Advent of Code 2017
===================

December 3rd
------------

You come across an experimental new kind of memory stored on an infinite
two-dimensional grid.

Each square on the grid is allocated in a spiral pattern starting at a
location marked 1 and then counting up while spiraling outward. For
example, the first few squares are allocated like this:

::

   17  16  15  14  13
   18   5   4   3  12
   19   6   1   2  11
   20   7   8   9  10
   21  22  23---> ...

While this is very space-efficient (no squares are skipped), requested
data must be carried back to square 1 (the location of the only access
port for this memory system) by programs that can only move up, down,
left, or right. They always take the shortest path: the Manhattan
Distance between the location of the data and square 1.

For example:

-  Data from square 1 is carried 0 steps, since it’s at the access port.
-  Data from square 12 is carried 3 steps, such as: down, left, left.
-  Data from square 23 is carried only 2 steps: up twice.
-  Data from square 1024 must be carried 31 steps.

How many steps are required to carry the data from the square identified
in your puzzle input all the way to the access port?

Analysis
~~~~~~~~

I freely admit that I worked out the program I wanted to write using
graph paper and some Python doodles. There’s no point in trying to write
a Joy program until I’m sure I understand the problem well enough.

The first thing I did was to write a column of numbers from 1 to n (32
as it happens) and next to them the desired output number, to look for
patterns directly:

::

   1  0
   2  1
   3  2
   4  1
   5  2
   6  1
   7  2
   8  1
   9  2
   10 3
   11 2
   12 3
   13 4
   14 3
   15 2
   16 3
   17 4
   18 3
   19 2
   20 3
   21 4
   22 3
   23 2
   24 3
   25 4
   26 5
   27 4
   28 3
   29 4
   30 5
   31 6
   32 5

There are four groups repeating for a given “rank”, then the pattern
enlarges and four groups repeat again, etc.

::

           1 2
         3 2 3 4
       5 4 3 4 5 6
     7 6 5 4 5 6 7 8
   9 8 7 6 5 6 7 8 9 10

Four of this pyramid interlock to tile the plane extending from the
initial “1” square.

::

            2   3   |    4  5   |    6  7   |    8  9
         10 11 12 13|14 15 16 17|18 19 20 21|22 23 24 25

And so on.

We can figure out the pattern for a row of the pyramid at a given “rank”
:math:`k`:

:math:`2k - 1, 2k - 2, ..., k, k + 1, k + 2, ..., 2k`

or

:math:`k + (k - 1), k + (k - 2), ..., k, k + 1, k + 2, ..., k + k`

This shows that the series consists at each place of :math:`k` plus some
number that begins at :math:`k - 1`, decreases to zero, then increases
to :math:`k`. Each row has :math:`2k` members.

Let’s figure out how, given an index into a row, we can calculate the
value there. The index will be from 0 to :math:`k - 1`.

Let’s look at an example, with :math:`k = 4`:

::

   0 1 2 3 4 5 6 7
   7 6 5 4 5 6 7 8

.. code:: ipython2

    k = 4

Subtract :math:`k` from the index and take the absolute value:

.. code:: ipython2

    for n in range(2 * k):
        print abs(n - k),


.. parsed-literal::

    4 3 2 1 0 1 2 3


Not quite. Subtract :math:`k - 1` from the index and take the absolute
value:

.. code:: ipython2

    for n in range(2 * k):
        print abs(n - (k - 1)),


.. parsed-literal::

    3 2 1 0 1 2 3 4


Great, now add :math:`k`\ …

.. code:: ipython2

    for n in range(2 * k):
        print abs(n - (k - 1)) + k,


.. parsed-literal::

    7 6 5 4 5 6 7 8


So to write a function that can give us the value of a row at a given
index:

.. code:: ipython2

    def row_value(k, i):
        i %= (2 * k)  # wrap the index at the row boundary.
        return abs(i - (k - 1)) + k

.. code:: ipython2

    k = 5
    for i in range(2 * k):
        print row_value(k, i),


.. parsed-literal::

    9 8 7 6 5 6 7 8 9 10


(I’m leaving out details of how I figured this all out and just giving
