text:
User:Egm6321.f11.team4/HW6

= Problem R*6.1 Solve Nonhomogeneous L2-ODE-CC = From [[media:Pea1.f11.mtg33.djvu|Mtg 33-1]]

Given
Equation (1) p.32-5 is $$\displaystyle a_2y''+a_1y'+a_0y=F(t) $$. Equation (1) p.16-5 is $$\displaystyle f_{tt}+2pf_{ty}+p^2f_{yy}=g_{tp}+pg_{yp}-g_y $$. Equation (2) p.16-5 is $$\displaystyle f_{tp}+pf_{yp}+2f_y=g_{pp} $$.

Find
$$\displaystyle \mathbf {1)} $$ Find the PDEs that govern the integrating factor $$\displaystyle h(t,y) $$ for equation (1) p.32-5. Can you solve these PDEs for $$\displaystyle h(t,y) $$? $$\displaystyle \mathbf {2.1)} $$ Find $$\displaystyle (\bar a_1, \ \bar a_0) $$ in terms of $$\displaystyle (a_0, \ a_1, \ a_2) $$. $$\displaystyle \mathbf {2.2)} $$ Find the quadratic equation for $$\displaystyle \alpha $$. $$\displaystyle \mathbf {2.3)} $$ Find the reduced-order equation. $$\displaystyle \mathbf {2.4)} $$ Use the IFM to solve (3) p.33-3. $$\displaystyle \mathbf {2.5)} $$ Show that $$\displaystyle \alpha \beta = \frac{a_0}{a_2}, \ \ \alpha + \beta = \frac{a_1}{a_2} $$. $$\displaystyle \mathbf {2.6)} $$ Deduce the particular solution $$\displaystyle y_p(t) $$ for general excitation $$\displaystyle F(t) $$. $$\displaystyle \mathbf {2.7)} $$ Verify result with table of particular solutions for: $$\displaystyle F(t)=t\exp(bt) $$. $$\displaystyle \mathbf {2.8)} $$ Solve the nonhomogeneous L2-ODE-CC (1) p.32-5 with $$\displaystyle F(t)=\tanh t $$. For the coefficients $$\displaystyle (a_0, \ a_1, \ a_2) $$, consider two different characteristic equations: $$\displaystyle \mathbf {2.8.1)} $$ $$\displaystyle (r+1)(r-2)=0 $$. $$\displaystyle \mathbf {2.8.2)} $$ $$\displaystyle (r-4)^2=0 $$. $$\displaystyle \mathbf {2.9)} $$ Determine the fundametal period of undamped free vibration, and plot the solution for the excitation $$\displaystyle F(t)=\tanh t $$ for about 5 periods, assuming zero initial conditions.

Solution Part 1
$$\displaystyle \mathbf {1)} $$ We need equation (1) p.32-5 to satisfy the first exactness condition for N2-ODEs, $$\displaystyle g(t,y,p)+f(t,y,p)y''=0 $$. Hence, $$\displaystyle f(t,y,p)=h(t,y)*a_2 $$ $$\displaystyle g(t,y,p)=h(t,y)*[a_1y'+a_0y-F(t)] $$. Using the second exactness condition {equations (1)-(2) p. 16-5}, we obtain the following PDEs: $$\displaystyle a_2{h_{tt}}+2pa_2h_{ty}+p^2a_2h_{yy}=a_1h_t+\cancel{pa_1h_y}-\cancel{a_1ph_y}-a_0h-a_0yh_y-F(t)h_y $$. $$\displaystyle 2a_2h_y=0 $$. Combine the two equations to obtain $$\displaystyle a_2h_{tt}-a_1h_t+a_0h=0 $$. The PDEs reduce to a single ODE that cannot be solved.

Solution Part 2.1
We are given the trial solution as $$\displaystyle h(t)=\exp(\alpha t) $$. So, $$\displaystyle \int \exp(\alpha t)[a_2y''+a_1y'+a_0y]dt=\int \exp(\alpha t) F(t)dt $$. Then, $$\displaystyle \int \exp(\alpha t)[a_2y+a_1y'+a_0y]dt=\exp(\alpha t)[\cancel{\bar a_2}y+\bar a_1y'+\bar a_0y] $$. Differentiate the RHS to obtain, $$\displaystyle \int \exp(\alpha t)[a_2y+a_1y'+a_0y]dt=\int \exp(\alpha t)[\bar a_1y+(\bar a_1\alpha+\bar a_0)y'+(\bar a_0\alpha)y]dt $$. Equate the two sides to obtain the three expressions below. $$\displaystyle a_2=\bar a_1 $$. $$\displaystyle a_1=\bar a_1 \alpha+\bar a_0 $$. $$\displaystyle a_0=\bar a_0 \alpha $$. Rearrange the second and third equations to yield $$\displaystyle \bar a_1= a_2 $$. $$\displaystyle \bar a_0=a_1-\alpha a_2 $$. $$\displaystyle \bar a_0=\frac{a_0}{\alpha} $$.

Solution Part 2.2
Using the last two equations, we can obtain the quadratic equation for $$\displaystyle \alpha $$. $$\displaystyle a_1-\alpha a_2=\frac{a_0}{\alpha} $$. $$\displaystyle \alpha*[a_1-\alpha a_2]=a_0 $$ $$\displaystyle \alpha a_1-\alpha^2 a_2=a_0 $$ $$\displaystyle a_2 \alpha^2 - a_1 \alpha + a_0 = 0 $$

Solution Part 2.3
The reduced-order equation is obtained using equations (1) p.33-2 and (2) p.33-1. $$\displaystyle \exp(\alpha t)[\bar a_1y'+\bar a_0y]=\int \exp(\alpha t)F(t)dt $$ $$\displaystyle \bar a_1y'+\bar a_0y=\exp(-\alpha t)\int \exp(\alpha t)F(t)dt $$

Solution Part 2.4
Next, we will use the IFM to solve the reduced-order equation. $$\displaystyle \bar h(t)=\exp[\int \beta dt]=\exp(\beta t) $$ where $$\displaystyle \beta=\frac{\bar a_0}{\bar a_1} $$ $$\displaystyle y(t)=\frac{1}{\bar h(t)}\int^t \bar h(s)B(s)ds $$ Divide the result of Part 2.3 by $$\displaystyle \bar a_1 $$ to obtain B(s). $$\displaystyle y'+\beta y = \frac{\exp(-\alpha t)}{\bar a_1} \int \exp(\alpha t)F(t)dt = B(s) $$ Then we can combine $$\displaystyle h(t), \ B(s) $$ into $$\displaystyle y(t) $$ to find $$\displaystyle y(t)=\frac{1}{[\exp(\beta t)]}\int^t [\exp(\beta s)] [\frac{\exp(-\alpha s)}{\bar a_1} \int \exp(\alpha t)F(t)dt] ds $$. $$\displaystyle y(t)=\frac{1}{\bar a_1[\exp(\beta t)]}\int^t [\exp((\beta-\alpha) s)] \left[ \int \exp(\alpha t)F(t)dt \right] ds $$

Solution Part 2.5
Rearrange $$\displaystyle \beta=\frac{\bar a_0}{\bar a_1} $$ to $$\displaystyle \bar a_0=\beta\bar a_1 $$. Substitute $$\displaystyle a_2=\bar a_1 $$ to $$\displaystyle \bar a_0=a_2\beta $$. That's equal to $$\displaystyle \frac{a_0}{\alpha} $$ and after rearranging one obtains $$\displaystyle \alpha\beta=\frac{a_0}{a_2} $$. To reach the other equality, just set the equation equal to $$\displaystyle a_1-\alpha a_2 $$ instead of $$\displaystyle \frac{a_0}{\alpha} $$. $$\displaystyle a_2\beta=a_1-\alpha a_2 $$. Rearranging yields $$\displaystyle \beta+\alpha=\frac{a_1}{a_2} $$.

Solution Part 2.6
Referencing R5.7, the below equation is given for an Euler L2-ODE-CC. $$\displaystyle y(x) = C_1 x e^{\lambda x} + C_2 e^{\lambda x} $$. After careful inspection, one can deduce that the particular solution is the last equation for Part 2.4 WITHOUT the constants of integration. $$\displaystyle y(t)=\frac{1}{\bar a_1[\exp(\beta t)]}\int^t [\exp((\beta-\alpha) s)] \left[ \int \exp(\alpha t)F(t)dt \right] ds $$

Solution Part 2.7
$$\displaystyle \int e^{(\alpha t)} F(t) dt $$ $$\displaystyle F(t) = t e^{bt} $$ $$\displaystyle \int t e^{(\alpha+b)t} dt $$ $$\displaystyle \frac{e^{(\beta-\alpha)t} ( b t + \alpha t - 1 )}{(\alpha + b)^2} $$ $$\displaystyle \int e^{(\beta-\alpha)s} \left[ e^{(\alpha+b)s} \frac{(bs+\alpha s-1)}{(\alpha + b)^2} \right] ds $$ $$\displaystyle \int e^{(\beta+b)s} \frac{(bs+\alpha s-1)}{(\alpha + b)^2} ds $$ $$\displaystyle e^{(\beta+b)t} \frac{\frac{s(\alpha+b)}{b+\beta} - \frac{\alpha+2b+\beta} {(b+\beta)^2}} {(\alpha + b)^2} $$ $$\displaystyle C_3 = \frac{\alpha+b}{b+\beta} $$ $$\displaystyle C_4 = \frac{\alpha+2b+\beta}{(b+\beta)^2} $$ $$\displaystyle C_5 = (\alpha+b)^2 $$ $$\displaystyle e^{(\beta+b)t} \frac{C_3t - C_4}{C_5} $$ $$\displaystyle y_P(t) = e^{bt} \frac{1}{a_2} \frac{C_3t - C_4}{C_5} $$ $$\displaystyle y_P(t) = C_{P1} t e^{bt} + C_{P2} e^{bt} $$ $$\displaystyle C_{P0} = \frac{C_3}{a_2 C_5} $$ $$\displaystyle C_{P1} = \frac{-C_4}{a_2 C_5} $$ NOTE: The following two integrals were performed using Wolfram|Alpha. integrate t*exp((\alpha+b)*t) dt integrate (exp((\beta+b)*s)*((b*s+\alpha*s-1)/((\alpha+b)^2)) ds Now compare to the Table. $$\displaystyle f(t) = t e^{bt} $$ $$\displaystyle y_P(t) = x^m (\sum_{i=0}^n c_i t_i) e^{bt} $$ $$\displaystyle y_P(t) = \cancel{x^m} C_{P0} \cancel{t^0} e^{bt} + \cancel{x^m} t C_{P1} e^{bt} $$ $$\displaystyle y_P(t) = C_{P0} e^{bt} + t C_{P1} e^{bt}  $$ Other than the arbitrary constant numbering convention, they match.

