizing |
 * moves, "R2 T^". Then suppose we find that "Part 2 Case 10" is our situation for positioning |
 * cubie #3. The customizing moves are "Rv F^", so we make these moves. Having gotten all     |
 * three edge cubies where we want them, we do the three-way shuffle, "Down-slice, TOP twice, |
 * Up-slice, TOP twice", or "2R^ Rv T2 2R^ Rv T2". How do we "undo" the customization?        |                                            |                                                                                |
 * Answer for the example: the entire customization, both part one and part two, is this:     |
 * "R2 T^ Rv F^". So, to undo this. the moves are: "Fv R^ Tv R2".                             |
 * Common sense should allow you to correctly position all twelve edge cubies, using all of   |
 * the directions which I have given you in this chapter. However, you will need to be        |
 * careful, especially in selecting ways to "customize", and in "undoing" your customizations. |
 * Careful attention to the details of the 3-way shuffle may help you orient all edge cubies  |
 * as you go along, but Chapter Nine, "Rubik's Maneuver -- How to Flip Two Edge Cubies", is   |
 * provided just in case some edge cubies do not have the correct orientation.                |
 * 1) If you suceeded in getting all twelve edge cubies properly located, there are only          #
 * 2) ((2 to the 12th power) / 2) = 4,096 / 2 = 2,046 ways these edge cubies can be oriented. We  #
 * 3) are closing in on the final, complete solution -- all siz sides -- of the Cube!             #
 * |   |    | #3 |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ none needed for part two.        |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 3.               /                                                             |
 * |   |    |    |    | #3 |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "R^ F^".                         |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 4.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * | #3 | L |    | F  |    | R  |    | K  | #3 |  Customizing Moves:               |
 * ++++++++++ "L2 Fv".                         |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 5.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  | #3 | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "Fv".                            |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 6.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  | #3 | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "F^".                            |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 7.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  | #3 | K  |    |  Customizing Moves:               |
 * ++++++++++ "R2 F^".                         |
 * |   |    |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 8.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "L^ Fv".                         |
 * | #3 |   |    |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 9.               /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "F2".                            |
 * |   |    | #3 |    |    |    | #1 |                                        |
 * __ This is where we want #3.                               |
 * Part 2 Case 10.              /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "Rv F^".                         |
 * |   |    |    |    | #3 |    | #1 |                                        |
 * DIAGRAM 8-8. Positioning Cubie #3 Without Disturbing #2 Or #1                              |
 * When edge cubies #1, #2, and #3 are in position, do the three-way shuffle of edge cubies,  |
 * just as we developed it earlier in this chapter. Then undo the customization process you   |
 * used to get the three edge cubies into position. Be sure to undo part two (the part which  |
 * positioned edge cubie #3) first, then undo part one (the part which positioned edge        |
 * cubie #2) second.                                                                          |
 * Let's just check to see if you really understand how to "undo" the "customization" process |
 * whenn there are two parts to it, such as we have in this chapter. Here is an example.      |
 * Suppose, after manipulating the entire Cube to get edge cubie #1 into position, we find    |
 * that "Part 1 Case 11" is our situation, according to diagram 8-7. We apply the customizing |
 * moves, "R2 T^". Then suppose we find that "Part 2 Case 10" is our situation for positioning |
 * cubie #3. The customizing moves are "Rv F^", so we make these moves. Having gotten all     |
 * three edge cubies where we want them, we do the three-way shuffle, "Down-slice, TOP twice, |
 * Up-slice, TOP twice", or "2R^ Rv T2 2R^ Rv T2". How do we "undo" the customization?        |                                            |                                                                                |
 * Answer for the example: the entire customization, both part one and part two, is this:     |
 * "R2 T^ Rv F^". So, to undo this. the moves are: "Fv R^ Tv R2".                             |
 * Common sense should allow you to correctly position all twelve edge cubies, using all of   |
 * the directions which I have given you in this chapter. However, you will need to be        |
 * careful, especially in selecting ways to "customize", and in "undoing" your customizations. |
 * Careful attention to the details of the 3-way shuffle may help you orient all edge cubies  |
 * as you go along, but Chapter Nine, "Rubik's Maneuver -- How to Flip Two Edge Cubies", is   |
 * provided just in case some edge cubies do not have the correct orientation.                |
 * 1) If you suceeded in getting all twelve edge cubies properly located, there are only          #
 * 2) ((2 to the 12th power) / 2) = 4,096 / 2 = 2,046 ways these edge cubies can be oriented. We  #
 * 3) are closing in on the final, complete solution -- all siz sides -- of the Cube!             #
 * __ This is where we want #3.                               |
 * Part 2 Case 10.              /                                                             |
 * |   |    |    |    |    |    | #2 |                                        |
 * |   | L  |    | F  |    | R  |    | K  |    |  Customizing Moves:               |
 * ++++++++++ "Rv F^".                         |
 * |   |    |    |    | #3 |    | #1 |                                        |
 * DIAGRAM 8-8. Positioning Cubie #3 Without Disturbing #2 Or #1                              |
 * When edge cubies #1, #2, and #3 are in position, do the three-way shuffle of edge cubies,  |
 * just as we developed it earlier in this chapter. Then undo the customization process you   |
 * used to get the three edge cubies into position. Be sure to undo part two (the part which  |
 * positioned edge cubie #3) first, then undo part one (the part which positioned edge        |
 * cubie #2) second.                                                                          |
 * Let's just check to see if you really understand how to "undo" the "customization" process |
 * whenn there are two parts to it, such as we have in this chapter. Here is an example.      |
 * Suppose, after manipulating the entire Cube to get edge cubie #1 into position, we find    |
 * that "Part 1 Case 11" is our situation, according to diagram 8-7. We apply the customizing |
 * moves, "R2 T^". Then suppose we find that "Part 2 Case 10" is our situation for positioning |
 * cubie #3. The customizing moves are "Rv F^", so we make these moves. Having gotten all     |
 * three edge cubies where we want them, we do the three-way shuffle, "Down-slice, TOP twice, |
 * Up-slice, TOP twice", or "2R^ Rv T2 2R^ Rv T2". How do we "undo" the customization?        |                                            |                                                                                |
 * Answer for the example: the entire customization, both part one and part two, is this:     |
 * "R2 T^ Rv F^". So, to undo this. the moves are: "Fv R^ Tv R2".                             |
 * Common sense should allow you to correctly position all twelve edge cubies, using all of   |
 * the directions which I have given you in this chapter. However, you will need to be        |
 * careful, especially in selecting ways to "customize", and in "undoing" your customizations. |
 * Careful attention to the details of the 3-way shuffle may help you orient all edge cubies  |
 * as you go along, but Chapter Nine, "Rubik's Maneuver -- How to Flip Two Edge Cubies", is   |
 * provided just in case some edge cubies do not have the correct orientation.                |
 * 1) If you suceeded in getting all twelve edge cubies properly located, there are only          #
 * 2) ((2 to the 12th power) / 2) = 4,096 / 2 = 2,046 ways these edge cubies can be oriented. We  #
 * 3) are closing in on the final, complete solution -- all siz sides -- of the Cube!             #
 * careful, especially in selecting ways to "customize", and in "undoing" your customizations. |
 * Careful attention to the details of the 3-way shuffle may help you orient all edge cubies  |
 * as you go along, but Chapter Nine, "Rubik's Maneuver -- How to Flip Two Edge Cubies", is   |
 * provided just in case some edge cubies do not have the correct orientation.                |
 * 1) If you suceeded in getting all twelve edge cubies properly located, there are only          #
 * 2) ((2 to the 12th power) / 2) = 4,096 / 2 = 2,046 ways these edge cubies can be oriented. We  #
 * 3) are closing in on the final, complete solution -- all siz sides -- of the Cube!             #
 * 1) ((2 to the 12th power) / 2) = 4,096 / 2 = 2,046 ways these edge cubies can be oriented. We  #
 * 2) are closing in on the final, complete solution -- all siz sides -- of the Cube!             #