the relevent bits. It took a little while to zero in of the aspects of
the pattern that were important for the task.)

Finding the rank and offset of a number.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now that we can compute the desired output value for a given rank and
the offset (index) into that rank, we need to determine how to find the
rank and offset of a number.

The rank is easy to find by iteratively stripping off the amount already
covered by previous ranks until you find the one that brackets the
target number. Because each row is :math:`2k` places and there are
:math:`4` per rank each rank contains :math:`8k` places. Counting the
initial square we have:

:math:`corner_k = 1 + \sum_{n=1}^k 8n`

I’m not mathematically sophisticated enough to turn this directly into a
formula (but Sympy is, see below.) I’m going to write a simple Python
function to iterate and search:

.. code:: ipython2

    def rank_and_offset(n):
        assert n >= 2  # Guard the domain.
        n -= 2  # Subtract two,
                # one for the initial square,
                # and one because we are counting from 1 instead of 0.
        k = 1
        while True:
            m = 8 * k  # The number of places total in this rank, 4(2k).
            if n < m:
                return k, n % (2 * k)
            n -= m  # Remove this rank's worth.
            k += 1

.. code:: ipython2

    for n in range(2, 51):
        print n, rank_and_offset(n)


.. parsed-literal::

    2 (1, 0)
    3 (1, 1)
    4 (1, 0)
    5 (1, 1)
    6 (1, 0)
    7 (1, 1)
    8 (1, 0)
    9 (1, 1)
    10 (2, 0)
    11 (2, 1)
    12 (2, 2)
    13 (2, 3)
    14 (2, 0)
    15 (2, 1)
    16 (2, 2)
    17 (2, 3)
    18 (2, 0)
    19 (2, 1)
    20 (2, 2)
    21 (2, 3)
    22 (2, 0)
    23 (2, 1)
    24 (2, 2)
    25 (2, 3)
    26 (3, 0)
    27 (3, 1)
    28 (3, 2)
    29 (3, 3)
    30 (3, 4)
    31 (3, 5)
    32 (3, 0)
    33 (3, 1)
    34 (3, 2)
    35 (3, 3)
    36 (3, 4)
    37 (3, 5)
    38 (3, 0)
    39 (3, 1)
    40 (3, 2)
    41 (3, 3)
    42 (3, 4)
    43 (3, 5)
    44 (3, 0)
    45 (3, 1)
    46 (3, 2)
    47 (3, 3)
    48 (3, 4)
    49 (3, 5)
    50 (4, 0)


.. code:: ipython2

    for n in range(2, 51):
        k, i = rank_and_offset(n)
        print n, row_value(k, i)


.. parsed-literal::

    2 1
    3 2
    4 1
    5 2
    6 1
    7 2
    8 1
    9 2
    10 3
    11 2
    12 3
    13 4
    14 3
    15 2
    16 3
    17 4
    18 3
    19 2
    20 3
    21 4
    22 3
    23 2
    24 3
    25 4
    26 5
    27 4
    28 3
    29 4
    30 5
    31 6
    32 5
    33 4
    34 3
    35 4
    36 5
    37 6
    38 5
    39 4
    40 3
    41 4
    42 5
    43 6
    44 5
    45 4
    46 3
    47 4
    48 5
    49 6
    50 7


Putting it all together
~~~~~~~~~~~~~~~~~~~~~~~

.. code:: ipython2

    def row_value(k, i):
        return abs(i - (k - 1)) + k
    
    
    def rank_and_offset(n):
        n -= 2  # Subtract two,
                # one for the initial square,
                # and one because we are counting from 1 instead of 0.
        k = 1
        while True:
            m = 8 * k  # The number of places total in this rank, 4(2k).
            if n < m:
                return k, n % (2 * k)
            n -= m  # Remove this rank's worth.
            k += 1
    
    
    def aoc20173(n):
        if n <= 1:
            return 0
        k, i = rank_and_offset(n)
        return row_value(k, i)

.. code:: ipython2

    aoc20173(23)




.. parsed-literal::

    2



.. code:: ipython2

    aoc20173(23000)




.. parsed-literal::

    105



.. code:: ipython2

    aoc20173(23000000000000)




.. parsed-literal::

    4572225



Sympy to the Rescue
===================

Find the rank for large numbers
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Using e.g. Sympy we can find the rank directly by solving for the roots
of an equation. For large numbers this will (eventually) be faster than
iterating as ``rank_and_offset()`` does.

