## \-\-- Day 15: Dueling Generators \-\--

Here, you encounter a pair of dueling
generators.
The generators, called *generator A* and *generator B*, are trying to
agree on a sequence of numbers. However, one of them is malfunctioning,
and so the sequences don\'t always match.

As they do this, a *judge* waits for each of them to generate its next
value, compares the lowest 16 bits of both values, and keeps track of
the number of times those parts of the values match.

The generators both work on the same principle. To create its next
value, a generator will take the previous value it produced, multiply it
by a *factor* (generator A uses `16807`; generator B uses `48271`), and
then keep the remainder of dividing that resulting product by
`2147483647`. That final remainder is the value it produces next.

To calculate each generator\'s first value, it instead uses a specific
starting value as its \"previous value\" (as listed in your puzzle
input).

For example, suppose that for starting values, generator A uses `65`,
while generator B uses `8921`. Then, the first five pairs of generated
values are:

    --Gen. A--  --Gen. B--
       1092455   430625591
    1181022009  1233683848
     245556042  1431495498
    1744312007   137874439
    1352636452   285222916

In binary, these pairs are (with generator A\'s value first in each
pair):

    00000000000100001010101101100111
    00011001101010101101001100110111

    01000110011001001111011100111001
    01001001100010001000010110001000

    00001110101000101110001101001010
    01010101010100101110001101001010

    01100111111110000001011011000111
    00001000001101111100110000000111

    01010000100111111001100000100100
    00010001000000000010100000000100

Here, you can see that the lowest (here, rightmost) 16 bits of the third
value match: `1110001101001010`. Because of this one match, after
processing these five pairs, the judge would have added only `1` to its
total.

To get a significant sample, the judge would like to consider *40
million* pairs. (In the example above, the judge would eventually find a
total of `588` pairs that match in their lowest 16 bits.)

After 40 million pairs, *what is the judge\'s final count*?

Your puzzle answer was `573`.

The first half of this puzzle is complete! It provides one gold star: \*

## \-\-- Part Two \-\-- {#part2}

In the interest of trying to align a little better, the generators get
more picky about the numbers they actually give to the judge.

They still generate values in the same way, but now they only hand a
value to the judge when it meets their *criteria*:

-   Generator A looks for values that are multiples of `4`.
-   Generator B looks for values that are multiples of `8`.

Each generator functions completely *independently*: they both go
through values entirely on their own, only occasionally handing an
acceptable value to the judge, and otherwise working through the same
sequence of values as before until they find one.

The judge still waits for each generator to provide it with a value
before comparing them (using the same comparison method as before). It
keeps track of the order it receives values; the first values from each
generator are compared, then the second values from each generator, then
the third values, and so on.

Using the example starting values given above, the generators now
produce the following first five values each:

    --Gen. A--  --Gen. B--
    1352636452  1233683848
    1992081072   862516352
     530830436  1159784568
    1980017072  1616057672
     740335192   412269392

These values have the following corresponding binary values:

    01010000100111111001100000100100
    01001001100010001000010110001000

    01110110101111001011111010110000
    00110011011010001111010010000000

    00011111101000111101010001100100
    01000101001000001110100001111000

    01110110000001001010100110110000
    01100000010100110001010101001000

    00101100001000001001111001011000
    00011000100100101011101101010000

Unfortunately, even though this change makes more bits similar on
average, none of these values\' lowest 16 bits match. Now, it\'s not
until the 1056th pair that the judge finds the first match:

    --Gen. A--  --Gen. B--
    1023762912   896885216

    00111101000001010110000111100000
    00110101011101010110000111100000

This change makes the generators much slower, and the judge is getting
impatient; it is now only willing to consider *5 million* pairs. (Using
the values from the example above, after five million pairs, the judge
would eventually find a total of `309` pairs that match in their lowest
16 bits.)

After 5 million pairs, but using this new generator logic, *what is the
judge\'s final count*?

Answer:

Although it hasn\'t changed, you can still [get your puzzle
input](15/input).