Solution Part 2.8.1
$$\displaystyle y(t)=\frac{1}{\cancel{\bar a_1}[\exp(\beta t)]}\int^t [\exp((\beta-\alpha) s)] \left[ \int \exp(\alpha t)F(t)dt \right] ds $$ $$\displaystyle \int e^{(\alpha t)} F(t) dt $$ $$\displaystyle F(t) = \tanh{t} $$ $$\displaystyle \int e^{(\alpha t)} [\tanh{t}] dt $$ $$\displaystyle y(t)''-y(t)'-2=\tanh(t) $$ $$\displaystyle a_2y(t)''+a_1y(t)'+a_0=\tanh(t) $$ $$\displaystyle \alpha=-1, \ \beta=2 $$ $$\displaystyle e^{-t} - 2 \tan^{-1}(e^{-t}) $$ $$\displaystyle \int^t e^{3s} \left[ e^{-s} - 2 \tan^{-1}(e^s) \right] ds $$ $$\displaystyle \frac{1}{6}[e^{2s}+2\log(e^{2s}+1)-4e^{3s}(tan^{-1}(e^{-s}))] $$ $$\displaystyle y_P(t) = \frac{1}{6} \frac{[e^{2s}+2\log(e^{2s}+1)-4e^{3s}(tan^{-1}(e^{-s}))]}{e^{-2t}} $$ NOTE: Wolfram|Alpha was used to determine the following integrals: integrate (e^(-t)*tanh(t)) dt integrate (e^(3s))(e^(-s)-2 tan^(-1)(e^(-s))) ds

Solution Part 2.8.2
$$\displaystyle \alpha=4, \ \beta=4 $$ $$\displaystyle y(t)=\frac{1}{\cancel{\bar a_1}[\exp(\beta t)]}\int^t [\exp((\beta-\alpha) s)] \left[ \int \exp(\alpha t)F(t)dt \right] ds $$ $$\displaystyle y(t)=e^{-4t} \int^t \cancel{e^{(\beta-\alpha)s}} \left[ \int e^{4t} F(t) dt \right] ds $$ $$\displaystyle \int e^{4t} F(t) dt $$ $$\displaystyle F(t) = \tanh{t} $$ $$\displaystyle \int e^{4t} [\tanh{t}] dt $$ $$\displaystyle -e^{2t} + \frac{e^{4t}}{4} + \log(e^{2t}+1) $$ $$\displaystyle \frac{1}{16} \left[ -8[Li_2 (-e^{2t})] - 8 e^{2t} + e^{4t} \right] $$ $$\displaystyle y_P(t) = \frac{e^{-4}}{16} \left[ -8[Li_2 (-e^{2t})] - 8 e^{2t} + e^{4t} \right] $$ where $$\displaystyle Li_2(-e^{2t})] $$ is the polylogarithm function NOTE: Wolfram|Alpha was used to determine the following integrals: integrate ((e^(4t))(\tanh(t))) dt integrate (-e^(2t)+(e^(4t)/4)+log(e^(2t)+1)) dt

Author
Egm6341.f11.team4.allen 10:56, 16 November 2011 (UTC)

= Problem R*6.2 - Show (1)p.34-6 agrees with King 2003 (1.6)p.8 = From [[media:Pea1.f11.mtg35.djvu|Mtg 35-1]]

Given
Equation (1) p.34-6 is $$\displaystyle y_P(x)=u_1(x)\int\frac{1}{h(x)}\left[\int h(x)\frac{f(x)}{u_1(x)}\,dx\right]\,dx$$.

Find
$$\displaystyle \mathbf {A)} $$ Show that Equation (1) p.34-6 agrees with King 2003 (1.6) p.8 which is $$\displaystyle y_P(x)=\int^x f(s)\left[\frac{u_1(s)u_2(x)-u_1(x)u_2(s)}{W(s)}\right]\,ds $$ with $$\displaystyle W := u_1 u_2' - u_2 u_1' = \begin{vmatrix} \ u_1 & u_2 \\ u_1' & u_2' \end{vmatrix}$$. $$\displaystyle \mathbf {HINT} \left( \frac{u_2}{u_1} \right)' = \frac{1}{h} $$ $$\displaystyle \mathbf {B)} $$ Also, discuss the feasibility of the following choices for variation of parameters: $$\displaystyle \mathbf {B.1)} y(x) = U(x) \pm u_1(x) $$. $$\displaystyle \mathbf {B.2)} y(x) = U(x) / u_1(x) $$. $$\displaystyle \mathbf {B.3)} y(x) = u_1(x) / U(x) $$.

Solution
$$\displaystyle \mathbf {A)} $$ Starting with the HINT, we'll apply the following basic differentiation rule: $$\displaystyle \frac{d}{dx}\left[\frac{u}{v}\right]=\frac{vu'-uv'}{v^2}$$ to find $$\displaystyle \frac{d}{dx}\left[\frac{u_2}{u_1}\right] = \frac{u_1u_2'-u_2u_1'}{u_1^2} $$. So, $$\displaystyle h = \frac{u_1^2}{u_1u_2'-u_2u_1'} $$. Next, we'll use integration by parts, $$\displaystyle \int u dv = uv-\int v du $$ with the following: $$\displaystyle u = \int h(x) \frac{f(x)}{u_1(x)} dx $$. $$\displaystyle dv = \frac{1}{h} = \frac{u_1u_2'-u_2u_1'}{u_1^2} dx $$. $$\displaystyle v = \frac{u_2}{u_1} $$. $$\displaystyle du = \frac{h(x)f(x)}{u_1(x)} dx $$. $$\displaystyle y_P(x) = u_1(x)\int\frac{1}{h(x)}\left[\int h(x)\frac{f(x)}{u_1(x)}\,dx\right]\,dx$$ $$\displaystyle y_P(x) = u_1(x) \int u dv $$ $$\displaystyle y_P(x) = u_1(x) [uv - \int v du] $$ $$\displaystyle y_P(x) = u_1(x) \left[ \frac{u_2(x)}{u_1(x)} \int h(s) \frac{f(s)}{u_1(s)}ds - \int \frac{u_2(s)}{u_1(s)} \frac{h(s)f(s)}{u_1(s)} ds \right] $$ $$\displaystyle y_P(x) = \cancel{u_1(x)} \frac{u_2(x)}{\cancel{u_1(x)}} \int \frac{u_1(s)^\cancel{2}}{u_1(s)u_2(s)'-u_2(s)u_1(s)'} \frac{f(s)}{\cancel{u_1(s)}}ds - u_1(x) \int \frac{u_2(s)}{\cancel{u_1(s)}} \frac{\cancel{u_1(s)^2}}{u_1(s)u_2(s)'-u_2(s)u_1(s)'} \frac{f(s)}{\cancel{u_1(s)}} ds $$ $$\displaystyle y_P(x) = \int^x f(s) \frac{u_1(s)u_2(x)}{u_1(s)u_2(s)'-u_2(s)u_1(s)'} ds - \int^x f(s) \frac{u_1(x)u_2(s)}{u_1(s)u_2(s)'-u_2(s)u_1(s)'} ds $$ $$\displaystyle y_P(x) = \int^x f(s) \frac{u_1(s)u_2(x)-u_1(x)u_2(s)}{u_1(s)u_2(s)'-u_2(s)u_1(s)'} ds $$ $$\displaystyle y_P(x)=\int^x f(s)\left[\frac{u_1(s)u_2(x)-u_1(x)u_2(s)}{W(s)}\right]\,ds $$ with $$\displaystyle W := u_1 u_2' - u_2 u_1' = \begin{vmatrix} \ u_1 & u_2 \\ u_1' & u_2' \end{vmatrix}$$. $$\displaystyle \mathbf {B.1)} y(x) = U(x) \pm u_1(x) $$. $$\displaystyle [y = U(x) \pm u_1(x)] * a_0(x) $$. $$\displaystyle [y' = U'(x) \pm u_1'(x)] * a_1(x) $$. $$\displaystyle [y = U(x) \pm u_1(x)] * a_2(x) $$. $$\displaystyle a_0y+a_1y'+y = U[a_0]+U'[a_1]+U'' + $$ left over terms without U and its derivatives. $$\displaystyle \mathbf {B.2)} y(x) = U(x) / u_1(x) $$. $$\displaystyle \left[ y = U(x) / u_1(x) \right] * a_0(x) $$. $$\displaystyle \left[ y' = \frac{u_1(x)U'(x)-U(x)u_1'(x)}{u_1(x)^2} \right] * a_1(x) $$. $$\displaystyle \left[ y = \frac{2U(x)u_1'(x)^2-u_1(x)[U(x)u_1(x)+2u_1'(x)U'(x)]+u_1(x)^2U''(x)}{u_1(x)^3} \right] * a_2(x) $$. $$\displaystyle \mathbf {B.3)} y(x) = u_1(x) / U(x) $$. $$\displaystyle \left[ y = u_1(x) / U(x) \right] * a_0(x) $$. $$\displaystyle \left[ y' = \frac{U(x)u_1'(x)-u_1(x)U'(x)}{U(x)^2} \right] * a_1(x) $$. $$\displaystyle \left[ y = \frac{2u_1(x)U'(x)^2-U(x)[u_1(x)U(x)+2U'(x)u_1'(x)]+U(x)^2u_1''(x)}{U(x)^3} \right] * a_2(x) $$. None of these alternative trial solutions are feasible because they don't produce the desired homogeneous equation that cancels out to simplify the equation. Also, there are several terms that do not contain $$\displaystyle U(x) $$ which were a requirement as well.

Author
Egm6341.f11.team4.allen 06:00, 16 November 2011 (UTC)

= Problem R*6.3 - Invalid solution = From [[media:Pea1.f11.mtg35.djvu|Mtg 35-4]]

Given
The first homogeneous solution: Trial solution: Characteristic equation:

Find
Explain why $$\displaystyle r_2(x)$$ is not a valid root, i.e., $$\displaystyle u_2(x) = e^{xr_{2}(x)}$$ is not a valid solution.

Solution
- Solved on our own Validity of $$\displaystyle u_2(x) = e^{xr_2(x)}$$ can be shown by substituting $$\displaystyle y= u_2(x) $$ into the Eq. (4.1). Before the substitution, compute the first and the second derivatives of $$\displaystyle y $$. Substitute $$\displaystyle y, y^{'}, y^{''}$$ in the Eq. (3.6) into the Eq. (3.1), - As shown in the Eq. (3.7), $$\displaystyle u_2(x) $$ is not a valid homogeneous solution.

Author
Contributed by Shin

= Problem R*6.4 - 2nd homogeneous solution by variation of parameters = From [[media:Pea1.f11.mtg35.djvu|Mtg 35-4]]

Given
Application: King 2003 p.28

Find
For the L2-ODE-VC Eq. (4.1), select a valid homogeneous solution, and call it $$\displaystyle u_1 $$. Find the 2nd homegeneous solution $$\displaystyle u_2(x) $$ by variation of parameters, and compare to $$\displaystyle e^{xr_2(x)} $$.

Solution
- Solved on our own Let trial solution be: Then, characteristic equation is: After factoring, the Eq. (4.3) becomes: Therefore, the first homogeneous solution is: Since it was proven that $$\displaystyle r_2(x) $$ is not a valid root in the problem R*6.3, $$\displaystyle r_2(x) $$ is ignored. The second homogeneous solution can be obtained using the equation (4) in the class note [[media:Pea1.f11.mtg34.djvu|34-5]]. The Eq. (4.7) depicts the equation (4) in the class note [[media:Pea1.f11.mtg34.djvu|34-5]]. The term $$\displaystyle a_1(x)$$ is determined by changing the form of the Eq. (4.1) as follows. First, the term $$\displaystyle \frac{1}{h(x)} $$ is first computed in order to solve the Eq. (4.7). Substitute $$\displaystyle a_1(x) $$ in the Eq. (4.9) into the Eq. (4.8), then compute $$\displaystyle \frac{1}{h(x)} $$, Then, substitute $$\displaystyle \frac{1}{h(x)} $$ from the Eq. (4.10) into the Eq. (4.7), - Since the second homogeneous solution has the form of $$ \displaystyle u_2(x) = -x $$, $$ \displaystyle u_2(x) = e^{x r_2(x)} $$ is not valid solution.

Author
Contributed by Shin

= Problem R*6.5 - Homogeneous solutions of L2-ODE-VC = From [[media:Pea1.f11.mtg36.djvu|Mtg 36-3]]

Given
A L2-ODE-VC

Find
For the L2-ODE-VC Eq. (5.1), find 2 homogeneous solutions $$\displaystyle u_1(x) $$ and $$\displaystyle u_2(x) $$ using the trial solution $$\displaystyle y(x) = e^{r(x)} $$.