.. code:: ipython2

    from sympy import floor, lambdify, solve, symbols
    from sympy import init_printing
    init_printing() 

.. code:: ipython2

    k = symbols('k')

Since

:math:`1 + 2 + 3 + ... + N = \frac{N(N + 1)}{2}`

and

:math:`\sum_{n=1}^k 8n = 8(\sum_{n=1}^k n) = 8\frac{k(k + 1)}{2}`

We want:

.. code:: ipython2

    E = 2 + 8 * k * (k + 1) / 2  # For the reason for adding 2 see above.
    
    E




.. math::

    4 k \left(k + 1\right) + 2



We can write a function to solve for :math:`k` given some :math:`n`\ …

.. code:: ipython2

    def rank_of(n):
        return floor(max(solve(E - n, k))) + 1

First ``solve()`` for :math:`E - n = 0` which has two solutions (because
the equation is quadratic so it has two roots) and since we only care
about the larger one we use ``max()`` to select it. It will generally
not be a nice integer (unless :math:`n` is the number of an end-corner
of a rank) so we take the ``floor()`` and add 1 to get the integer rank
of :math:`n`. (Taking the ``ceiling()`` gives off-by-one errors on the
rank boundaries. I don’t know why. I’m basically like a monkey doing
math here.) =-D

It gives correct answers:

.. code:: ipython2

    for n in (9, 10, 25, 26, 49, 50):
        print n, rank_of(n)


.. parsed-literal::

    9 1
    10 2
    25 2
    26 3
    49 3
    50 4


And it runs much faster (at least for large numbers):

.. code:: ipython2

    %time rank_of(23000000000000)  # Compare runtime with rank_and_offset()!


.. parsed-literal::

    CPU times: user 68 ms, sys: 8 ms, total: 76 ms
    Wall time: 73.8 ms




.. math::

    2397916



.. code:: ipython2

    %time rank_and_offset(23000000000000)


.. parsed-literal::

    CPU times: user 308 ms, sys: 0 ns, total: 308 ms
    Wall time: 306 ms




.. math::

    \left ( 2397916, \quad 223606\right )



After finding the rank you would still have to find the actual value of
the rank’s first corner and subtract it (plus 2) from the number and
compute the offset as above and then the final output, but this overhead
is partially shared by the other method, and overshadowed by the time it
(the other iterative method) would take for really big inputs.

The fun thing to do here would be to graph the actual runtime of both
methods against each other to find the trade-off point.

It took me a second to realize I could do this…
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Sympy is a *symbolic* math library, and it supports symbolic
manipulation of equations. I can put in :math:`y` (instead of a value)
and ask it to solve for :math:`k`.

.. code:: ipython2

    y = symbols('y')

.. code:: ipython2

    g, f = solve(E - y, k)

The equation is quadratic so there are two roots, we are interested in
the greater one…

.. code:: ipython2

    g




.. math::

    - \frac{1}{2} \sqrt{y - 1} - \frac{1}{2}



.. code:: ipython2

    f




.. math::

    \frac{1}{2} \sqrt{y - 1} - \frac{1}{2}



Now we can take the ``floor()``, add 1, and ``lambdify()`` the equation
to get a Python function that calculates the rank directly.

.. code:: ipython2

    floor(f) + 1




.. math::

    \lfloor{\frac{1}{2} \sqrt{y - 1} - \frac{1}{2}}\rfloor + 1



.. code:: ipython2

    F = lambdify(y, floor(f) + 1)

.. code:: ipython2

    for n in (9, 10, 25, 26, 49, 50):
        print n, int(F(n))


.. parsed-literal::

    9 1
    10 2
    25 2
    26 3
    49 3
    50 4


It’s pretty fast.

.. code:: ipython2

    %time int(F(23000000000000))  # The clear winner.


