--- Day 16: Packet Decoder --- ------------------------------ As you leave the cave and reach open waters, you receive a transmission from the Elves back on the ship. The transmission was sent using the Buoyancy Interchange Transmission System (BITS), a method of packing numeric expressions into a binary sequence. Your submarine's computer has saved the transmission in [hexadecimal](https://en.wikipedia.org/wiki/Hexadecimal) (your puzzle input). The first step of decoding the message is to convert the hexadecimal representation into binary. Each character of hexadecimal corresponds to four bits of binary data: ``` 0 = 0000 1 = 0001 2 = 0010 3 = 0011 4 = 0100 5 = 0101 6 = 0110 7 = 0111 8 = 1000 9 = 1001 A = 1010 B = 1011 C = 1100 D = 1101 E = 1110 F = 1111 ``` The BITS transmission contains a single *packet* at its outermost layer which itself contains many other packets. The hexadecimal representation of this packet might encode a few extra `0` bits at the end; these are not part of the transmission and should be ignored. Every packet begins with a standard header: the first three bits encode the packet *version*, and the next three bits encode the packet *type ID*. These two values are numbers; all numbers encoded in any packet are represented as binary with the most significant bit first. For example, a version encoded as the binary sequence `100` represents the number `4`. Packets with type ID `4` represent a *literal value*. Literal value packets encode a single binary number. To do this, the binary number is padded with leading zeroes until its length is a multiple of four bits, and then it is broken into groups of four bits. Each group is prefixed by a `1` bit except the last group, which is prefixed by a `0` bit. These groups of five bits immediately follow the packet header. For example, the hexadecimal string `D2FE28` becomes: ``` 110100101111111000101000 VVVTTTAAAAABBBBBCCCCC ``` Below each bit is a label indicating its purpose: * The three bits labeled `V` (`110`) are the packet version, `6`. * The three bits labeled `T` (`100`) are the packet type ID, `4`, which means the packet is a literal value. * The five bits labeled `A` (`10111`) start with a `1` (not the last group, keep reading) and contain the first four bits of the number, `0111`. * The five bits labeled `B` (`11110`) start with a `1` (not the last group, keep reading) and contain four more bits of the number, `1110`. * The five bits labeled `C` (`00101`) start with a `0` (last group, end of packet) and contain the last four bits of the number, `0101`. * The three unlabeled `0` bits at the end are extra due to the hexadecimal representation and should be ignored. So, this packet represents a literal value with binary representation `011111100101`, which is `2021` in decimal. Every other type of packet (any packet with a type ID other than `4`) represent an *operator* that performs some calculation on one or more sub-packets contained within. Right now, the specific operations aren't important; focus on parsing the hierarchy of sub-packets. An operator packet contains one or more packets. To indicate which subsequent binary data represents its sub-packets, an operator packet can use one of two modes indicated by the bit immediately after the packet header; this is called the *length type ID*: * If the length type ID is `0`, then the next *15* bits are a number that represents the *total length in bits* of the sub-packets contained by this packet. * If the length type ID is `1`, then the next *11* bits are a number that represents the *number of sub-packets immediately contained* by this packet. Finally, after the length type ID bit and the 15-bit or 11-bit field, the sub-packets appear. For example, here is an operator packet (hexadecimal string `38006F45291200`) with length type ID `0` that contains two sub-packets: ``` 00111000000000000110111101000101001010010001001000000000 VVVTTTILLLLLLLLLLLLLLLAAAAAAAAAAABBBBBBBBBBBBBBBB ``` * The three bits labeled `V` (`001`) are the packet version, `1`. * The three bits labeled `T` (`110`) are the packet type ID, `6`, which means the packet is an operator. * The bit labeled `I` (`0`) is the length type ID, which indicates that the length is a 15-bit number representing the number of bits in the sub-packets. * The 15 bits labeled `L` (`000000000011011`) contain the length of the sub-packets in bits, `27`. * The 11 bits labeled `A` contain the first sub-packet, a literal value representing the number `10`. * The 16 bits labeled `B` contain the second sub-packet, a literal value representing the number `20`. After reading 11 and 16 bits of sub-packet data, the total length indicated in `L` (27) is reached, and so parsing