Solution
- Solved on our own Let trial solution be: Then, characteristic equation is: Upon factoring we get, First homogeneous solution is thus: Since $$\displaystyle r_2(x) $$ is not a constant, therefore it's not a valid root and we have only one valid root, i.e. $$\displaystyle r_1 = 2 $$. $$\displaystyle u_2(x) $$ can be obtained using procedure, Variation of parameters, as given in equation (4) in class notes [[media:Pea1.f11.mtg34.djvu|34-5]]. Therefore we have,

h(x) is given by, $$\displaystyle a_1(x)$$ is determined from Eq. (5.1) written in following form, Therefore $$\displaystyle h(x) $$ is given by,

Now substituting $$\displaystyle \frac{1}{h(x)} $$ from the Eq. (5.10) into the Eq. (5.7), - The homogenous solutions for Eq. (5.1) are, $$ \displaystyle u_1(x) = e^{2x} $$ and $$ \displaystyle u_2(x) = e^{2x} \frac {(x+1)^2}{2} $$.

Author
Contributed by Ankush

= Problem R*6.6 - Find the homogeneous L2-ODE-VC = From [[media:Pea1.f11.mtg36.djvu|Mtg 36-10]]

Given
Trial solution

Characteristic equation

Find
The homogeneous L2-ODE-VC

Solution
According to Eq.6.1, we can get So we have Rearrange the Eq.6.6, we obtain Hence, we get the following equations Solving the equations above, we have Therefore, the L2-ODE-VC is

Author
Contributed by Kexin Ren

= Problem R*6.7 - Show 37-4 (2) using variation of parameters = From [[media:Pea1.f11.mtg37.djvu|Mtg 37-4]]

Find
Show that 2nd homogeneous solutions of the Legendre equation can be obtained from the 1st homogeneous solutions by variation os parameters.

Solution
- Solved on our own Given should be

The Legendre function is

For a second order equation, n=2.

The general from of a homogeneous L2-ODE-VC is

The general solution form is

U(x) is unknown. $$u_1(x)=P_2(x)$$. So solve $$u_2(x)$$ with $$u_1(x)$$.

Substitute (7.7),(7.8),(7.9) into (7.5)

To simplify, set $$a_2=1$$.

Set Z=U' to reduce the order,

Divide $$u_1(x)$$.

To solve (7.15), use IFM to make the equation exact. Then it will be easier to solve the equation.

Multipliy h to the equation (7.15)

At (7.13), we set $$a_2=1$$ to simplify the process. So,

From (7.16), (7.23) and (7.1),

From (7.21),

Author
= Problem R*6.8 - Solve Non-homogeneous L2-ODE-VCs = From [[media:Pea1.f11.mtg37.djvu|Mtg 37-4]]

Given
Non-homogeneous L2-ODE-VCs:

Find
a) Find the 1st homogeneous solution by trial solution. b) Find the complete solution by variation of parameters with $$\displaystyle f(x)=e^{\alpha x} $$

Solution
- Solved on our own

Solution of (8.1)
Assume a trial solution: $$\displaystyle u_1(x)=e^{rx} $$. So the characteristic equation is:

From Problem R*6.3 we know that $$\displaystyle r(x)= \frac{1}{x-1} $$ is not a valid root. So $$\displaystyle r=1 $$ and thus the 1st homogeneous solution is:

Assume the full solution of the form:

then:

Change (8.1) into the form:

where: $$\displaystyle a_2=x-1 $$, $$\displaystyle a_1= -x $$ and $$\displaystyle a_0= 1 $$. So:

We get:

Now in order to solve $$\displaystyle U $$, assume $$\displaystyle Z:=U' $$. So:

Use IFM to make the equation exact:

and:

Then we get:

So the complete solution is:

And the 2nd homogeneous solution is:

Solution of (8.2)
Assume a trial solution: $$\displaystyle u_1(x)=x^c \sin x $$. So the characteristic equation is:

So $$\displaystyle c=-1 $$ and thus the 1st homogeneous solution is:

Just like what we did above, assume the full solution of the form:

So we get:

where: $$\displaystyle a_2=x $$, $$\displaystyle a_1= 2 $$ and $$\displaystyle a_0= x $$.

Again, assume $$\displaystyle Z:=U' $$ and use IFM, then we can get the full solution:

So the complete solution is:

And the 2nd homogeneous solution is:

Author
YuChen

= Problem R6.9 - Non-homogeneous Legendre equation = From [[media:Pea1.f11.mtg37.djvu|Mtg 37-5]]

Find
Find the final solution y(x) by variation of parameters.

Solution
- Solved on our own The given function is

The general solution form is

U(x) is unknown. $$u_1(x)=P_2(x)$$.

$$u_1(x)$$ is a homogeneous solution, From (7.23), (7.24), (7.25) From (9.5),(9.6),(9.7)

To simplify, set $$a_2=1$$.

Set Z=U' to reduce the order,

To solve (.15), use IFM to make the equation exact.

From (9.6), (9.9), (9.17),

As a result, the second homogeneous solution is

Author
=Contributing Members=

title: User:Egm6321.f11.team4/HW6


text:
Rubik's Cube/Ignoring Details

This is where Chapter 04 has been installed. Sorry, folks! Several of the diagrams got messed up in translation!

Ray Calvin Baker 03:38, 10 November 2011 (UTC)

