AdventOfCode/2024/24/24.md

## \-\-- Day 24: Crossed Wires \-\--

You and The Historians arrive at the edge of a [large
grove](/2022/day/23) somewhere in the jungle. After the last incident,
the Elves installed a small device that monitors the fruit. While The
Historians search the grove, one of them asks if you can take a look at
the monitoring device; apparently, it\'s been malfunctioning recently.

The device seems to be trying to produce a number through some boolean
logic gates. Each gate has two inputs and one output. The gates all
operate on values that are either *true* (`1`) or *false* (`0`).

-   `AND` gates output `1` if *both* inputs are `1`; if either input is
    `0`, these gates output `0`.
-   `OR` gates output `1` if *one or both* inputs is `1`; if both inputs
    are `0`, these gates output `0`.
-   `XOR` gates output `1` if the inputs are *different*; if the inputs
    are the same, these gates output `0`.

Gates wait until both inputs are received before producing output; wires
can carry `0`, `1` or no value at all. There are no loops; once a gate
has determined its output, the output will not change until the whole
system is reset. Each wire is connected to at most one gate output, but
can be connected to many gate inputs.

Rather than risk getting shocked while tinkering with the live system,
you write down all of the gate connections and initial wire values (your
puzzle input) so you can consider them in relative safety. For example:

    x00: 1
    x01: 1
    x02: 1
    y00: 0
    y01: 1
    y02: 0

    x00 AND y00 -> z00
    x01 XOR y01 -> z01
    x02 OR y02 -> z02

Because gates wait for input, some wires need to start with a value (as
inputs to the entire system). The first section specifies these values.
For example, `x00: 1` means that the wire named `x00` starts with the
value `1` (as if a gate is already outputting that value onto that
wire).

The second section lists all of the gates and the wires connected to
them. For example, `x00 AND y00 -> z00` describes an instance of an
`AND` gate which has wires `x00` and `y00` connected to its inputs and
which will write its output to wire `z00`.

In this example, simulating these gates eventually causes `0` to appear
on wire `z00`, `0` to appear on wire `z01`, and `1` to appear on wire
`z02`.

Ultimately, the system is trying to produce a *number* by combining the
bits on all wires starting with `z`. `z00` is the least significant bit,
then `z01`, then `z02`, and so on.

In this example, the three output bits form the binary number `100`
which is equal to the decimal number `4`.

Here\'s a larger example:

    x00: 1
    x01: 0
    x02: 1
    x03: 1
    x04: 0
    y00: 1
    y01: 1
    y02: 1
    y03: 1
    y04: 1

    ntg XOR fgs -> mjb
    y02 OR x01 -> tnw
    kwq OR kpj -> z05
    x00 OR x03 -> fst
    tgd XOR rvg -> z01
    vdt OR tnw -> bfw
    bfw AND frj -> z10
    ffh OR nrd -> bqk
    y00 AND y03 -> djm
    y03 OR y00 -> psh
    bqk OR frj -> z08
    tnw OR fst -> frj
    gnj AND tgd -> z11
    bfw XOR mjb -> z00
    x03 OR x00 -> vdt
    gnj AND wpb -> z02
    x04 AND y00 -> kjc
    djm OR pbm -> qhw
    nrd AND vdt -> hwm
    kjc AND fst -> rvg
    y04 OR y02 -> fgs
    y01 AND x02 -> pbm
    ntg OR kjc -> kwq
    psh XOR fgs -> tgd
    qhw XOR tgd -> z09
    pbm OR djm -> kpj
    x03 XOR y03 -> ffh
    x00 XOR y04 -> ntg
    bfw OR bqk -> z06
    nrd XOR fgs -> wpb
    frj XOR qhw -> z04
    bqk OR frj -> z07
    y03 OR x01 -> nrd
    hwm AND bqk -> z03
    tgd XOR rvg -> z12
    tnw OR pbm -> gnj

After waiting for values on all wires starting with `z`, the wires in
this system have the following values:

    bfw: 1
    bqk: 1
    djm: 1
    ffh: 0
    fgs: 1
    frj: 1
    fst: 1
    gnj: 1
    hwm: 1
    kjc: 0
    kpj: 1
    kwq: 0
    mjb: 1
    nrd: 1
    ntg: 0
    pbm: 1
    psh: 1
    qhw: 1
    rvg: 0
    tgd: 0
    tnw: 1
    vdt: 1
    wpb: 0
    z00: 0
    z01: 0
    z02: 0
    z03: 1
    z04: 0
    z05: 1
    z06: 1
    z07: 1
    z08: 1
    z09: 1
    z10: 1
    z11: 0
    z12: 0

Combining the bits from all wires starting with `z` produces the binary
number `0011111101000`. Converting this number to decimal produces
`2024`.

