commit d49ebe2fbc20345b35358f6d402cfb11160f87aa · hauleth.dev/advent-of-code

+42 -40

2021/day19.livemd

···

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       ```elixir

     

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       # File.read!("day19.txt")

     

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       input =

     

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       -
         File.read!("day19.txt")

     

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         |> String.split("\n\n")

     

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         |> Enum.map(fn scanner ->

     

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           scanner

     
···

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       34

     

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       ```

     

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       $$

     

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       \left\{

     

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       \begin{alignat*}{3}

     

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       b_{x_1} r_x + b_{x_0} = a_{x_1} \\

     

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       b_{x_2} r_x + b_{x_0} = a_{x_2}

     

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       \end{alignat*}

     

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       \right.

     

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       \left\{

     

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       \begin{alignat*}{3}

     

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       b_{y_1} r_y + b_{y_0} = a_{y_1} \\

     

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       b_{y_2} r_y + b_{y_0} = a_{y_2} \\

     

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       \end{alignat*}

     

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       \right.

     

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       \left\{

     

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       \begin{alignat*}{3}

     

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       b_{z_1} r_z + b_{z_0} = a_{z_1} \\

     

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       b_{z_2} r_z + b_{z_0} = a_{z_2} \\

     

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       \end{alignat*}

     

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       \right.

     

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       $$

     
···

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       This mean that we need to solve:

     

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       $$

     

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       B_x x + b_0 = a_x \\

     

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       B_y y + b_0 = a_y \\

     

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       B_z z + b_0 = a_z

     

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       $$

     

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       Where:

     
···

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       {b_1}_d & 1 \\

     

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       {b_2}_d & 1 \\

     

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       \end{bmatrix}

     

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       d = \begin{bmatrix}

     

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       r_d \\

     

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       b_{d_0}

     

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       \end{bmatrix}

     

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       a_d = \begin{bmatrix}

     

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       {a_1}_d \\

     
···

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       $$

     

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       By applying [Cramer's Rule](https://en.wikipedia.org/wiki/Cramer%27s_rule) we can solve these

     

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       linear equations with:

     

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       $$

     

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       e_d = \frac{

     

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       \begin{vmatrix}

     

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       {b_1}_d & 1 \\

     

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       {b_2}_d & 1

     

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       \end{vmatrix}

     

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       }{

     

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       \begin{vmatrix}

     

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       {a_1}_d & 1 \\

     

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       {a_2}_d & 1

     

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       \end{vmatrix}

     

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       } = \frac{{b_1}_d - {b_2}_d}{{a_1}_d - {a_2}_d} \\

     

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       b_{d_0} = \frac{

     

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         \begin{vmatrix}

     

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       {a_1}_d & {b_1}_d \\

     

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       {a_2}_d & {b_2}_d

     

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       \end{vmatrix}

     

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       }{

     

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         \begin{vmatrix}

     

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       {a_1}_d & 1 \\

     

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       {a_2}_d & 1

     

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       \end{vmatrix}

     

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       } = \frac{{a_1}_d {b_2}_d - {b_1}_d {a_2}_d}{{a_1}_d - {a_2}_d}

     

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       $$

     

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       ```elixir

     
···

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           b_2 = reaxe(select_common(p0, p2), axes)

     

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           transform =

     

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             for i <- ~w[x y z]a, into: %{} do

     

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               a_1 = a_1[i]

     

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               a_2 = a_2[i]

     

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               b_1 = b_1[i]

     

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               b_2 = b_2[i]

     

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               det_b = b_1 - b_2

     

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               r = div(a_1 - a_2, det_b)

     

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               b_0 = div(b_1 * a_2 - a_1 * b_2, det_b)

     

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               {i, {r, b_0}}

     

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             end

     

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           new_points =

     
···

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       <!-- livebook:{"output":true} -->

     

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       ```

     

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       -
       {:module, Day19, <<70, 79, 82, 49, 0, 0, 26, ...>>, {:pairs, 1}}

     

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       ```

     

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       ```elixir

     
···

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       <!-- livebook:{"output":true} -->

     

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       ```

     

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       -
       0.002822

     

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       1: {5, 25}

     

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       -
       0.017052

     

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       2: {5, 20}

     

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       0.0355

     

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       3: {7, 13}

     

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       0.134655

     

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       4: {6, 7}

     

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       0.229952

     

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       5: {5, 2}

     

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       0.288351

     

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       6: {1, 1}

     

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       0.076665

     

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       7: {1, 0}

     

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       -
       0.071942

     

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       ```

     

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       <!-- livebook:{"output":true} -->

···

       9
        
       ```elixir

     

       10
        
       # File.read!("day19.txt")

     

       11
        
       input =

     

       12
       +
         File.read!("2021/day19.txt")

     

       13
        
         |> String.split("\n\n")

     

       14
        
         |> Enum.map(fn scanner ->

     

       15
        
           scanner

     
···

       34
        
       34

     

       35
        
       ```

     

       36
        
       

     

       37
       +
       To find rotation of point $b_1$ and $b_2$ around shift point $b_0$

     

       38
       +
       

     

       39
        
       $$

     

       40
        
       \left\{

     

       41
        
       \begin{alignat*}{3}

     

       42
       +
       {b_1}_x r_x + {b_0}_x = {a_1}_x \\

     

       43
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       {b_2}_x r_x + {b_0}_x = {a_2}_x

     

       44
        
       \end{alignat*}

     

       45
        
       \right.

