AoC 2021 Day 08: Seven Segment Search

Dated Mar 7, 2022; last modified on Mon, 07 Mar 2022

Day 8 - Advent of Code 2021. . Accessed Mar 7, 2022.

Part I Description

You barely reach the safety of the cave when the whale smashes into the cave mouth, collapsing it. Sensors indicate another exit to this cave at a much greater depth, so you have no choice but to press on.

As your submarine slowly makes its way through the cave system, you notice that the four-digit seven-segment displays in your submarine are malfunctioning; they must have been damaged during the escape. You’ll be in a lot of trouble without them, so you’d better figure out what’s wrong.

Each digit of a seven-segment display is rendered by turning on or off any of the seven segments named a through g:

  0:      1:      2:      3:      4:
 aaaa    ....    aaaa    aaaa    ....
b    c  .    c  .    c  .    c  b    c
b    c  .    c  .    c  .    c  b    c
 ....    ....    dddd    dddd    dddd
e    f  .    f  e    .  .    f  .    f
e    f  .    f  e    .  .    f  .    f
 gggg    ....    gggg    gggg    ....

  5:      6:      7:      8:      9:
 aaaa    aaaa    aaaa    aaaa    aaaa
b    .  b    .  .    c  b    c  b    c
b    .  b    .  .    c  b    c  b    c
 dddd    dddd    ....    dddd    dddd
.    f  e    f  .    f  e    f  .    f
.    f  e    f  .    f  e    f  .    f
 gggg    gggg    ....    gggg    gggg

So, to render a 1, only segments c and f would be turned on; the rest would be off. To render a 7, only segments a, c, and f would be turned on.

The problem is that the signals which control the segments have been mixed up on each display. The submarine is still trying to display numbers by producing output on signal wires a through g, but those wires are connected to segments randomly. Worse, the wire/segment connections are mixed un separately for each four-digit display! (All of the digits within a display use the same connections, though.)

So, you might know that only signal wires b and g are turned on, but that doesn’t mean segments b and g are turned on: the only digit that uses two segments is 1, so it must mean segments c and f are meant to be on. With just that information, you still can’t tell which wire (b/g) goes to which segment (c/f). For that, you’ll need to collect more information.

For each display, you watch the changing signals for a while, make a note of all ten unique signal patterns you see, and then write down a single four-digit output value (your puzzle input). Using the signal patterns, you should be able to work out which patterns corresponds to which digit.

For example, here’s what you might see in a single entry in your notes:

acedgfb cdfbe gcdfa fbcad dab cefabd cdfgeb eafb cagedb ab | cdfeb fcadb cdfeb cdbaf

Each entry consists of ten unique signal patterns, a | delimiter, and finally the four-digit output value. Within an entry, the same wire/segment connections are used (but you don’t know what the connections actually are). The unique signal patterns correspond to the ten different ways the submarine tries to render a digit using the current wire/segment connections. Because 7 is the only digit that uses three segments, dab in the above example means that to render 7, signal lines d, a, and b are on. Because 4 is the only digit that uses four segments, eafb means that to render a 4, signal lines e, a, f, and b are on.

Using this information, you should be able to work out which combination of signal wires corresponds to each of the ten digits. Then, you can decode the four digit output value. Unfortunately, in the above example, all of the digits in the output value cdfeb fcadb cdfeb cdbaf use five segments and are more difficult to deduce.

For now, focus on the easy digits. Because the digits 1, 4, 7, and 8 each use a unique number of segments, you should be able to tell which combinations of signals correspond to those digits.

In the output values (the part after the | on each line), how many times do digits 1, 4, 7, or 8 appear?

{-#  OPTIONS_GHC -Wall  #-}
{-#  LANGUAGE RecordWildCards  #-}

module AoC2021.SevenSegmentSearch

import qualified Data.IntMap as IntMap
import qualified Data.IntSet as IntSet
import qualified Data.Map as Map
import Data.Foldable (Foldable(foldl'))
import Data.Char (ord)

Input Representation

The line be cfbegad cbdgef fgaecd cgeb fdcge agebfd fecdb fabcd edb | fdgacbe cefdb cefbgd gcbe has cfbegad matching with fdgacbe in the output value, so I need a representation that allows those two to be linked. Sorting the characters is sufficient as it gives abcdefg in both cases.