.. parsed-literal::

    CPU times: user 0 ns, sys: 0 ns, total: 0 ns
    Wall time: 11.9 µs




.. math::

    2397916



Knowing the equation we could write our own function manually, but the
speed is no better.

.. code:: ipython2

    from math import floor as mfloor, sqrt
    
    def mrank_of(n):
        return int(mfloor(sqrt(23000000000000 - 1) / 2 - 0.5) + 1)

.. code:: ipython2

    %time mrank_of(23000000000000)


.. parsed-literal::

    CPU times: user 0 ns, sys: 0 ns, total: 0 ns
    Wall time: 12.9 µs




.. math::

    2397916



Given :math:`n` and a rank, compute the offset.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now that we have a fast way to get the rank, we still need to use it to
compute the offset into a pyramid row.

.. code:: ipython2

    def offset_of(n, k):
        return (n - 2 + 4 * k * (k - 1)) % (2 * k)

(Note the sneaky way the sign changes from :math:`k(k + 1)` to
:math:`k(k - 1)`. This is because we want to subract the
:math:`(k - 1)`\ th rank’s total places (its own and those of lesser
rank) from our :math:`n` of rank :math:`k`. Substituting :math:`k - 1`
for :math:`k` in :math:`k(k + 1)` gives :math:`(k - 1)(k - 1 + 1)`,
which of course simplifies to :math:`k(k - 1)`.)

.. code:: ipython2

    offset_of(23000000000000, 2397916)




.. math::

    223606



So, we can compute the rank, then the offset, then the row value.

.. code:: ipython2

    def rank_of(n):
        return int(mfloor(sqrt(n - 1) / 2 - 0.5) + 1)
    
    
    def offset_of(n, k):
        return (n - 2 + 4 * k * (k - 1)) % (2 * k)
    
    
    def row_value(k, i):
        return abs(i - (k - 1)) + k
    
    
    def aoc20173(n):
        k = rank_of(n)
        i = offset_of(n, k)
        return row_value(k, i)

.. code:: ipython2

    aoc20173(23)




.. math::

    2



.. code:: ipython2

    aoc20173(23000)




.. math::

    105



.. code:: ipython2

    aoc20173(23000000000000)




.. math::

    4572225



.. code:: ipython2

    %time aoc20173(23000000000000000000000000)  # Fast for large values.


.. parsed-literal::

    CPU times: user 0 ns, sys: 0 ns, total: 0 ns
    Wall time: 20 µs




.. math::

    2690062495969



A Joy Version
=============

At this point I feel confident that I can implement a concise version of
this code in Joy. ;-)

.. code:: ipython2

    from notebook_preamble import J, V, define

``rank_of``
~~~~~~~~~~~

::

      n rank_of
   ---------------
         k

The translation is straightforward.

::

   int(floor(sqrt(n - 1) / 2 - 0.5) + 1)

   rank_of == -- sqrt 2 / 0.5 - floor ++

.. code:: ipython2

    define('rank_of == -- sqrt 2 / 0.5 - floor ++')

``offset_of``
~~~~~~~~~~~~~

::

      n k offset_of
   -------------------
            i

   (n - 2 + 4 * k * (k - 1)) % (2 * k)

A little tricky…

::

   n k dup 2 *
   n k k 2 *
   n k k*2 [Q] dip %
   n k Q k*2 %

   n k dup --
   n k k --
   n k k-1 4 * * 2 + -
   n k*k-1*4     2 + -
   n k*k-1*4+2       -
   n-k*k-1*4+2

   n-k*k-1*4+2 k*2 %
   n-k*k-1*4+2%k*2

Ergo:

::

   offset_of == dup 2 * [dup -- 4 * * 2 + -] dip %

.. code:: ipython2

    define('offset_of == dup 2 * [dup -- 4 * * 2 + -] dip %')

``row_value``
~~~~~~~~~~~~~

::

      k i row_value
   -------------------
           n

   abs(i - (k - 1)) + k

   k i over -- - abs +
   k i k    -- - abs +
   k i k-1     - abs +
   k i-k-1       abs +
   k |i-k-1|         +
   k+|i-k-1|

.. code:: ipython2

    define('row_value == over -- - abs +')

``aoc2017.3``
~~~~~~~~~~~~~