 +-+
 * HOW TO FIND YOUR VERY OWN PERSONAL WAYS TO SOLVE RUBIK'S CUBE                              |
 * (Preliminary April 20, 2007 version)            |
 * by Mr. Ray Calvin Baker                         |
 * This is a FREE educational resource             |
 * Chapter Four - - - - - - - - - - Ignoring Details -- Moving Corner Cubies                  |
 * Goal One is to get all eight corner cubies properly located at the eight corners of the    |
 * Cube. Our "warming up" exercises positioned four of the corner cubies. Now don't be        |
 * alarmed, but we may have to temporarily give up some of this apparent progress in order to |
 * make real and lasting progress toward the goal of getting all six sides of the Cube        |
 * correct. I propose to set the Cube aside for a little while, do some "paper and pencil"    |
 * work to plan some moves, and then apply some carefully planned moves to the Cube.          |
 * In this chapter, I will first describe how I found an essential move (swapping two corner  |
 * cubies). Then, I will show how to use this swap, if necessary, to locate all eight corner  |
 * cubies. Next, after we have thought things through, I will show you another way to organize |
 * a plan. Finally, after everything is laid out for you, I will invite you to pick up your   |
 * Cube once more, and put all eight corner cubies where they belong.                         |
 * YOUR CUBE SHOULD BE ON THE TABLE! YOUR HANDS SHOULD BE ON PAPER AND PENCIL! For now, we are |
 * THINKING about strategic ways to deal with the Cube, not moving cubies!                    |
 * 1) If you were able to position the four corner cubies of the TOP layer correctly, either as   #
 * 2) described in chapter three, or using your own methods, there are now only 4 * 3 * 2 * 1     #
 * 3) = 24 ways to arrange the remaining four corner cubies. I will show you all 24 of these,     #
 * 4) later in this chapter.                                                                      #
 * The best way I have discovered so far on how to find ways to shuffle the corner cubies is  |
 * to make a lot of copies of a simplified diagram of the Cube on scratch paper, then try     |
 * things until something useful shows up. (The brackets "[ ]" above each diagram are used to |
 * indicate which corner cubie is hidden at the BOTTOM BACK LEFT "BKL" location of the Cube.  |
 * The parentheses "" indicate spaces where you can fill in an indication of which cubie   |
 * occupies that corner of the Cube. Use the space between the diagrams to record the         |
 * operation you tried. You may also number the moves, if you want to.)                       |
 * [ ]                 [ ]                  [ ]                   [ ]           |
 * _ _              _  _              _  _               _  _               |
 * _    _        _     _        _     _         _     _            |
 * |       |   __   |        |   __   |        |   __    |        |   __       |
 * _ |  _   #1   _  |  _   #2   _  |  _   #3    _  |  _   #4       |
 * ... and so on.... |
 * This is a sample of part of a work sheet, such as I use to plan moves for corner cubies. |
 * DIAGRAM 4-1. Blank Diagrams for Planning Moves                                           |
 * The first diagrams I made looked like this:                                                |
 * [h]                 [h]                  [h]                                |
 * _ (a) _             _ (a) _              _ (a) _                                    |
 * (b) _    _ (c)      (e) _     _ (c)      (e) _     _ (f)                                |
 * |   (d)    |   Fv   |    (b)    |   R^   |    (c)    |   F^                            |
 * (e) _ |  _ (f)  #1  (g) _  |  _ (f)  #2  (g) _  |  _ (d)  #3                            |
 * (g)       -->       (d)        -->       (b)        -->                           |
 * This is just an                 I began                                                 |
 * arbitrary way                   exploring                                               |
 * to label the                    with a few                                              |
 * corner cubies.                  arbitrary moves.                                        |
 * [h]                 [h]                  [h]                                |
 * _ (a) _             _ (a) _              _ (g) _                                    |
 * (c) _    _ (f)      (c) _     _ (b)      (b) _     _ (c)                                |
 * |   (b)    |   Rv   |    (g)    |   T2   |    (a)    |                                 |
 * (e) _ |  _ (d)  #4  (e) _  |  _ (f)  #5  (e) _  |  _ (f)                                |
 * (g)       -->       (d)        -->       (d)                                      |
 * Comparing this to         T2 looked like an                        |
 * the starting position,    interesting thing                        |
 * b and c have traded       to try, and it was!                      |
 * places, while e and f                                              |
 * are where they started.                                            |
 * DIAGRAM 4-2. I Found a Three-Way Swap                                                   |
 * To evaluate whether or not a series of moves is useful, we need to compare the original    |
 * position of the corner cubies with their final position. Sometimes, we can find "cycles"   |
 * that return many cubies to their original positions, while moving a few cubies in a        |
 * predictable pattern.                                                                       |
 * Comparing this last diagram with the first, we find that five cubies are back at their     |
 * starting position, while three cubies, a, d, and g, have moved. We can diagram this:       |
 * a --> d --> g --> a. (Do you see how and why I wrote this down?) This is called a "cycle"  |
 * (of length three), and it helps us predict what would happen if we did "Fv R^ F^ Rv T2"    |
 * twice, or three times, or more; and we don't have to work out all of the messy details to  |
 * see this! P.S. This is a useful recipe for moving exactly three corner cubies any way we   |
 * want to, if the three corner cubies are in exactly the proper positions. The idea discussed |
 * in Chapter Six, "Customize Your Moves -- Commutation", will allow you to shuffle any three |
 * cubies.                                                                                |
 * Flush with success at finding a useful result so soon, I guessed that I could find a way to |
 * interchange exactly two corner cubies just as easily. I tried to move one of the three     |
 * cubies "out of the way" with a move of the LEFT side, then I tried using the cycle I had   |
 * discovered.                                                                                |
 * YOUR CUBE SHOULD BE ON THE TABLE! YOUR HANDS SHOULD BE ON PAPER AND PENCIL!                |
 * After "Fv R^ F^ Rv T2", the diagrams of the imaginary Cube we are THINKING about looks like |
 * this.                                                                                |
 * [h]                 [g]                  [g]                   [g]          |
 * _ (g) _             _ (b) _              _ (b) _               _ (b) _              |
 * (b) _    _ (c)      (e) _     _ (c)      (h) _     _ (c)       (h) _     _ (f)          |
 * |   (a)    |   L^   |    (a)    |   Fv   |    (e)    |   R^    |    (c)    |   F^      |
 * (e) _ |  _ (f)  #6  (h) _  |  _ (f)  #7  (d) _  |  _ (f)  #8   (d) _  |  _ (a)  #9      |
 * (d)       -->       (d)        -->       (a)        -->        (e)        -->     |
 * Continuing an experiment....                                           |
 * [g]                 [g]                  [g]                   [g]          |
 * _ (b) _             _ (b) _              _ (d) _               _ (d) _              |
 * (c) _    _ (f)      (c) _     _ (e)      (e) _     _ (c)       (h) _  |  _ (c)          |
 * |   (e)    |   Rv   |    (d)    |   T2   |    (b)    |   Fv    |    (e)    |   R^      |
 * (h) _ |  _ (a)  10  (h) _  |  _ (f)  11  (h) _  |  _ (f)  12   (a) _  |  _ (f)  13      |
 * (d)       -->       (a)        -->       (a)        -->        (b)        -->     |
 * [g]                 [g]                  [g]                   [g]          |
 * _ (d) _             _ (d) _              _ (d) _               _ (a) _              |
 * (h) _    _ (f)      (c) _     _ (f)      (c) _     _ (e)       (e) _  |  _ (c)          |
 * |   (c)    |   F^   |    (e)    |   Rv   |    (a)    |   T2    |    (d)    |   Lv      |
 * (a) _ |  _ (b)  14  (h) _  |  _ (b)  15  (h) _  |  _ (f)  16   (h) _  |  _ (f)  17      |
 * (e)       -->       (a)        -->       (b)        -->        (b)        -->     |
 * [h]                                      [h]                                |
 * _ (g) _                                  _ (a) _                                    |
 * (a) _    _ (c)                           (b) _     _ (c)                                |
 * |   (d)    |                             |    (d)    |                                 |
 * (e) _ |  _ (f)                           (e) _  |  _ (f)                                |
 * (b)                                      (g)                                      |
 * Final position                           Original position                              |
 * .. that seemed to result in nothing useful.                                             |
 * DIAGRAM 4-3. An Experiment Which Seemed Useless                                         |
 * This was disappointing! Not every idea of mine worked! Sometimes I had to scribble for an  |
 * hour or two on scratch paper to find a way to do something important and necessary. But    |
 * after searching for a way to interchange exactly two corner cubies, I eventually found     |
 * three more moves that really did accomplish what I needed.                                 |
 * [h]                 [h]                  [h]                   [h]          |
 * _ (g) _             _ (g) _              _ (a) _               _ (a) _              |
 * (a) _    _ (c)      (a) _     _ (d)      (b) _     _ (g)       (b) _     _ (c)          |
 * |   (d)    |   Rv   |    (b)    |   Tv   |    (d)    |   R^    |    (g)    |           |
 * (e) _ |  _ (f)  18  (e) _  |  _ (c)  19  (e) _  |  _ (c)  20   (e) _  |  _ (f)          |
 * (b)       -->       (f)        -->       (f)        -->        (d)                |
 * DIAGRAM 4-4. Three More Moves Provide a Useful Two-Cubie Interchange                    |
 * YOUR CUBE SHOULD BE ON THE TABLE! YOUR HANDS SHOULD BE ON PAPER AND PENCIL!                |
 * Here is a summary of this entire process we have been thinking about.                      |
 * [h]    First,   Fv R^ F^ Rv T2                 [h]    Notice that corner    |
 * _ (a) _        Second,  L^                     _ (a) _        cubies d and g have   |
 * (b) _ T  _ (c)     Third,   Fv R^ F^ Rv T2     (b) _  T  _ (c)    traded places,        |
 * | F (d) R  |      Fourth,  Fv R^ F^ Rv T2      |  F (g) R  |                           |
 * (e) _ |  _ (f)     Fifth,   Lv                 (e) _  |  _ (f)    These are in the      |
 * (g)          Sixth,   Rv Tv R^                 (d)          FRT and BFR           |
 * Original position                              Final position     locations.            |
 * DIAGRAM 4-5. Summary of FRONT RIGHT TOP and BOTTOM FRONT RIGHT SWAP                     |
 * Call this the "FRONT RIGHT TOP and BOTTOM FRONT RIGHT SWAP"!                               |
 * Now that I know a way to interchange two corner cubies, I have completed the difficult part |
 * of the strategic planning which will allow me to accomplish Goal One. Surely you can find  |
 * easier, better, and faster ways to position the corner cubies! (Hint: Why do a three-cycle |
 * twice? Isn't it easier and quicker to do it backwards once instead?)                       |
 * CAUTION! When you try to use a procedure such as this, or any others to be discovered in   |
 * the following chapters, it is EXTREMELY IMPORTANT that EVERY DETAIL and EVERY MOVE be      |
 * EXACTLY CORRECT! Otherwise, you may mess up parts of the Cube you thought you had finished. |
 * You may have to start over! (Oh, well! It's happened to me, LOTS of times!)                |
 * Hey! Not so fast! What if the corner cubies I want to interchange are NOT in the FRONT     |
 * RIGHT TOP and BOTTOM FRONT RIGHT locations? This sounds like it could be a serious problem, |
 * since we correctly positioned all corner cubies of the TOP layer according to directions in |
 * the previous chapter. It is quite possible that two (or more!) corner cubies in the BOTTOM |
 * layer are not correctly positioned for this interchange to work properly.                  |
