Advent of Code

--- 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 626.

--- Part Two ---

   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?

   Your puzzle answer was 306.

   Both parts of this puzzle are complete! They provide two gold stars: **

   At this point, all that is left is for you to admire your advent calendar.

   If you still want to see it, you can get your puzzle input.

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