::

      n aoc2017.3
   -----------------
           m

   n dup rank_of
   n k [offset_of] dupdip
   n k offset_of k
   i             k swap row_value
   k i row_value
   m

.. code:: ipython2

    define('aoc2017.3 == dup rank_of [offset_of] dupdip swap row_value')

.. code:: ipython2

    J('23 aoc2017.3')


.. parsed-literal::

    2


.. code:: ipython2

    J('23000 aoc2017.3')


.. parsed-literal::

    105


.. code:: ipython2

    V('23000000000000 aoc2017.3')


.. parsed-literal::

                                                        . 23000000000000 aoc2017.3
                                         23000000000000 . aoc2017.3
                                         23000000000000 . dup rank_of [offset_of] dupdip swap row_value
                          23000000000000 23000000000000 . rank_of [offset_of] dupdip swap row_value
                          23000000000000 23000000000000 . -- sqrt 2 / 0.5 - floor ++ [offset_of] dupdip swap row_value
                          23000000000000 22999999999999 . sqrt 2 / 0.5 - floor ++ [offset_of] dupdip swap row_value
                       23000000000000 4795831.523312615 . 2 / 0.5 - floor ++ [offset_of] dupdip swap row_value
                     23000000000000 4795831.523312615 2 . / 0.5 - floor ++ [offset_of] dupdip swap row_value
                      23000000000000 2397915.7616563076 . 0.5 - floor ++ [offset_of] dupdip swap row_value
                  23000000000000 2397915.7616563076 0.5 . - floor ++ [offset_of] dupdip swap row_value
                      23000000000000 2397915.2616563076 . floor ++ [offset_of] dupdip swap row_value
                                 23000000000000 2397915 . ++ [offset_of] dupdip swap row_value
                                 23000000000000 2397916 . [offset_of] dupdip swap row_value
                     23000000000000 2397916 [offset_of] . dupdip swap row_value
                                 23000000000000 2397916 . offset_of 2397916 swap row_value
                                 23000000000000 2397916 . dup 2 * [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
                         23000000000000 2397916 2397916 . 2 * [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
                       23000000000000 2397916 2397916 2 . * [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
                         23000000000000 2397916 4795832 . [dup -- 4 * * 2 + -] dip % 2397916 swap row_value
    23000000000000 2397916 4795832 [dup -- 4 * * 2 + -] . dip % 2397916 swap row_value
                                 23000000000000 2397916 . dup -- 4 * * 2 + - 4795832 % 2397916 swap row_value
                         23000000000000 2397916 2397916 . -- 4 * * 2 + - 4795832 % 2397916 swap row_value
                         23000000000000 2397916 2397915 . 4 * * 2 + - 4795832 % 2397916 swap row_value
                       23000000000000 2397916 2397915 4 . * * 2 + - 4795832 % 2397916 swap row_value
                         23000000000000 2397916 9591660 . * 2 + - 4795832 % 2397916 swap row_value
                          23000000000000 22999994980560 . 2 + - 4795832 % 2397916 swap row_value
                        23000000000000 22999994980560 2 . + - 4795832 % 2397916 swap row_value
                          23000000000000 22999994980562 . - 4795832 % 2397916 swap row_value
                                                5019438 . 4795832 % 2397916 swap row_value
                                        5019438 4795832 . % 2397916 swap row_value
                                                 223606 . 2397916 swap row_value
                                         223606 2397916 . swap row_value
                                         2397916 223606 . row_value
                                         2397916 223606 . over -- - abs +
                                 2397916 223606 2397916 . -- - abs +
                                 2397916 223606 2397915 . - abs +
                                       2397916 -2174309 . abs +
                                        2397916 2174309 . +
                                                4572225 . 


::

     rank_of == -- sqrt 2 / 0.5 - floor ++
   offset_of == dup 2 * [dup -- 4 * * 2 + -] dip %
   row_value == over -- - abs +

   aoc2017.3 == dup rank_of [offset_of] dupdip swap row_value