The ten signal patterns are in no particular order, so a [String] will do. The output values do not need to be in any particular order, so a [String] will also do.

data SevenSegmentDisplay = SevenSegmentDisplay{
    uniquePatterns :: [IntSet.IntSet], outputValues :: [IntSet.IntSet]} deriving Show

However, the solution for Part II makes use of set operations, e.g. intersection and subtraction, and therefore, using a [Set Char] instead of a [String] makes more sense. Even better, an [IntSet] .

Part I Solution

numActiveSegmentsToDigits :: IntMap.IntMap [Int]
numActiveSegmentsToDigits = IntMap.fromList
    [(6, [0, 6, 9]), (2, [1]), (5, [2, 3, 5]), (4, [4]), (3, [7]), (7, [8])]

nonAmbiguousLengths :: IntSet.IntSet
--  There is also `Map.keysSet` but that returns an
nonAmbiguousLengths = IntSet.fromList $ IntMap.keys $
    IntMap.filter (\t -> length t == 1) numActiveSegmentsToDigits

The containers package provides IntMap and IntSet in addition to the general Map and Set data structures. This distinction is motivated by ’s work on finite maps that are based on ’s Patricia trees, instead of the usual base of balanced binary search trees. While both bases have fast lookups and inserts, Patricia trees have fast merges of two containers.

I’ve been getting the vibe that Haskell is more explicit in its connection to academia, e.g. foundational papers being linked from API docs, and library writers and maintainers being faculty in CS departments.

numOf1478AppearancesInOutput :: [SevenSegmentDisplay] -> Int
numOf1478AppearancesInOutput = foldr f 0 where
    f :: SevenSegmentDisplay -> Int -> Int
    f SevenSegmentDisplay{ outputValues=outputs } prevSum =
        prevSum + length (
            (\s -> IntSet.member (IntSet.size s) nonAmbiguousLengths)

Pattern-matching using SevenSegmentDisplay{ outputValues=outputValues } leads to a Wname-shadowing HLint warning on the second outputValues. notes that either of the NamedFieldPuns or RecordWildCards extensions allows shadowing of field names. However, adding either language extension results in HLint warning that the LANGUAGE pragma is unused.

Compared to other Part I’s, this one felt too straightforward. Most of the difficulty was in using parsec to parse the input line .

Part II Description

Part II might feature additional information to distinguish additional digits. Maybe the output values follow some pattern, e.g. the digits are always increasing from right to left, not all digits are possible for a given output, etc.

Update: I was wrong. We do have enough information to deduce all of the digits. I don’t see how this is always possible.

Through a little deduction, you should now be able to determine the remaining digits. Consider again the first example above:

acedgfb cdfbe gcdfa fbcad dab cefabd cdfgeb eafb cagedb ab | cdfeb fcadb cdfeb cdbaf

After some careful analysis, the mapping between signal wires and segments only make sense in the following configuration:

e    a
e    a
g    b
g    b

So, the unique signal patterns would correspond to the following digits:

acedgfb: 8
cdfbe: 5
gcdfa: 2
fbcad: 3
dab: 7
cefabd: 9
cdfgeb: 6
eafb: 4
cagedb: 0
ab: 1

Then, the four digits of the output value can be decoded:

cdfeb: 5
fcadb: 3
cdfeb: 5
cdbaf: 3

Therefore, the output value for this entry is 5353.

For each entry, determine all of the wire/segment connections and decode the four-digit output values. What do you get if you add up all of the output values?

Part II Solution

In the non-jumbled up case, the matching of digits to segments is:

1 -   c  f
4 -  bcd f
7 - a c  f
8 - abcdefg

0 - abc efg
2 - a cde g
3 - a cd fg
5 - ab d fg
6 - ab defg
9 - abcd fg

From Part I, the jumbled representations of 1, 4, 7, and 8 can be identified by counting the number of segments. We can go further and note that there are shared segments, for example, 0 and 1 both have segments c and f. So, solving Part II comes down to building up from the base provided by 1, 4, 7, and 8.

The union of the active segments in 147 is abcdf, but that doesn’t seem helpful. 8 isn’t helpful because it has all of the segments active. The union of 1x’s segments (for \(x \in [4, 7, 8]\)) doesn’t seem helpful because it will be the same as the active segments for \(x\). The union of 47’s active segments is abcdf, which also doesn’t correspond to a digit.