 * Fortunately, there is an easy solution for this problem, if the corner cubies which need to |
 * be interchanged are adjacent to eache other. (Some examples are: BFR and BFL, BFL and BKL, |
 * and BKL and BKR.) Just use some combination of three-layer moves (like, for example, 3T^,  |
 * 3Tv, 3R^, 3Rv, 3F^, and 3Fv) to rotate the entire Cube until the corner cubies you wish to |
 * switch ARE in the FRONT RIGHT TOP and BOTTOM FRONT RIGHT positions. Then apply the series  |
 * of moves which interchanges those to corner cubies! (This principle will be dicussed more  |
 * thoroughly in Chapter Six, "Customize Your Moves -- Commutation".)                         |
 * What can you do if the corner cubies to be interchanged are at opposite corners of the     |
 * BOTTOM layer?                                                                              |
 * YOUR CUBE SHOULD STILL BE ON THE TABLE!                                                    |
 * You had matched up some        After "3R2" flips the                                     |
 * of the corner cubies on        entire Cube, you MIGHT                                    |
 * the BACK, FRONT, LEFT,         find 2 corner cubies in       If that is what you find,   |
 * RIGHT, and TOP sides.          the correct FRT and KLT       you should be able to       |
 * locations.                   do this!                    |
 * _ * _                          _ * _                         _ * _             |
 * _ * _TOP_ * _                  _ * _klt_ * _                 _ * _klt_ * _         |
 * _ * _ ? _ * _ ? _ * _          _ * _   _ * _   _ * _         _ * _   _ * _   _ * _     |
 * * _TOP_ * _TOP_ * _TOP_ *      * _krt * _   _  * _flt_ *     * _flt_ * _   _ * _krt_ *   |
 * |  * _ ? _ * _ | _ *   |       |   * _   _ * _   _ *   |     |   * _   _ * _   _ *   |   |
 * | F |  * _TOP_ *   | R |       |krt|   * _frt_ *   |flt|     |flt|   * _frt_ *   |krt|   |
 * * _ | ? |  *   | ? | _ *  3R2  * _ |   |   *   |   | _ *     * _ |   |   *   |   | _ *   |
 * |  * _ | F | R | _ *   |  -->  |   * _ |frt|frt| _ *   |     |   * _ |frt|frt| _ *   |   |
 * | ? |  * _ | _ *   | ? |       |   |   * _ | _ *   |   |     |   |   * _ | _ *   |   |   |
 * * _ | F |  *   | R | _ *       * _ |"k"|   *   |"r"| _ *     * _ | F |   *   | R | _ *   |
 * |  * _ | ? | ? | _ *   |       |   * _ |   |   | _ *   |     |   * _ |   |   | _ *   |   |
 * | ? |  * _ | _ *   | ? |       |"k"|   * _ | _ *   |"r"|     | F |   * _ | _ *   | R |   |
 * * _ | ? |  *   | ? | _ *       * _ |   |   *   |   | _ *     * _ |   |   *   |   | _ *   |
 * * _ | ? | ? | _ *              * _ |"k"|"r"| _ *             * _ | F | R | _ *       |
 * * _ | _ *                      * _ | _ *                     * _ | _ *           |
 * What you had at the end        ONE POSSIBILITY after you     What you want.              |
 * of chapter three. (BACK        turn the Cube upside down.    Notice that you had to      |
 * and LEFT sides show a          (I have used "k" and "r"      exchange cubies marked      |
 * similar pattern.)              to indicate that these        "krt" and "klt".            |
 * (This is a repeat of           colors match correctly.)                                  |
 * diagram 3-1B.)                                                                          |
 * DIAGRAM 4-6A.                  DIAGRAM 4-6B.                 DIAGRAM 4-6C.               |
 * DIAGRAM 4-6. How Do You Swap Diagonally Opposite Cubies?                                 |
 * I told you, "You should be able to do this!". Here is "the hard way". (I am going to shrink |
 * the diagrams a bit, leaving out "irrelevant details", so that I can fit more diagrams on   |
 * the page.) Remember, at this stage, we are interested in getting corner cubies to the      |
 * correct LOCATIONS. Their orientation is irrelevant.                                        |
 * YOUR CUBE SHOULD STILL BE ON THE TABLE!                                                    |
 * Here is the plan, using only rotations of the entire Cube, and our newly-discovered SWAP   |
 * sequence. Three SWAPS of adjacent corner cubies results in the SWAp of two diagonally      |
 * opposite corner cubies. (Parentheses, "(" and ")", are used to highlight the pair of corner |
 * cubies about to be SWAPped.)                                                               |
 * 1klt                 1klt                    1klt               1klt              |
 * (2krt    3flt        4frt     3flt)        (4frt     2krt        3flt    2krt          |
 * 4frt)               (2krt                  3flt)                4frt              |
 * DIAGRAM 4-7A.        DIAGRAM 4-7B.         DIAGRAM 4-7C.         DIAGRAM 4-7D.         |
 * DIAGRAM 4-7. Plan for SWAPping Diagonally Opposite Cubies the Hard Way                 |
 * (We are still ignoring the edge cubies at this stage of the solution.)                     |
 * _ * _                _ * _                  _ * _                 _ * _            |
 * _ * _klt_ * _        _ * _ l _ * _          _ * _ l _ * _         _ * _klt_ * _        |
 * * _krt_ T _flt_ *    * _ l _ l _klt_ * DO   * _ l _ l _klt_ *     * _frt_ T _flt_ *      |
 * * _frt_ *  |     |   * _krt_ *   | THE  |   * _frt_ *   |     |   * _krt_ *   |          |
 * |  * _ |
 * |krt|  *   |flt| 3Fv | F |   *   |klt| FRT  | F |   *   |klt| 3F^ |frt|   *   |flt| 3R^  |
 * * _ |frt|frt| _ * --> * _ |krt|krt| _ * and * _ |frt|frt| _ * --> * _ |krt|krt| _ * -->  |
 * |  F _ | _ R   |     |   F _ | _ t   | BFR  |   F _ | _ t   |     |   F _ | _ R   |      |
 * | F |  *   | R |     | F |   *   |flt| SWAP | F |   *   |flt|     | F |   *   | R |      |
 * * _ | F | R | _ *    * _ |frt|frt| _ * ---> * _ |krt|krt| _ *     * _ | F | R | _ *      |
 * * _ | _ *            * _ | _ *              * _ | _ *             * _ | _ *          |
 * You worked hard in   I'm using "l" to       "krt" indicates       After rotating the     |
 * chapter three to     indicate the face      the cubie that        entire Cube, then      |
 * get four corner      which USED to be       will end up in        using the SWAP moves,  |
 * cubies correct, so   on the LEFT (but       the KRT position      we can return the      |
 * I'll show those as   isn't any more         when we are done      Cube to this           |
 * "F" and "R"! (You    because I turned       (we hope!).           position.              |
 * can't see "B", "K"   the Cube).        See                        "frt" and "krt" have   |
 * or "L" -- they're                    diagram                      been SWAPped.          |
 * OK, too.)                              4-5                                               |
 * DIAGRAM 4-8A.        DIAGRAM 4-8B.          DIAGRAM 4-8C.         DIAGRAM 4-8D.          |
 * _ * _                 _ * _                 _ * _                 _ * _            |
 * _ * _klt_ * _         _ * _klt_ * _         _ * _klt_ * _         _ * _klt_ * _        |
 * * _frt_ T _ k _ * DO  * _frt_ T _ k _ *     * _frt_ T _krt_ *     * _ l _ T _krt_ *      |   |                                                                                |
 * * _flt_ *  | THE  |   * _krt_ *   |     |   * _flt_ *   |     |   * _frt_ *   | THE  |
 * |frt|  *   | R | FRT  |frt|   *   | R | 3Rv |frt|   *   |krt| 3Fv | F |   *   |krt| FRT  |
 * * _ |flt|flt| _ * and * _ |krt|krt| _ * --> * _ |flt|flt| _ * --> * _ |frt|frt| _ * and  |
 * |  F _ | _ R   | BFR  |   F _ | _ R   |     |   F _ | _ R   |     |   F _ | _ R   | BFR  |
 * | F |  *   | R | SWAP | F |   *   | R |     | F |   *   | R |     | F |   *   | R | SWAP |
 * * _ |krt|krt| _ * ---> * _ |flt|flt| _ *    * _ | F | R | _ *     * _ |flt|flt| _ * -->  |
 * * _ | _ *             * _ | _ *             * _ | _ *             * _ | _ *          |
 * This may not     See                        ... now the "krt"                       See  |
 * look like much  diagram                     cubie is in place,                  siagram  |
 * progress yet,    4-5                        and we know how to                      4-5  |
 * but ...                                     SWAP "frt" and "flt".                        |
 * DIAGRAM 4-8E.         DIAGRAM 4-8F.         DIAGRAM 4-8G.         DIAGRAM 4-8H.          |
 * _ * _                 _ * _                                                        |
 * _ * _klt_ * _         _ * _klt_ * _                                                    |
 * * _ l _ T _krt_ *     * _flt_ T _krt_ *                                                  |                                                                                |   |   * _flt_ *   |      |   * _frt_ *   |                                                  |
 * | F |  *   |krt| 3F^  |flt|   *   |krt|                                                  |
 * * _ |flt|flt| _ * --? * _ |frt|frt| _ *                                                  |
 * |  F _ | _ R   |      |   F _ | _ R   |                                                  |
 * | F |  *   | R |      | F |   *   | R |                                                  |
 * * _ |frt|frt| _ *     * _ | F | R | _ *                                                  |
 * * _ | _ *             * _ | _ *                                                      |
 * Rotate the Cube       ... and now we have all eight                                      |
 * back to position...   corner cubiea at their proper locations.                           |
 * DIAGRAM 4-8I.         DIAGRAM 4-8J.                                                      |
 * DIAGRAM 4-8. Ten Stages in SWAP Diagonally Opposite Cubies the Hard Way                  |
 * That was a real mental workout! I have shown several examples of ways we can use the SWAP, |
 * and, if you are clever, you should be able to get all eight corner cubies into their proper |
 * locations.                                                                                |
 * We still need to orient all eight corner cubies. Then we will be able to finish off all    |
 * twelve edge cubies. Believe me, you are about to learn how to do these things. You will    |
 * also learn a general principle which will make your planning and Cube turning MUCH simpler |
 * and easier (It's in Chapter Six)!                                                          |
 * I promised that I would show you all 24 possible ways the corner cubies on the VOTTOM layer |
 * of your Cube could be arranged, after you have finished the work in Chapter Three, "Some   |
 * Simple Moves -- Positioning Four Corner Cubies". This will complete all of the strategic   |
 * planning we will need to fininsh Chapter Four.                                             |
 * _ * _                          _ * _                         _ * _             |
 * _ * _TOP_ * _                  _ * _ ? _ * _           (LEFT)  * _???_ *  (BACK)   |
 * _ * _ ? _ * _ ? _ * _          _ * _ ? _ * _   _ * _         _ * _     *     _ * _     |
 * * _TOP_ * _TOP_ * _TOP_ *      * _ ? _ * _"b"_ * _ ? _ *     * _???_ *  TOP  * _???_ *   |
 * |  * _ ? _ * _ | _ *   |       |   * _ ? _ * _ ? _ *   |         *     _ * _     *       |
 * | F |  * _TOP_ *   | R |       | ? |   * _ ? _ *   | ? |      FRONT  * _???_ *  RIGHT    |
 * * _ | ? |  *   | ? | _ *  3R2  * _ | ? |   *   | ? | _ *                 *               |
 * |  * _ | F | R | _ *   |  -->  |   * _ | ? | ? | _ *   |     This diagram indicates the  |
 * | ? |  * _ | _ *   | ? |       | ? |   * _ | _ *   | ? |     four corner cubies which    |
 * * _ | F |  *   | R | _ *       * _ |"k"|   *   |"r"| _ *     are now of interest to us.  |
 * |  * _ | ? | ? | _ *   |       |   * _ | ? | ? | _ *   |                                 |
 * | ? |  * _ | _ *   | ? |       |"k"|   * _ | _ *   |"r"|            (L) ??? (K)          |
 * * _ | ? |  *   | ? | _ *       * _ | ? |   *   | ? | _ *             ??? T ???           |
 * * _ | ? | ? | _ *              * _ |"k"|"r"| _ *                 F  ???  R           |
 * * _ | _ *                      * _ | _ *                     Condensed           |
 * *                              *                          version            |
 * DIAGRAM 4-9A.                  DIAGRAM 4-9B.                 DIAGRAM 4-9C.               |
 * DIAGRAM 4-9. Preparing to Locate the Final Four Corner Cubies                            |
 * PICK UP YOUR CUBE! It should have the TOP side showing your chosen color on at least five  |
 * colored labels, as shown in diagram 4-9A. Turn your Cube upside down as shown in diagrams  |
 * 4-9A and 4-9B.                                                                             |
 * I am going to refocus on what is now the FRONT, BACK, LEFT, RIGHT, and TOP, so I'll show   |
 * you how things are now named, and I'll show you what we are looking for. Then I'll tell you |
 * what we need to do for each of the 24 possibilities.                                       |
