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                                 {:year [8]2023}

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--- Day 20: Pulse Propagation ---

   With your help, the Elves manage to find the right parts and fix all of
   the machines. Now, they just need to send the command to boot up the
   machines and get the sand flowing again.

   The machines are far apart and wired together with long cables. The cables
   don't connect to the machines directly, but rather to communication
   modules attached to the machines that perform various initialization tasks
   and also act as communication relays.

   Modules communicate using pulses. Each pulse is either a high pulse or a
   low pulse. When a module sends a pulse, it sends that type of pulse to
   each module in its list of destination modules.

   There are several different types of modules:

   Flip-flop modules (prefix %) are either on or off; they are initially off.
   If a flip-flop module receives a high pulse, it is ignored and nothing
   happens. However, if a flip-flop module receives a low pulse, it flips
   between on and off. If it was off, it turns on and sends a high pulse. If
   it was on, it turns off and sends a low pulse.

   Conjunction modules (prefix &) remember the type of the most recent pulse
   received from each of their connected input modules; they initially
   default to remembering a low pulse for each input. When a pulse is
   received, the conjunction module first updates its memory for that input.
   Then, if it remembers high pulses for all inputs, it sends a low pulse;
   otherwise, it sends a high pulse.

   There is a single broadcast module (named broadcaster). When it receives a
   pulse, it sends the same pulse to all of its destination modules.

   Here at Desert Machine Headquarters, there is a module with a single
   button on it called, aptly, the button module. When you push the button, a
   single low pulse is sent directly to the broadcaster module.

   After pushing the button, you must wait until all pulses have been
   delivered and fully handled before pushing it again. Never push the button
   if modules are still processing pulses.

   Pulses are always processed in the order they are sent. So, if a pulse is
   sent to modules a, b, and c, and then module a processes its pulse and
   sends more pulses, the pulses sent to modules b and c would have to be
   handled first.

   The module configuration (your puzzle input) lists each module. The name
   of the module is preceded by a symbol identifying its type, if any. The
   name is then followed by an arrow and a list of its destination modules.
   For example:

 broadcaster -> a, b, c
 %a -> b
 %b -> c
 %c -> inv
 &inv -> a

   In this module configuration, the broadcaster has three destination
   modules named a, b, and c. Each of these modules is a flip-flop module (as
   indicated by the % prefix). a outputs to b which outputs to c which
   outputs to another module named inv. inv is a conjunction module (as
   indicated by the & prefix) which, because it has only one input, acts like
   an inverter (it sends the opposite of the pulse type it receives); it
   outputs to a.

   By pushing the button once, the following pulses are sent:

 button -low-> broadcaster
 broadcaster -low-> a
 broadcaster -low-> b
 broadcaster -low-> c
 a -high-> b
 b -high-> c
 c -high-> inv
 inv -low-> a
 a -low-> b
 b -low-> c
 c -low-> inv
 inv -high-> a

   After this sequence, the flip-flop modules all end up off, so pushing the
   button again repeats the same sequence.

   Here's a more interesting example:

 broadcaster -> a
 %a -> inv, con
 &inv -> b
 %b -> con
 &con -> output

   This module configuration includes the broadcaster, two flip-flops (named
   a and b), a single-input conjunction module (inv), a multi-input
   conjunction module (con), and an untyped module named output (for testing
   purposes). The multi-input conjunction module con watches the two
   flip-flop modules and, if they're both on, sends a low pulse to the output
   module.

   Here's what happens if you push the button once:

 button -low-> broadcaster
 broadcaster -low-> a
 a -high-> inv
 a -high-> con
 inv -low-> b
 con -high-> output
 b -high-> con
 con -low-> output

   Both flip-flops turn on and a low pulse is sent to output! However, now
   that both flip-flops are on and con remembers a high pulse from each of
   its two inputs, pushing the button a second time does something different:

 button -low-> broadcaster
 broadcaster -low-> a
 a -low-> inv
 a -low-> con
 inv -high-> b
 con -high-> output

   Flip-flop a turns off! Now, con remembers a low pulse from module a, and
   so it sends only a high pulse to output.

   Push the button a third time:

 button -low-> broadcaster
 broadcaster -low-> a
 a -high-> inv
 a -high-> con
 inv -low-> b
 con -low-> output
 b -low-> con
 con -high-> output

   This time, flip-flop a turns on, then flip-flop b turns off. However,
   before b can turn off, the pulse sent to con is handled first, so it
   briefly remembers all high pulses for its inputs and sends a low pulse to
   output. After that, flip-flop b turns off, which causes con to update its
   state and send a high pulse to output.

   Finally, with a on and b off, push the button a fourth time:

 button -low-> broadcaster
 broadcaster -low-> a
 a -low-> inv
 a -low-> con
 inv -high-> b
 con -high-> output

   This completes the cycle: a turns off, causing con to remember only low
   pulses and restoring all modules to their original states.

   To get the cables warmed up, the Elves have pushed the button 1000 times.
   How many pulses got sent as a result (including the pulses sent by the
   button itself)?

   In the first example, the same thing happens every time the button is
   pushed: 8 low pulses and 4 high pulses are sent. So, after pushing the
   button 1000 times, 8000 low pulses and 4000 high pulses are sent.
   Multiplying these together gives 32000000.

   In the second example, after pushing the button 1000 times, 4250 low
   pulses and 2750 high pulses are sent. Multiplying these together gives
   11687500.

   Consult your module configuration; determine the number of low pulses and
   high pulses that would be sent after pushing the button 1000 times,
   waiting for all pulses to be fully handled after each push of the button.
   What do you get if you multiply the total number of low pulses sent by the
   total number of high pulses sent?

   Your puzzle answer was 791120136.

--- Part Two ---

   The final machine responsible for moving the sand down to Island Island
   has a module attached named rx. The machine turns on when a single low
   pulse is sent to rx.

   Reset all modules to their default states. Waiting for all pulses to be
   fully handled after each button press, what is the fewest number of button
   presses required to deliver a single low pulse to the module named rx?

   Your puzzle answer was 215252378794009.

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

   At this point, you should [16]return to your Advent calendar and try
   another puzzle.

   If you still want to see it, you can [17]get your puzzle input.

   You can also [Shareon [18]Twitter [19]Mastodon] this puzzle.

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