of this packet stops. As another example, here is an operator packet (hexadecimal string `EE00D40C823060`) with length type ID `1` that contains three sub-packets: ``` 11101110000000001101010000001100100000100011000001100000 VVVTTTILLLLLLLLLLLAAAAAAAAAAABBBBBBBBBBBCCCCCCCCCCC ``` * The three bits labeled `V` (`111`) are the packet version, `7`. * The three bits labeled `T` (`011`) are the packet type ID, `3`, which means the packet is an operator. * The bit labeled `I` (`1`) is the length type ID, which indicates that the length is a 11-bit number representing the number of sub-packets. * The 11 bits labeled `L` (`00000000011`) contain the number of sub-packets, `3`. * The 11 bits labeled `A` contain the first sub-packet, a literal value representing the number `1`. * The 11 bits labeled `B` contain the second sub-packet, a literal value representing the number `2`. * The 11 bits labeled `C` contain the third sub-packet, a literal value representing the number `3`. After reading 3 complete sub-packets, the number of sub-packets indicated in `L` (3) is reached, and so parsing of this packet stops. For now, parse the hierarchy of the packets throughout the transmission and *add up all of the version numbers*. Here are a few more examples of hexadecimal-encoded transmissions: * `8A004A801A8002F478` represents an operator packet (version 4) which contains an operator packet (version 1) which contains an operator packet (version 5) which contains a literal value (version 6); this packet has a version sum of `*16*`. * `620080001611562C8802118E34` represents an operator packet (version 3) which contains two sub-packets; each sub-packet is an operator packet that contains two literal values. This packet has a version sum of `*12*`. * `C0015000016115A2E0802F182340` has the same structure as the previous example, but the outermost packet uses a different length type ID. This packet has a version sum of `*23*`. * `A0016C880162017C3686B18A3D4780` is an operator packet that contains an operator packet that contains an operator packet that contains five literal values; it has a version sum of `*31*`. Decode the structure of your hexadecimal-encoded BITS transmission; *what do you get if you add up the version numbers in all packets?* --- Part Two --- ---------------- Now that you have the structure of your transmission decoded, you can calculate the value of the expression it represents. Literal values (type ID `4`) represent a single number as described above. The remaining type IDs are more interesting: * Packets with type ID `0` are *sum* packets - their value is the sum of the values of their sub-packets. If they only have a single sub-packet, their value is the value of the sub-packet. * Packets with type ID `1` are *product* packets - their value is the result of multiplying together the values of their sub-packets. If they only have a single sub-packet, their value is the value of the sub-packet. * Packets with type ID `2` are *minimum* packets - their value is the minimum of the values of their sub-packets. * Packets with type ID `3` are *maximum* packets - their value is the maximum of the values of their sub-packets. * Packets with type ID `5` are *greater than* packets - their value is *1* if the value of the first sub-packet is greater than the value of the second sub-packet; otherwise, their value is *0*. These packets always have exactly two sub-packets. * Packets with type ID `6` are *less than* packets - their value is *1* if the value of the first sub-packet is less than the value of the second sub-packet; otherwise, their value is *0*. These packets always have exactly two sub-packets. * Packets with type ID `7` are *equal to* packets - their value is *1* if the value of the first sub-packet is equal to the value of the second sub-packet; otherwise, their value is *0*. These packets always have exactly two sub-packets. Using these rules, you can now work out the value of the outermost packet in your BITS transmission. For example: * `C200B40A82` finds the sum of `1` and `2`, resulting in the value `*3*`. * `04005AC33890` finds the product of `6` and `9`, resulting in the value `*54*`. * `880086C3E88112` finds the minimum of `7`, `8`, and `9`, resulting in the value `*7*`. * `CE00C43D881120` finds the maximum of `7`, `8`, and `9`, resulting in the value `*9*`. * `D8005AC2A8F0` produces `1`, because `5` is less than `15`. * `F600BC2D8F` produces `0`, because `5` is not greater than `15`. * `9C005AC2F8F0` produces `0`, because `5` is not equal to `15`. * `9C0141080250320F1802104A08` produces `1`, because `1` + `3` = `2` * `2`. *What do you get if you evaluate the expression represented by your hexadecimal-encoded BITS transmission?*