 * First, the names (and abbreviations) of the LOCATIONS for the last four corner cubies are  |
 * shown in diagram 4-10.                                                                     |
 * _ * _                          The last four corner cubies        |
 * (LEFT)     _ - BACK  - _      (BACK)           are indicated like this:           |
 * * _LEFT TOP  _ *                                                        |
 * -      - _KLT_ -       -                 klt -- the cubie which belongs     |
 * _ * _            *             _ * _                  at location KLT             |
 * _ - FRONT - _    -       -     _ - BACK  - _                                          |
 * * _LEFT  TOP _ *      TOP      * _RIGHT  TOP _ *     flt -- the cubie which belongs     |
 * |  - _FLT_ -       -       -       - _KRT_ -   |            at location FLT             |
 * *            _ * _             *                                                |
 * |          -     _ - FRONT - _     -           |     krt -- the cubie which belongs     |
 * FRONT    * _RIGHT  TOP _ *     RIGHT                  at location KRT             |
 * |                  - _FRT_ -                   |                                        |
 * * (This is an enlargement    frt -- the cubie which belongs     |
 * | of part of diagram 4-9C.)         at location KLT             |
 * DIAGRAM 4-10. Introducing More Abbreviated Diagrams                                    |
 * Now, as promised, here are the 24 possbilities of what you will see on what is now the TOP |
 * of your Cube (diagrams 4-11A through 4-14F)                                                |
 * (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.         |
 * DIAGRAM 4-11. The First Set of Six Possibilities                                           |
 * (L) krt (K)  : (L) frt (K) : (L) krt (K) : (L) flt (K)    : (L) flt (K)   : (L) frt (K)    |
 * klt T frt   :  klt T krt  :  klt T flt  :  klt T krt     :  klt T frt    :  klt T flt     |
 * F flt  R    :  F  flt  R  :  F  frt  R  :  F  frt  R     :  F  krt  R    :  F  krt  R     |
 * :            :             :  (3=cycle)     :  (diagonal    :  (3-cycle)     |
 * :            :             :                :   corners)    :                |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-12A.       : 4-12B.      : 4-12C.      : 4-12D.         : 4-12E.        : 4-12F.         |
 * DIAGRAM 4-12. The Second Set of Siz Possibilities                                          |
 * For diagrams 4-12A through 4-12F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-11A through 4-11F, then follow the directions for the correct diagram.     |
 * (L) frt (K)  : (L) krt (K) : (L) flt (K) : (L) krt (K)    : (L) frt (K)   : (L) flt (K)    |
 * krt T flt   :  frt T flt  :  krt T frt  :  flt T frt     :  flt T krt    :  frt T krt     |
 * F klt  R    :  F  klt  R  :  F  klt  R  :  F  klt  R     :  F  klt  R    :  F  klt  R     |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-13A.       : 4-13B.      : 4-13C.      : 4-13D.         : 4-13E.        : 4-13F.         |
 * DIAGRAM 4-13. The Third Set of Six Possibilities                                           |
 * For diagrams 4-13A through 4-13F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-12A through 4-12F. Follow the directions for those diagrams.               |
 * (L) flt (K)  : (L) flt (K) : (L) frt (K) : (L) frt (K)    : (L) krt (K)   : (L) krt (K)    |
 * frt T klt   :  krt T klt  :  flt T klt  :  krt T klt     :  frt T klt    :  flt T klt     |
 * F krt  R    :  F  frt  R  :  F  krt  R  :  F  flt  R     :  F  flt  R    :  F  frt  R     |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-14A.       : 4-14B.      : 4-14C.      : 4-14D.         : 4-14E.        : 4-14F.         |
 * DIAGRAM 4-14. The Final Set of Six Possibilities                                           |
 * For diagrams 4-14A through 4-14F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-13A through 4-13F. Tricky, huh? I think you see a pattern!                 |
 * Some of these moves may have altered the orientation of some corner cubies, but that is not |
 * a problem. Only the LOCATION of the eight CORNER CUBIES matters now, but the locations MUST |
 * BE CORRECT before you proceed any further. If the location of any corner cubie is          |
 * incorrect, you will have to go back and try again.                                         |
 * 1) For you arithmetic buffs -- there are now only ((3 to the 8th power) * (12 factorial) *     #
 * 2) (2 to the 12th power) / 12) = (6,561 * 479,001,600 * 4,096 / 12) = 1,072,718,335,180,800    #
 * 3) (That's one quadrillion, seventy-two trillion, seven hundred eighteen billion, three        #
 * 4) hundred thirty-five million, one hundred eighty thousand, eight hundred ) possible          #
 * 5) arrangements of the cubies left.                                                            #
 * First, the names (and abbreviations) of the LOCATIONS for the last four corner cubies are  |
 * shown in diagram 4-10.                                                                     |
 * _ * _                          The last four corner cubies        |
 * (LEFT)     _ - BACK  - _      (BACK)           are indicated like this:           |
 * * _LEFT TOP  _ *                                                        |
 * -      - _KLT_ -       -                 klt -- the cubie which belongs     |
 * _ * _            *             _ * _                  at location KLT             |
 * _ - FRONT - _    -       -     _ - BACK  - _                                          |
 * * _LEFT  TOP _ *      TOP      * _RIGHT  TOP _ *     flt -- the cubie which belongs     |
 * |  - _FLT_ -       -       -       - _KRT_ -   |            at location FLT             |
 * *            _ * _             *                                                |
 * |          -     _ - FRONT - _     -           |     krt -- the cubie which belongs     |
 * FRONT    * _RIGHT  TOP _ *     RIGHT                  at location KRT             |
 * |                  - _FRT_ -                   |                                        |
 * * (This is an enlargement    frt -- the cubie which belongs     |
 * | of part of diagram 4-9C.)         at location KLT             |
 * DIAGRAM 4-10. Introducing More Abbreviated Diagrams                                    |
 * Now, as promised, here are the 24 possbilities of what you will see on what is now the TOP |
 * of your Cube (diagrams 4-11A through 4-14F)                                                |
 * (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.         |
 * DIAGRAM 4-11. The First Set of Six Possibilities                                           |
 * (L) krt (K)  : (L) frt (K) : (L) krt (K) : (L) flt (K)    : (L) flt (K)   : (L) frt (K)    |
 * klt T frt   :  klt T krt  :  klt T flt  :  klt T krt     :  klt T frt    :  klt T flt     |
 * F flt  R    :  F  flt  R  :  F  frt  R  :  F  frt  R     :  F  krt  R    :  F  krt  R     |
 * :            :             :  (3=cycle)     :  (diagonal    :  (3-cycle)     |
 * :            :             :                :   corners)    :                |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-12A.       : 4-12B.      : 4-12C.      : 4-12D.         : 4-12E.        : 4-12F.         |
 * DIAGRAM 4-12. The Second Set of Siz Possibilities                                          |
 * For diagrams 4-12A through 4-12F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-11A through 4-11F, then follow the directions for the correct diagram.     |
 * (L) frt (K)  : (L) krt (K) : (L) flt (K) : (L) krt (K)    : (L) frt (K)   : (L) flt (K)    |
 * krt T flt   :  frt T flt  :  krt T frt  :  flt T frt     :  flt T krt    :  frt T krt     |
 * F klt  R    :  F  klt  R  :  F  klt  R  :  F  klt  R     :  F  klt  R    :  F  klt  R     |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-13A.       : 4-13B.      : 4-13C.      : 4-13D.         : 4-13E.        : 4-13F.         |
 * DIAGRAM 4-13. The Third Set of Six Possibilities                                           |
 * For diagrams 4-13A through 4-13F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-12A through 4-12F. Follow the directions for those diagrams.               |
 * (L) flt (K)  : (L) flt (K) : (L) frt (K) : (L) frt (K)    : (L) krt (K)   : (L) krt (K)    |
 * frt T klt   :  krt T klt  :  flt T klt  :  krt T klt     :  frt T klt    :  flt T klt     |
 * F krt  R    :  F  frt  R  :  F  krt  R  :  F  flt  R     :  F  flt  R    :  F  frt  R     |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-14A.       : 4-14B.      : 4-14C.      : 4-14D.         : 4-14E.        : 4-14F.         |
 * DIAGRAM 4-14. The Final Set of Six Possibilities                                           |
 * For diagrams 4-14A through 4-14F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-13A through 4-13F. Tricky, huh? I think you see a pattern!                 |
 * Some of these moves may have altered the orientation of some corner cubies, but that is not |
 * a problem. Only the LOCATION of the eight CORNER CUBIES matters now, but the locations MUST |
 * BE CORRECT before you proceed any further. If the location of any corner cubie is          |
 * incorrect, you will have to go back and try again.                                         |
 * 1) For you arithmetic buffs -- there are now only ((3 to the 8th power) * (12 factorial) *     #
 * 2) (2 to the 12th power) / 12) = (6,561 * 479,001,600 * 4,096 / 12) = 1,072,718,335,180,800    #
 * 3) (That's one quadrillion, seventy-two trillion, seven hundred eighteen billion, three        #
 * 4) hundred thirty-five million, one hundred eighty thousand, eight hundred ) possible          #
 * 5) arrangements of the cubies left.                                                            #
 * DIAGRAM 4-13. The Third Set of Six Possibilities                                           |
 * For diagrams 4-13A through 4-13F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-12A through 4-12F. Follow the directions for those diagrams.               |
 * (L) flt (K)  : (L) flt (K) : (L) frt (K) : (L) frt (K)    : (L) krt (K)   : (L) krt (K)    |
 * frt T klt   :  krt T klt  :  flt T klt  :  krt T klt     :  frt T klt    :  flt T klt     |
 * F krt  R    :  F  frt  R  :  F  krt  R  :  F  flt  R     :  F  flt  R    :  F  frt  R     |
 * DIAGRAM      : DIAGRAM     : DIAGRAM     : DIAGRAM        : DIAGRAM       : DIAGRAM        |
 * 4-14A.       : 4-14B.      : 4-14C.      : 4-14D.         : 4-14E.        : 4-14F.         |
 * DIAGRAM 4-14. The Final Set of Six Possibilities                                           |
 * For diagrams 4-14A through 4-14F, simply turn the TOP layer (Tv) and compare the result    |
 * with diagrams 4-13A through 4-13F. Tricky, huh? I think you see a pattern!                 |
 * Some of these moves may have altered the orientation of some corner cubies, but that is not |
 * a problem. Only the LOCATION of the eight CORNER CUBIES matters now, but the locations MUST |
 * BE CORRECT before you proceed any further. If the location of any corner cubie is          |
 * incorrect, you will have to go back and try again.                                         |
 * 1) For you arithmetic buffs -- there are now only ((3 to the 8th power) * (12 factorial) *     #
 * 2) (2 to the 12th power) / 12) = (6,561 * 479,001,600 * 4,096 / 12) = 1,072,718,335,180,800    #
 * 3) (That's one quadrillion, seventy-two trillion, seven hundred eighteen billion, three        #
 * 4) hundred thirty-five million, one hundred eighty thousand, eight hundred ) possible          #
 * 5) arrangements of the cubies left.                                                            #
 * Some of these moves may have altered the orientation of some corner cubies, but that is not |
 * a problem. Only the LOCATION of the eight CORNER CUBIES matters now, but the locations MUST |
 * BE CORRECT before you proceed any further. If the location of any corner cubie is          |
 * incorrect, you will have to go back and try again.                                         |
 * 1) For you arithmetic buffs -- there are now only ((3 to the 8th power) * (12 factorial) *     #
 * 2) (2 to the 12th power) / 12) = (6,561 * 479,001,600 * 4,096 / 12) = 1,072,718,335,180,800    #
 * 3) (That's one quadrillion, seventy-two trillion, seven hundred eighteen billion, three        #
 * 4) hundred thirty-five million, one hundred eighty thousand, eight hundred ) possible          #
 * 5) arrangements of the cubies left.                                                            #
 * 1) hundred thirty-five million, one hundred eighty thousand, eight hundred ) possible          #
 * 2) arrangements of the cubies left.                                                            #