Simulate the system of gates and wires. *What decimal number does it
output on the wires starting with `z`?*

Your puzzle answer was `51715173446832`.

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

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

After inspecting the monitoring device more closely, you determine that
the system you\'re simulating is trying to *add two binary numbers*.

Specifically, it is treating the bits on wires starting with `x` as one
binary number, treating the bits on wires starting with `y` as a second
binary number, and then attempting to add those two numbers together.
The output of this operation is produced as a binary number on the wires
starting with `z`. (In all three cases, wire `00` is the least
significant bit, then `01`, then `02`, and so on.)

The initial values for the wires in your puzzle input represent *just
one instance* of a pair of numbers that sum to the wrong value.
Ultimately, *any* two binary numbers provided as input should be handled
correctly. That is, for any combination of bits on wires starting with
`x` and wires starting with `y`, the sum of the two numbers those bits
represent should be produced as a binary number on the wires starting
with `z`.

For example, if you have an addition system with four `x` wires, four
`y` wires, and five `z` wires, you should be able to supply any four-bit
number on the `x` wires, any four-bit number on the `y` numbers, and
eventually find the sum of those two numbers as a five-bit number on the
`z` wires. One of the many ways you could provide numbers to such a
system would be to pass `11` on the `x` wires (`1011` in binary) and
`13` on the `y` wires (`1101` in binary):

    x00: 1
    x01: 1
    x02: 0
    x03: 1
    y00: 1
    y01: 0
    y02: 1
    y03: 1

If the system were working correctly, then after all gates are finished
processing, you should find `24` (`11+13`) on the `z` wires as the
five-bit binary number `11000`:

    z00: 0
    z01: 0
    z02: 0
    z03: 1
    z04: 1

Unfortunately, your actual system needs to add numbers with many more
bits and therefore has many more wires.

Based on forensic analysis of scuff marks and
scratches on the device, you can tell that there are exactly *four*
pairs of gates whose output wires have been *swapped*. (A gate can only
be in at most one such pair; no gate\'s output was swapped multiple
times.)

For example, the system below is supposed to find the bitwise `AND` of
the six-bit number on `x00` through `x05` and the six-bit number on
`y00` through `y05` and then write the result as a six-bit number on
`z00` through `z05`:

    x00: 0
    x01: 1
    x02: 0
    x03: 1
    x04: 0
    x05: 1
    y00: 0
    y01: 0
    y02: 1
    y03: 1
    y04: 0
    y05: 1

    x00 AND y00 -> z05
    x01 AND y01 -> z02
    x02 AND y02 -> z01
    x03 AND y03 -> z03
    x04 AND y04 -> z04
    x05 AND y05 -> z00

However, in this example, two pairs of gates have had their output wires
swapped, causing the system to produce wrong answers. The first pair of
gates with swapped outputs is `x00 AND y00 -> z05` and
`x05 AND y05 -> z00`; the second pair of gates is `x01 AND y01 -> z02`
and `x02 AND y02 -> z01`. Correcting these two swaps results in this
system that works as intended for any set of initial values on wires
that start with `x` or `y`:

    x00 AND y00 -> z00
    x01 AND y01 -> z01
    x02 AND y02 -> z02
    x03 AND y03 -> z03
    x04 AND y04 -> z04
    x05 AND y05 -> z05

In this example, two pairs of gates have outputs that are involved in a
swap. By sorting their output wires\' names and joining them with
commas, the list of wires involved in swaps is `z00,z01,z02,z05`.

Of course, your actual system is much more complex than this, and the
gates that need their outputs swapped could be *anywhere*, not just
attached to a wire starting with `z`. If you were to determine that you
need to swap output wires `aaa` with `eee`, `ooo` with `z99`, `bbb` with
`ccc`, and `aoc` with `z24`, your answer would be
`aaa,aoc,bbb,ccc,eee,ooo,z24,z99`.

Your system of gates and wires has *four* pairs of gates which need
their output wires swapped - *eight* wires in total. Determine which
four pairs of gates need their outputs swapped so that your system
correctly performs addition; *what do you get if you sort the names of
the eight wires involved in a swap and then join those names with
commas?*

Answer:

Although it hasn\'t changed, you can still [get your puzzle
input](24/input).