     

       46
        
       \left\{

     

       47
        
       \begin{alignat*}{3}

     

       48
       +
       {b_1}_y r_y + {b_0}_y = {a_1}_y \\

     

       49
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       {b_2}_y r_y + {b_0}_y = {a_2}_y \\

     

       50
        
       \end{alignat*}

     

       51
        
       \right.

     

       52
        
       \left\{

     

       53
        
       \begin{alignat*}{3}

     

       54
       +
       {b_1}_z r_z + {b_0}_z = {a_1}_z \\

     

       55
       +
       {b_2}_z r_z + {b_0}_z = {a_2}_z \\

     

       56
        
       \end{alignat*}

     

       57
        
       \right.

     

       58
        
       $$

     
···

       61
        
       This mean that we need to solve:

     

       62
        
       

     

       63
        
       $$

     

       64
       +
       B_x r_x + b_0 = a_x \\

     

       65
       +
       B_y r_y + b_0 = a_y \\

     

       66
       +
       B_z r_z + b_0 = a_z

     

       67
        
       $$

     

       68
        
       

     

       69
        
       Where:

     
···

       73
        
       {b_1}_d & 1 \\

     

       74
        
       {b_2}_d & 1 \\

     

       75
        
       \end{bmatrix}

     

       76
       +
       b_0 = \begin{bmatrix}

     

       77
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       {b_0}_d \\

     

       78
       +
       {b_0}_d

     

       79
        
       \end{bmatrix}

     

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       a_d = \begin{bmatrix}

     

       81
        
       {a_1}_d \\

     
···

       84
        
       $$

     

       85
        
       

     

       86
        
       By applying [Cramer's Rule](https://en.wikipedia.org/wiki/Cramer%27s_rule) we can solve these

     

       87
       +
       linear equations with (we are looking for $b_0$ and $r$):

     

       88
        
       

     

       89
        
       $$

     

       90
       +
       r_d = \frac{

     

       91
        
       \begin{vmatrix}

     

       92
       +
       {a_1}_d & 1 \\

     

       93
       +
       {a_2}_d & 1

     

       94
        
       \end{vmatrix}

     

       95
        
       }{

     

       96
        
       \begin{vmatrix}

     

       97
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       {b_1}_d & 1 \\

     

       98
       +
       {b_2}_d & 1

     

       99
        
       \end{vmatrix}

     

       100
       +
       } = \frac{{a_1}_d - {a_2}_d}{{b_1}_d - {b_2}_d} \\

     

       101
        
       b_{d_0} = \frac{

     

       102
        
         \begin{vmatrix}

     

       103
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       {b_1}_d & {a_1}_d \\

     

       104
       +
       {b_2}_d & {a_2}_d

     

       105
        
       \end{vmatrix}

     

       106
        
       }{

     

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         \begin{vmatrix}

     

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       {b_1}_d & 1 \\

     

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       {b_2}_d & 1

     

       110
        
       \end{vmatrix}

     

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       } = \frac{{b_1}_d {a_2}_d - {a_1}_d {b_2}_d}{{b_1}_d - {b_2}_d}

     

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       $$

     

       113
        
       

     

       114
        
       ```elixir

     
···

       135
        
           b_2 = reaxe(select_common(p0, p2), axes)

     

       136
        
       

     

       137
        
           transform =

     

       138
       +
             for d <- ~w[x y z]a, into: %{} do

     

       139
       +
               a_1 = a_1[d]

     

       140
       +
               a_2 = a_2[d]

     

       141
       +
               b_1 = b_1[d]

     

       142
       +
               b_2 = b_2[d]

     

       143
        
               det_b = b_1 - b_2

     

       144
        
               r = div(a_1 - a_2, det_b)

     

       145
        
       

     

       146
        
               b_0 = div(b_1 * a_2 - a_1 * b_2, det_b)

     

       147
        
       

     

       148
       +
               {d, {r, b_0}}

     

       149
        
             end

     

       150
        
       

     

       151
        
           new_points =

     
···

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       <!-- livebook:{"output":true} -->

     

       213
        
       

     

       214
        
       ```

     

       215
       +
       {:module, Day19, <<70, 79, 82, 49, 0, 0, 29, ...>>, {:pairs, 1}}

     

       216
        
       ```

     

       217
        
       

     

       218
        
       ```elixir

     
···

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       <!-- livebook:{"output":true} -->

     

       285
        
       

     

       286
        
       ```

     

       287
       +
       0.005503

     

       288
        
       1: {5, 25}

     

       289
       +
       0.029354

     

       290
        
       2: {5, 20}

     

       291
       +
       0.067284

     

       292
        
       3: {7, 13}

     

       293
       +
       0.195126

     

       294
        
       4: {6, 7}

     

       295
       +
       0.292686

     

       296
        
       5: {5, 2}

     

       297
       +
       0.374403

     

       298
        
       6: {1, 1}

     

       299
       +
       0.074645

     

       300
        
       7: {1, 0}

     

       301
       +
       0.090199

     

       302
        
       ```

     

       303
        
       

     

       304
        
       <!-- livebook:{"output":true} -->