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

title: Rubik's Cube/Ignoring Details


text:
Rubik's Cube/Keeping Track of Details

This is where Chapter 05 has been installed. Some diagrams may have been messed up in translation. I will try to fix them as I convert from Top, Bottom, and bacK to Up, Down, and Back.

Ray Calvin Baker 03:43, 10 November 2011 (UTC)

 +-+ +-+
 * 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 resources                      |
 * Chapter Five - - - - - - - - - - Keeping Track of Lots of Details -- Rotating Corner Cubies |
 * Goal Two is to get all of the corner cubies properly oriented in their proper corners of   |
 * the Cube.                                                                                |
 * We will need to consider more details when we try to orient the corner cubies, without     |
 * messing up where they are placed. For now, we are still ignoring the edge cubies. So, let's |
 * start with a diagram of the corner cubies of Rubik's Cube. (Yes, I have still left out some |
 * details which are not essential for what I am trying to do.) I am using arbitrary letters  |
 * and numbers to identify the labels on the visible faces of each corner cubie. And yes, this |
 * is another chapter where we use pencil and paper more than Cube turning.                   |
 * This diagram is like an isometric drawing, with some lines left out. I tried to emphasize  |
 * the three sides of the visible corner cubies, and I added some notes, "(1)", "(C)", "(5)", |
 * "(G)", "(e)", "(f)", and "(g)" to indicate the hidden labels on several of the corner      |
 * cubies. Later on, I will show you several other types of diagrams that I used; I hope a few |
 * of them will also help you.                                                                |
 * d               a, b, c, and d are on the TOP side.                       |
 * a   TOP    c                                                                    |
 * (1)        b        (C)     A, B, E, and F are on the FRONT side.                     |
 * A              3                                                                  |
 * B  2              2, 3, 6, and 7 are on the RIGHT side.                     |
 * (5) FRONT     RIGHT (G)                                                               |
 * E              7        (Items indicated in parentheses are                       |
 * F  6               actually hidden, around the corner.)                     |
 * (e)        (g)                                                                   |
 * (f)        Position 0                                                      |
 * DIAGRAM 5-1. Diagram for Orienting Corner Cubies -- Position Zero                     |
 * (There is a cubie in the back, which would bear the labels "H", "h", and "8".)             |
 * If we are not yet cubemeisters, we will need to do SOMETHING to get started. I propose that |
 * we explore what happens with two simple moves. As you will see, we may be able to find LOTS |
 * of clever operators with just a little bit of VERY CAREFUL analysis.                       |
 * So, here is our first (arbitary) move, Fv. Rotate the FRONT face clockwise one quarter     |
 * turn.                                                                                |
 * d                                                                         |
 * 5   TOP    c                                                                    |
 * (e)        1        (C)                                                               |
 * E              3                                                                  |
 * A  a                                                                        |
 * (f) FRONT     RIGHT (G)                                                               |
 * F              7                                                                  |
 * B  b                                                                        |
 * (6)        (g)                                                                   |
 * (2)        Position 1                                                      |
 * DIAGRAM 5-2. What FRONT Clockwise Did                                                 |
 * Our second arbitrary move is R^. Rotate the RIGHT face counterclockwise one quarter turn.  |
 * d                                                                         |
 * 5   TOP    G                                                                    |
 * (e)        C        (g)                                                               |
 * E              7                                                                  |
 * c  3                                                                        |
 * (f) FRONT     RIGHT (2)                                                               |
 * F              b                                                                  |
 * 1  a                                                                        |
 * (6)        (B)                                                                   |
 * (A)        Position 2                                                      |
 * DIAGRAM 5-3. What RIGHT Counterclockwise Did -- Position Two                          |
 * Now let's compare position 0 with position 2.                                              |
 * d                                    d                                    |
 * a   TOP    c                         5    TOP    G                              |
 * (1)        b        (C)              (e)         C        (g)                         |
 * A              3                     E               7                            |
 * B  2                                 c   3                                  |
 * (5) FRONT     RIGHT (G)    Fv R^     (f)  FRONT     RIGHT (2)                         |
 * E              7       >         F               b                            |
 * F  6                                 1   a                                  |
 * (e)        (g)                       (6)         (B)                             |
 * (f)                                  (A)                                   |
 * Position 0                           Position 2                               |
 * DIAGRAM 5-4. Comparing Position Zero With Position Two                                |
 * "A" started on the front side of the TOP LEFT FRONT cubie. "A" ended on the bottom side of |
 * the BOTTOM RIGHT FRONT cubie.                                                              |
 * "f" started on the bottom side of the BOTTOM RIGHT FRONT cubie. "f" ended on the left side |
 * of the BOTTOM LEFT FRONT cubie.                                                            |
 * "5" started on the left side of the BOTTOM LEFT FRONT cubie. "5" ended on the top side of  |
 * the TOP LEFT FRONT cubie.                                                                  |
 * We can summarize these facts like this: A -> f -> 5 -> a -> etc. The entire series is:     |
 * A -> f -> 5 -> a -> 6 -> e -> 1 -> F -> E -> A.                                            |
 * "a" started on the top side of the TOP LEFT FRONT cubie. But this means that if we repeat  |
 * these two moves three times, the TOP LEFT FRONT cube will be rotated counterclockwise by   |
 * 120 degrees. Other things will happen also, so we need to think things through.            |
 * "B" started on the front side of the TOP RIGHT FRONT cubie. "B" ended on the bottom side of |
 * the BOTTOM RIGHT BACK cubie.                                                               |
 * "g" started on the bottom side of the BOTTOM RIGHT BACK cubie. "g" ended on the back side  |
 * of the TOP RIGHT BACK cubie.                                                               |
 * "C" started on the back side of the TOP RIGHT BACK cubie. "C" ended on the top side of the |
 * TOP RIGHT FRONT cubie.                                                                     |
 * "b" started on the top side of the TOP RIGHT FRONT cubie. We can summarize these facts like |
 * this: B -> g -> C -> b -> etc. The entire series is:                                       |
 * B -> g -> C -> b -> 7 -> 3 -> 2 -> G -> c -> B.                                            |
 * Repeating the two moves three times will also rotate the TOP RIGHT FROMT cube clockwise by |
 * 120 degrees.                                                                               |
 * You should be able to verify these sequences:                                              |
 * 3 -> 2 -> G -> c -> B -> g -> C -> b -> 7 -> 3 -> 2, and                                   |
 * E -> A -> f -> 5, F -> E -> A -> f, and 7 -> 3 -> 2 -> G.                                  |
 * Notice that three repetitions of the two moves rotates the TOP RIGHT BACK cubie clockwise  |
 * by 120 degrees.                                                                            |
 * d                                    d                                    |
 * a   TOP    c                         5    TOP    G                              |
 * (1)        b        (C)              (e)         C        (g)                         |
 * A              3                     E               7                            |
 * B  2                                 c   3                                  |
 * (5) FRONT     RIGHT (G)    Fv R^     (f)  FRONT     RIGHT (2)    Fv R^                |
 * E              7       >         F               b       >                |
 * F  6                                 1   a                                  |
 * (e)        (g)                       (6)         (B)                             |
 * (f)                                  (A)                                   |
 * Position 0                           Position 2                               |
 * d                                    d                                    |
 * f   TOP    2                         A    TOP    3                              |
 * (6)        g        (B)              (a) +       B      - (c)                         |
 * F              b                     1       -       C                            |
 * G  7                                 2   b                                  |
 * (A) FRONT     RIGHT (3)    Fv R^     (E)  FRONT     RIGHT (7)                         |
 * 1              C       >         e               g                            |
 * e  5                           +     6   f     -                            |
 * (a)        (c)                       (5)    +    (G)                             |
 * (E)                                  (F)                                   |
 * "After Four Turns"                   "After Six Turns"                            |
 * DIAGRAM 5-5. After Six Turns                                                          |
 * In this last picture, "After Six Turns", I have marked clockwise rotations of corner cubies |
 * with "-" signs, and counterclockwise rotations with "+" signs. (The two cubies in the BACK |
 * LEFT column are not rotated at all. I can use the "0" symbol to indicate this, if I need   |
 * to.)                                                                                |
 * Hmmm. Three cubies are rotated clockwise, three are rotated counterclockwise, and two are  |
 * not rotated at all. Because we noticed that all cubies are returned to their stating       |
 * locations, we will, hopefully, not have to worry about messing up Goal One. Can we do      |
 * something useful with the information we have learned so far? Yes!                         |
 * Let's try to ignore "irrelevant" details, and concentrate on the patterns of rotation of   |
 * the corner cubies. I make a copy of the picture "After Six Turns" (with slightly different |
 * perspective), unroll it, then remove some details. NOTE: These operations are NOT something |
 * you can actually do on a real Cube -- these are conceptual drawings only, intended to help |
 * us visualize more of the surface of the Cube. We can NOT take a Cube apart in this fashion. |
 * but we CAN make drawings of all six sides.                                                 |
 * TOP               |
 * explodes;           |
 * _ *---* _         |
 * *  |corner |   *        |
 * _ * _                        _ * _                  /| ? |cubies | ? |\\      |
 * _ * _ d _ * _                _ * _ d _ * _             / * _ *stretch* _ * \\     |
 * _ * _ ? _ * _ ? _ * _        _ * _ ? _ * _ ? _ * _        /A |   *---*   | 3\\    |
 * * _ a _ * _TOP_ * _ c _ *    * _ A _ * _TOP_ * _ 3 _ *     * _ | ? |       | ? | _ *    |
 * |  * _ ? _ * _ ? _ *   |     |   * _ ? _ * _ ? _ *   |     |   * _ |   B   | _ *   |    |
 * | A |  * _ b _ *   | 3 |     | 1 |   * _ B _ *   | C |     | 1 |   * _   _ *   | C |    |
 * * _ | ? |  *   | ? | _ *     * _ | ? |   *   | ? | _ *     * _ | ? |   *   | ? | _ *    |
 * |  * _ | B | 2 | _ *   |     |   * _ | 2 | b | _ *   |     |   * _ | 2 | b | _ *   |    |
 * | ? |  * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |     | ? |   * _ | _ *   | ? |    |
 * * _ FRONT  *   RIGHT _ *     * _ FRONT   *   RIGHT _ *     * _ FRONT   *   RIGHT _ *    |
 * |  * _ | ? | ? | _ *   |     |   * _ | ? | ? | _ *   |     |   * _ | ? | ? | _ *   |    |
 * | E |  * _ | _ *   | 7 |     | e |   * _ | _ *   | g |     | e |   * _ | _ *   | g |    |
 * * _ | ? |  *   | ? | _ *     * _ | ? |   *   | ? | _ *     * _ | ? |   *   | ? | _ *    |
 * * _ | F | 6 | _ *            * _ | 6 | f | _ *          \\  * _ | 6 | f | _ *  /    |
 * * _ | _ *                    * _ | _ *               \\ |   * _ | _ *   | /     |
 * *                            *            BOTTOM  \\| ? |   *   | ? |/      |
 * explodes * _ |  F   | _ *        |
 * Position 0                   After Six Moves               Exploding the Cube           |
 * DIAGRAM 5-6A.                DIAGRAM 5-6B.                 DIAGRAM 5-6C.                |
 * RIP!              * | Opening   | *     | *                               * |   |
 * _  _              /| *  the      * |\    *d|\\                            /|d*   |
 * _ * \\ /  * _        * |/|   Cube's  |\| *   |\\| *                          * |/|   |
 * _ *  |   V   |   * _    /|?* |   surface | *?|\  | *?|\\                        /|?* |   |
 * ( _ | ? | _ - _ | ? | _ ) ( _/| *          * |\\_) * |\\|  \\_                  _/ |/| *   |
 * |  * _ *   |   * _ *   | |   *-|           |-*   | |\\| *     * _           _ *    * |/|   |
 * |  |   * _ | _ *   |   | |   |   * _   _ *   |   | | *?|\\ A  |   * _   _ *   | 3 /|?* | | |
 * * _ | ? |  -   | ? | _ * * _ | ? |   -   | ? | _ * *?|\\| * _ | ? |   -   | ? | _* |/|?* | |
 * |  * _ |       | _ *   | |   * _ |       | _ *   | |\\| *a|   * _ |   B   | _ *    |c* |/| |
 * |  |   * _   _ *   |   | |   |   * _   _ *   |   | | *L|\\| 1 |   * _   _ *   | C  |/|K* | |
 * * _ | ? |  *   | ? | _ * * _ | ? |   *   | ? | _ * * |\\|?* _ | ? |   *   | ? | _  * |/| * |
 * |  * _ |   |   | _ *   | |   * _ |   |   | _ *   | |\\| * |   * _ | 2 | b | _ *    |?* |/| |
 * | ? |  * _ | _ *   | ? | |   |   * _ | _ *   |   | | *?|\\| ? |   * _ | _ *   | ?  |/|?* | |
 * * _ FRONT  *   RIGHT _ * * _ FRONT   *   RIGHT _ * * |\\| * _ FRONT   *   RIGHT _  * |/| * |
 * |  * _ | ? | ? | _ *   | |   * _ |   |   | _ *   |  \\| *E|   * _ | ? | ? | _ *    |7* |/  |
 * |  |   * _ | _ *   |   | |   |   * _ | _ *   |   |   *?|\\| e |   * _ | _ *   | g  |/|?*   |
 * * _ | ? |  *   | ? | _ * * _ |   |   *   |   | _ *    \\| * _ | ? |   *   | ? | _  * |/    |
 * |  * _ |   |   | _ *   | |   * _ |   |   | _ *   |      *     * _ | 6 | f | _ *     *      |
 * |  |   * _ | _ *   |   | |   |   * _ | _ *   |   |       \\ 5  |   * _ | _ *   |  G /      |