If we go one level deeper, we might deduce an additional digit. With regard to x’s active segments, \(x \in [1, 4, 7, 8]\), being a subset of another number y’s active segments, \(y \in [0, 2, 3, 5, 6, 9]\):

1: 0, 3, 9
4: 9
7: 0, 3, 9

Nothing jumps out yet. Maybe looking at xs and ys that share segments might be illuminating? 1’s cf shares at least one active segment with all ys and therefore given that the union of 1x’s active segments is the same as that of x, proceeding further doesn’t seem useful. Maybe looking at non-shared segments between xs and ys helps? Nah, that wouldn’t provide additional info than what I got from the shared segments analysis.

Maybe I can do something with the ys:

0 - abc efg (6)
6 - ab defg (6)
9 - abcd fg (6)

2 - a cde g (5)
3 - a cd fg (5)
5 - ab d fg (5)

The union of 069’s active segments is abcdefg, and so is the union of 235’s active segments, so that’s not helpful.

The intersection of 069’s active segments is abfg, and the complement of this intersection is cde; nothing useful yet.

I’m mostly talking in set terminology, so maybe instead of using String to represent a display digit, I should have used a Set Char.

The intersection of 253’s active segments is adg, and the complement of this intersection is bcef; nothing useful yet.

The union of the previous two intersections (abfg and adg) is abdfg, and for once we have something useful as that equals to 5’s active segments! The complement of abdfg is ce, but that’s not useful.

Of 23’s active segments, the union is acdefg, the complement of the union is b, the intersection is acdg, and the complement of this intersection is bef. None of these look helpful.

The knowns and unknowns are currently:

8 - abcdefg (7)

1 -   c  f  (2)
4 -  bcd f  (4)
7 - a c  f  (3)
5 - ab d fg (5)

0 - abc efg (6)
6 - ab defg (6)
9 - abcd fg (6)

2 - a cde g (5)
3 - a cd fg (5)

I had already computed combinations of 147, but maybe there’s new info now that 5 is also known.

The unknowns, \([0, 6, 9, 2, 3]\) have at least 5 active segments, so taking operations which give at least 5 elements will be most useful.

The union of 15s active segments is abcdfg, which matches the active segments of 9. Sweet!

The union of 45’s active segments is abcdfg, which we’ve already determined to be 9. The union of 75’s active segments doesn’t yield anything new either.

The knowns and unknowns are currently:

8 - abcdefg (7)

1 -   c  f  (2)
4 -  bcd f  (4)
7 - a c  f  (3)
5 - ab d fg (5)
9 - abcd fg (6)

0 - abc efg (6)
6 - ab defg (6)

2 - a cde g (5)
3 - a cd fg (5)

The difference between 0 and 6 is that the former has a c, while the latter has a d. Subtracting 5 abdfg from 9 abcdfg gives c, and therefore distinguishes 0 from 6.

The difference between 2 and 3 is that the former has an e, while the latter has an f. The complement of the union of 14759’s active segments is e, and that can be used to distinguish 2 from 3.

Tedious exercise, but rewarding in the end. I feel like Sherlock Holmes.

has simpler logic though. For convenience, here is the non-scrambled mapping:

1 -   c  f  (2)
4 -  bcd f  (4)
7 - a c  f  (3)
8 - abcdefg (7)

0 - abc efg (6)
6 - ab defg (6)
9 - abcd fg (6)

2 - a cde g (5)
3 - a cd fg (5)
5 - ab d fg (5)

\([1, 4, 7, 8]\) are identifiable from their unique lengths. 6 is the 6-segment digit digit that does not include 1. 9 is the 6-segment digit that includes 4. 0 is the remaining 6-segment digit. 3 is the 5-segment digit that includes 1. 5 is the 5-segment digit that’s included in 6. 2 is the remaining 5-segment digit.