title: Rubik's Cube/Moving Edge Cubies


text:
Talk:Grub composting

Creating grub composting article
Hello. I'm being bold and creating an article on grub composting here (unless someone objects or has a better idea). The content will be moved from the Wikipedia article on Hermetia illucens. I believe this may be the best way to compost feces and putrid food waste, and convert these into animal feed. I'm trying to figure out when this can be done safely and how to design composting bins. The global impact this could have, especially in the developing world, is enormous.

There is a bit of edit warring coming from some rival business interests (they are linking to their sites, deleting links to rival sites) but they are also adding useful info. Some of it is Original Research or "unreliable" due to conflicts of interest. Some of it may fail Wikipedia's notability guideline. I'm hoping that here we could write it so that it is in a neutral POV where the various theories and research can be presented and reviewed by peers with full disclosure of everyone's financial interests.

The "debating" was mainly around who has the better bin design and who owns a patent or trademark to "BioPod". My interest is designing bins from reused or common materials. There are many research questions regarding how to design bins which will become clear as the article develops (here or elsewhere). It appears that Robert Olivier has has recently created an account at Wikipedia. He is a notable researcher and entrepreneur involved in the "debates". He has also expressed an interest in open research and so may be willing to help us according to the policies here.

Rudork 02:06, 26 November 2011 (UTC)

title: Talk:Grub composting