 * ( _ | ? |  *   | ? | _ ) ( _ |   |   *   |   | _ )         \\_ | ? |   *   | ? | _/        |
 * * _ |      | _ *         * _ |       | _ *                * _ |   F   | _ *            |
 * * _  _ *                 * _   _ *                        * _   _ *                |
 * Starting to Rip Apart     Separating the LEFT        We Are Unwrapping the Cube!           |
 * the Surface of the        Side From the BACK                                               |
 * Cube                      Side                                                             |
 * DIAGRAM 5-6D.             DIAGRAM 5-6E.              DIAGRAM 5-6F.                         |
 * DIAGRAM 5-6. Unwrapping the Cube                                                           |
 * There are several ways to show all six sides of the Cube on one diagram. Each way          |
 * introduces some kind of distortion, but sometimes we can live with that if we can gain more |
 * insight into how the Cube works.                                                           |
 * Examples:                                                                                |
 * Why not choose a way that emphasizes the details we are interested in? For now, we are     |
 * interested primarily in how the corner cubies can be rotated. So, let's take a diagram of  |
 * an "unwrapped" Cube and ignore the parts we don't need now. Later, we may want to emphasize |
 * some other details, instead.                                                               |
 * +---+---+---+      A completely unwrapped Cube                        |
 * --> |TOP|TOP|TOP|                                                         |
 * +---+---+---+      The distortions in this kind of                    |
 * |TOP|TOP|TOP|      diagram separate parts of some                     |
 * +---+---+---+      cubies, often by a large distance.                 |
 * |TOP|TOP|TOP|                                                         |
 * +---+---+---+---+---+---+---+---+---+---+---+---+          For example,           |
 * --> | L | L | L | F | F | F | R | R | R | K | K | K | <---     the three arrows       |
 * +---+---+---+---+---+---+---+---+---+---+---+---+          point to separated     |
 * | L | L | L | F | F | F | R | R | R | K | K | K |          parts of the           |
 * +---+---+---+---+---+---+---+---+---+---+---+---+          BACK LEFT TOP          |
 * | L | L | L | F | F | F | R | R | R | K | K | K |          corner cubie.          |
 * | B | B | B |                                                         |
 * +---+---+---+      This is a diagram of the Cube                      |
 * | B | B | B |      as we would like to see it                         |
 * +---+---+---+      when we are finished working                       |
 * | B | B | B |      with it.                                           |
 * DIAGRAM 5-7. A Completely Unwrapped Cube                                          |
 * (0)--+---+--(-)     A completely unwrapped Cube                        |
 * --> | 4 | ? | 3 |                                                         |
 * +---+---+---+      The distortions in this kind of                    |
 * | ? |TOP| ? |      diagram separate parts of some                     |
 * +---+---+---+      cubies, often by a large distance.                 |
 * | A | ? | B |                                                         |
 * (0)--+---+--(+)--+---+--(-)--+---+--(-)--+---+--(0)         For example,           |
 * --> | D | ? | a | 1 | ? | 2 | b | ? | C | c | ? | d | <---     the three arrows       |
 * +---+---+---+---+---+---+---+---+---+---+---+---+          point to separated     |
 * | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? |          parts of the           |
 * +---+---+---+---+---+---+---+---+---+---+---+---+          BACK LEFT TOP          |
 * |  | ? | E | e | ? | 6 | f | ? | g | 7 | ? |   |           corner cubie.          |
 * (0)--+---+--(+)--+---+--(+)--+---+--(-)--+---+--(0)                                |
 * | 5 | ? | F |                                                         |
 * +---+---+---+      This is what diagram 5-6F woulf look like          |
 * | ? | B | ? |      if we finished unwrapping it.                      |
 * |  | ? | G |       I emphasize the rotation of each corner            |
 * (0)--+---+--(-)     cubie with "(+)", "(-)", and "(0)" signs.          |
 * DIAGRAM 5-7. Unwrapped Cube After Six Moves                                       |
 * If we ignore portions of diagram 5-7 which do not pertain to the rotations of corner       |
 * cubies, we can make some simplified diagrams, which may be easier to work with.            |
 * | A | ? | B |                                                         |
 * (0)--+---+--(+)--+---+--(-)--+---+--(-)--+---+--(0)  First, we ignore portions     |
 * | D | ? | a | 1 | ? | 2 | b | ? | C | c | ? | d |   of the TOP and BOTTOM.        |
 * | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? |   Then, we ignore the lines     |
 * +---+---+---+---+---+---+---+---+---+---+---+---+   and the edge cubies.          |
 * |  | ? | E | e | ? | 6 | f | ? | g | 7 | ? |   |                                  |
 * (0)--+---+--(+)--+---+--(+)--+---+--(-)--+---+--(0)                                |
 * | 5 | ? | F |                                                         |
 * DIAGRAM 5-8A.                                                                      |
 * TOP                       For short, we can draw it like this:             |
 * 0      +      -       -       0                                                       |
 * D    a 1    2 b     C c     d       0 + - - 0                                        |
 * LEFT   FRONT  RIGHT   BACK         0 + + - 0                                        |
 * H    E e    6 f     g 7     h         [F] [K]  <-- These notes serve only to remind  |
 * 0      +      +       -       0                     us which are the FRONT and BACK   |
 * BOTTOM                                     sides of the Cube.                |
 * DIAGRAM 5-8B.                        DIAGRAM 5-8C.                                    |
 * DIAGRAM 5-8. Developing a New Type of Diagram                                         |
 * See if you can figure out the rather peculiar arithmetic of "rotate by 120 degrees".       |
 * "0" & "0" = "0"    "0" & "+" = "+"     "0" & "-" = "-"                                |
 * "+" & "0" = "+"    "+" & "+" = "-"     "+" & "-" = "0"                                |
 * "-" & "0" = "-"    "-" & "+" = "0"     "-" & "-" = "+"                                |
 * DIAGRAM 5-9. The Peculiar Arithmetic of Rotate by 120 Degrees                         |
 * Now I remove even more details from the stuff between "TOP" and "BOTTOM" above, and make   |
 * four copies. (By the way, what we are doing here is "paper and pencil" work, organized in a |
 * systematic fashion. It does not yet require that you do anything with a Cube.)             |
 * 0 + - - 0    0 + - - 0     0 + - - 0     0 + - - 0    (I used "[F]" and "[K]" to      |
 * 0 + + - 0    0 + + - 0     0 + + - 0     0 + + - 0     emphasize how the Cube         |
 * [F] [K]      [F] [K]       [F] [K]       [F] [K]     is oriented.)                  |
 * Now combine these (using "&" arithmetic) with rotated copies. (There is some important     |
 * information concerning "how to do this" as it is related to working with an actual Cube in |
 * Chapter Six, "Customize Your Moves -- Commutation". Please don't rush into things just     |
 * yet -- leave your Cube on the table just a little while longer.)                           |
 * 0 + - - 0    - 0 + - -     - - 0 + -     + - - 0 +                                    |
 * 0 + + - 0    - 0 + + -     + - 0 + +     + + - 0 +                                    |
 * [F] [K]      [F] [K]       [F] [K]       [F] [K]                                    |
 * ... and the results are:                                                                   |
 * 0 - + + 0    - + 0 + -     - 0 - 0 -     + + + + +                                    |
 * 0 - - + 0    - + - 0 -     + 0 + 0 +     - + - 0 -                                    |
 * [F] [K]      [F] [k]       [F] [K]       [F] [K]                                    |
 * These strange-looking diagrams really are coded solutions for problems involving the       |
 * orientation of the corner cubies of the Cube. Chapter Seven, "Finishing the Orientation of |
 * Corner Cubies", will explain how we can use these patterns, and many others, to get all    |
 * eight corners of the Cube perfectly oriented. But first, let me explain in Chapter Six,    |
 * "Customize Your Moves -- Commutation", how to "customize" any series of moves you can learn |
 * to solve lots and lots and lots of Cube problems.                                          |
 * Diagram 5-5 shows us that we can rotate several corner cubies without changing their       |
 * locations. In Chapter Seven, "Finishing the Orientation of Corner Cubies", we will find    |
 * some useful ways to orient the corner cubies of our partially unscrambled Cube. But first, |
 * we need to understand a very helpful principle that will guide all of our work. This will  |
 * be explained in Chapter Six, "Customize Your Moves -- Commutation".                        |
 * If we ignore portions of diagram 5-7 which do not pertain to the rotations of corner       |
 * cubies, we can make some simplified diagrams, which may be easier to work with.            |
 * | A | ? | B |                                                         |
 * (0)--+---+--(+)--+---+--(-)--+---+--(-)--+---+--(0)  First, we ignore portions     |
 * | D | ? | a | 1 | ? | 2 | b | ? | C | c | ? | d |   of the TOP and BOTTOM.        |
 * | ? | L | ? | ? | F | ? | ? | R | ? | ? | K | ? |   Then, we ignore the lines     |
 * +---+---+---+---+---+---+---+---+---+---+---+---+   and the edge cubies.          |
 * |  | ? | E | e | ? | 6 | f | ? | g | 7 | ? |   |                                  |
 * (0)--+---+--(+)--+---+--(+)--+---+--(-)--+---+--(0)                                |
 * | 5 | ? | F |                                                         |
 * DIAGRAM 5-8A.                                                                      |
 * TOP                       For short, we can draw it like this:             |
 * 0      +      -       -       0                                                       |
 * D    a 1    2 b     C c     d       0 + - - 0                                        |
 * LEFT   FRONT  RIGHT   BACK         0 + + - 0                                        |
 * H    E e    6 f     g 7     h         [F] [K]  <-- These notes serve only to remind  |
 * 0      +      +       -       0                     us which are the FRONT and BACK   |
 * BOTTOM                                     sides of the Cube.                |
 * DIAGRAM 5-8B.                        DIAGRAM 5-8C.                                    |
 * DIAGRAM 5-8. Developing a New Type of Diagram                                         |
 * See if you can figure out the rather peculiar arithmetic of "rotate by 120 degrees".       |
 * "0" & "0" = "0"    "0" & "+" = "+"     "0" & "-" = "-"                                |
 * "+" & "0" = "+"    "+" & "+" = "-"     "+" & "-" = "0"                                |
 * "-" & "0" = "-"    "-" & "+" = "0"     "-" & "-" = "+"                                |
 * DIAGRAM 5-9. The Peculiar Arithmetic of Rotate by 120 Degrees                         |
 * Now I remove even more details from the stuff between "TOP" and "BOTTOM" above, and make   |
 * four copies. (By the way, what we are doing here is "paper and pencil" work, organized in a |
 * systematic fashion. It does not yet require that you do anything with a Cube.)             |
 * 0 + - - 0    0 + - - 0     0 + - - 0     0 + - - 0    (I used "[F]" and "[K]" to      |
 * 0 + + - 0    0 + + - 0     0 + + - 0     0 + + - 0     emphasize how the Cube         |
 * [F] [K]      [F] [K]       [F] [K]       [F] [K]     is oriented.)                  |
 * Now combine these (using "&" arithmetic) with rotated copies. (There is some important     |
 * information concerning "how to do this" as it is related to working with an actual Cube in |
 * Chapter Six, "Customize Your Moves -- Commutation". Please don't rush into things just     |
 * yet -- leave your Cube on the table just a little while longer.)                           |
 * 0 + - - 0    - 0 + - -     - - 0 + -     + - - 0 +                                    |
 * 0 + + - 0    - 0 + + -     + - 0 + +     + + - 0 +                                    |
 * [F] [K]      [F] [K]       [F] [K]       [F] [K]                                    |
 * ... and the results are:                                                                   |
 * 0 - + + 0    - + 0 + -     - 0 - 0 -     + + + + +                                    |
 * 0 - - + 0    - + - 0 -     + 0 + 0 +     - + - 0 -                                    |
 * [F] [K]      [F] [k]       [F] [K]       [F] [K]                                    |
 * These strange-looking diagrams really are coded solutions for problems involving the       |
 * orientation of the corner cubies of the Cube. Chapter Seven, "Finishing the Orientation of |
 * Corner Cubies", will explain how we can use these patterns, and many others, to get all    |
 * eight corners of the Cube perfectly oriented. But first, let me explain in Chapter Six,    |
 * "Customize Your Moves -- Commutation", how to "customize" any series of moves you can learn |
 * to solve lots and lots and lots of Cube problems.                                          |
 * Diagram 5-5 shows us that we can rotate several corner cubies without changing their       |
 * locations. In Chapter Seven, "Finishing the Orientation of Corner Cubies", we will find    |
 * some useful ways to orient the corner cubies of our partially unscrambled Cube. But first, |
 * we need to understand a very helpful principle that will guide all of our work. This will  |
 * be explained in Chapter Six, "Customize Your Moves -- Commutation".                        |
 * 0 - + + 0    - + 0 + -     - 0 - 0 -     + + + + +                                    |
 * 0 - - + 0    - + - 0 -     + 0 + 0 +     - + - 0 -                                    |
 * [F] [K]      [F] [k]       [F] [K]       [F] [K]                                    |
 * These strange-looking diagrams really are coded solutions for problems involving the       |
 * orientation of the corner cubies of the Cube. Chapter Seven, "Finishing the Orientation of |
 * Corner Cubies", will explain how we can use these patterns, and many others, to get all    |
 * eight corners of the Cube perfectly oriented. But first, let me explain in Chapter Six,    |
 * "Customize Your Moves -- Commutation", how to "customize" any series of moves you can learn |
 * to solve lots and lots and lots of Cube problems.                                          |
 * Diagram 5-5 shows us that we can rotate several corner cubies without changing their       |
 * locations. In Chapter Seven, "Finishing the Orientation of Corner Cubies", we will find    |
 * some useful ways to orient the corner cubies of our partially unscrambled Cube. But first, |
 * we need to understand a very helpful principle that will guide all of our work. This will  |
 * be explained in Chapter Six, "Customize Your Moves -- Commutation".                        |
 * some useful ways to orient the corner cubies of our partially unscrambled Cube. But first, |
 * we need to understand a very helpful principle that will guide all of our work. This will  |
 * be explained in Chapter Six, "Customize Your Moves -- Commutation".                        |

title: Rubik's Cube/Keeping Track of Details