The major shortcoming of how I went about deducing the mapping was thinking in terms of unions and intersections, and not considering subsets. Once I found a working approach, I hastened to implement it and call it a day.

type Segment = IntSet.IntSet
type SegmentsMapping = Map.Map Segment Int

allSegmentsActive :: IntSet.IntSet
allSegmentsActive = (IntSet.fromList . map ord) "abcdefg"

deduceMappings :: [Segment] -> SegmentsMapping
deduceMappings input =
    Map.fromList [(repr0, 0), (repr1, 1), (repr2, 2), (repr3, 3), (repr4, 4), (repr5, 5), (repr6, 6), (repr7, 7), (repr8, 8), (repr9, 9)] where
        getSegmentsOfSize :: Int -> [Segment]
        getSegmentsOfSize n = filter (\s -> IntSet.size s == n) input

        repr1 = head (getSegmentsOfSize 2)
        repr4 = head (getSegmentsOfSize 4)
        repr7 = head (getSegmentsOfSize 3)
        repr8 = allSegmentsActive

        reprs069 = getSegmentsOfSize 6
        intersection069 = foldl' IntSet.intersection allSegmentsActive reprs069

        reprs235 = getSegmentsOfSize 5
        intersection235 = foldl' IntSet.intersection allSegmentsActive reprs235

        repr5 = IntSet.union intersection069 intersection235
        repr9 = IntSet.union repr1 repr5

        reprs06 = filter (/= repr9) reprs069
        difference95 = IntSet.difference repr9 repr5

        repr6 = head (filter (IntSet.disjoint difference95) reprs06)
        repr0 = head (filter (/= repr6) reprs06)

        union14579 = foldl' IntSet.union IntSet.empty [repr1, repr4, repr5, repr7, repr9]
        complement14579 = IntSet.difference allSegmentsActive union14579

        reprs23 = filter (/= repr5) reprs235
        repr3 = head (filter (IntSet.disjoint complement14579) reprs23)
        repr2 = head (filter (/= repr3) reprs23)

parseInt :: SegmentsMapping -> [Segment] -> Int
parseInt mapping = foldl' (\acc segment -> acc * 10 + (mapping Map.! segment)) 0

sumOfOutputValues :: [SevenSegmentDisplay] -> Int
sumOfOutputValues = foldl' (\acc display -> acc + extractOutputValue display) 0 where
    extractOutputValue :: SevenSegmentDisplay -> Int
    extractOutputValue SevenSegmentDisplay{ .. } = parseInt (deduceMappings uniquePatterns) outputValues

approaches the problem differently. They note that the search space is small enough, \(7! = 5{,}040\), that there is no need to use first-order logic .

Why is the search space \(7!\) instead of \(6!\), given that we already know the representations of \([1, 4, 7, 8]\)?

Answer: We’re looking at a seven-segment display.

Why \(7!\) though. \(7!\) is the number of ways \(7\) items can be ordered. I don’t understand why the search space also corresponds to this.

-- From

type Segment = Int
newtype Observation = Observation { view :: [Segment] }

type Wire = Int
newtype Digit = Digit [Wire] deriving (Eq, Ord)

So far I’ve been using data to define custom types; newtype is new to me.

The syntax and usage of newtype and data are identical. Replacing newtype with data will compile and most probably work, but data can only be replaced with newtype if the type has exactly one constructor with exactly one field inside it. The restrictions for newtype imply that the new type and the type of field are in direct correspondence (isomorphic), and therefore after the type is checked at compile time, at run time the two types can be treated essentially the same, without the overhead or indirection normally associated with a data constructor.

Note that there is more subtle reasoning on the nuance between newtype and data (e.g. why not use newtype everywhere you can?) in and in A Gentle Introduction to Haskell: Types, Again . This nuance doesn’t seem useful to me at this stage.

-- From

combine :: Observation -> Digit
combine = Digit . sort . view

This syntax is possible because Observation { view :: [Segment] } creates an accessor function called view of type Observation -> [Segment] . Segment and Wire are synonyms for Int, and are thus entirely compatible, and that’s why Digit can accept a [Segment] .

-- From

reference :: [Digit]
reference = map (combine . readObservation)
    [ "abcefg", "cf", "acdeg", "acdfg", "bcdf"
    , "abdfg", "abdefg", "acf", "abcdefg", "abcdfg"

solve :: [Observation] -> [Observation] -> [Int]
solve obsDigits obsDisplay =
    let permute p = map (combine . Observation . map (p !!) . view)
        Just perm = find ((== sort reference) . sort . flip permute obsDigits)
                    (permutations [0..6])
    in map (fromJust . (`elemIndex` reference)) (permute perm obsDisplay)

I don’t yet follow why solve works, and I don’t think dissecting it further is worthwhile. Maybe I can leave it at “exhaustive search is feasible for this problem”?

New API: flip :: (a -> b -> c) -> b -> a -> c, where flip f takes its (first) two arguments in the reverse order of f, e.g.

 >>> flip (++) "hello" "world"


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