Transcendental Algebra

This puzzle is written in the titular Transcendental Algebra; Googling the term should lead to a transcription and translation of its constructor Jakob Linzbach's work on the subject. Using the concepts presented in the work (or other references revealed later), you can start translating the introduction:

"I am a philosophical language. 103 years ago, Jakob Linzbach wrote about me. 21 years ago, a woman wrote my language puzzle. I will now talk about a language puzzle from every year besides 2016^[1]. One team language puzzle exists."

This confirms that Transcendental Algebra is the correct language to work with here—Jakob Linzbach published his book in 1921, 103 years before 2024.

The foreword in the transcription mentions that Transcendental Algebra was the subject of a problem written by Ksenia Gilyarova for the inaugural International Linguistics Olympiad (IOL) in 2003, 21 years before 2024. It is thus reasonable to assume that the "other language puzzles" mentioned are other problems in the IOL.

Each description describes the properties of a problem in the IOL:

• The first is a general description of the type of data in the problem's dataset—examples include a bunch of transitive sentences, some arithmetic equations, or a script made of dots.
• The second describes a relevant property of the language itself. Sometimes this is demonstrated using words or sentences in the dataset.
• The third is a word, phrase, or sentence present in its dataset, which is used in extraction. Take its position in that dataset, then index it into the problem's name, as presented on the IOL website.

The sentences are initially ordered by the problem's name. Each problem comes from a different year, one for every year in the IOL's history up to this point (besides 2016, as mentioned in the introduction); resort in that order to get the instruction HOLD BILINGUAL DIALOG.

Submitting a video or audio clip of two people having a conversation while speaking different languages will lead to a response containing the answer, SHENZHEN IO.

Appendix

Appendix A - Reading the Transcendental Algebra

Transcendental algebra consists primarily of "figures" (pictorial representations of a physical object) and "symbols" (representations of abstract quantities and relations), in addition to numbers, letters, and mathematical symbols. The book contains several examples that can be used to interpret the sentences.

The mathematical symbols operate as follows:

• +: serves as a conjunction operator. Besides being the word "and", it also acts as "with" and is used in describing nouns with adjectives, for instance.
• –: The opposite of addition, often translated "without". Often used as a negation operator (not a). Also used to accentuate particular parts of a figure.
• ×: used to combine concepts.
• ÷: essentially "of", used to specify a concept of reduced scope, such as by marking possession.
• a^b: From its role as iterated multiplication, exponentiation takes on the much more significant role of representing the flow of time, and as such is used for things like verbs. A figure exponent represents an active verb; the inverse of one is a passive verb. Exponents of 0,\frac{1}{n}\(\frac{1}{n}\), 1, n, and infinity take on special meaning relating to the evolution of the object.
• a log b: Expresses evolution, the process by which the first operand becomes the second.^[4]
• =/>/</≈/≠: "Logical judgements". = is used both as a copula ("be") and to specify direct objects.

This puzzle also uses letters a, b, c… and the grammar to isolate representations of specific parts of a sentence (e.g., "a : (a^b = c)" is "subject", while "b : a^b" is "transitive verb"). Likewise, while n is most often used for pluralization, here it is isolated on occasion to talk about the whole class of numbers.

Clarifications regarding symbols are discussed in footnotes within the parse.

Appendix B - The Parse

Many of the symbols and combinations of symbols used are found in the book. Several aren't; these are explained here.

Introduction

[Person]1 = [Language]([Brain] / [Head])^-∞

"Person 1 is a language of brain-transubstantiation."

→ "I^[5] am a philosophical language."

[Person]_{[22 VI 1874]}^[Pen] – 103[Earth] = [Person]₁

"Person born on 22 June 1874 writes without 103 Earths person 1."

→ "Jakob Linzbach^[6] wrote me 103 years ago."

[Woman]^[Pen] – 21[Earth] = [Language][Question]^∞ / [Person]₁

"Woman writes without 21 Earths language-question-transubstantiation of person 1."

→ "A woman^[7] wrote a language puzzle^[8] about me 21 years ago."

[Person]₁^[Language] + t⁰ = 1[Language][Question]^∞ / ([Earth] / 0 - [Earth]₂₀₁₆)

"Person 1 speaks with time-origin 1 language-question-transubstantiation of Earth of 0 without Earth 2016."

→ "I now talk about one language puzzle of every^[9] year besides 2016."

(1[Language][Question]^∞ / [Person]₁[Person]₂[Person]₃)¹

"One language-question-transubstantiation of company^[10] exists."

→ "There is one team round puzzle."

Adyghe

([Question]^∞ / [Language]₁) = n([Question]1 / ³√(xyz))

"Question-transubstantiation of language 1 is n question-itself of location^[11]."

→ "Language 1's puzzle is a bunch of questions about location."

([Language]^-1 / ³√(xyz))^1/[Chain] + ([Language]^-1 / [Walk]) - x

"Part of language of location is attached with part of language of action without-horizontal."

→ "The preposition morpheme is attached to the left of the verb morpheme."

There aren't many problems like Adyghe that talk entirely about location and nothing else. Earlier years are also generally less interesting to talk about, so the fact here is an interesting detail about the morphology.

[Man]₃^-y = (1_? = [Stool] - y)

"Man 3 without-heights thing^[12] what^[13] is stool without-height."

→ "What does he drop under the stool?" (Sentence 5)

Benabena

([Question]^∞ / [Language]₂) = n([Person]_n^b = [Person]_n)

"Question-transubstantiation of language 2 is n (person n b-s person n)."

"Language 2's puzzle is a bunch of transitive sentences, with pronouns on each side."

([Language]^-1 / 0t) = ([Language]^-1 / [Walk]) - x, ([Language]^-1 / +t) = ([Language]^-1 / [Walk]) + x

"Part of language of zero time^[14] is part of language of action without-horizontal. Part of language of positive time is part of language of action with-horizontal."

→ "The present-tense morpheme is to the left of the verb morpheme, while the future-tense morpheme is to the right of the verb morpheme."

Also not much to talk about with this one, if only because it doesn't provide any interesting morphology to talk about.

2[Person]₂^{[Circle][Through]} + t = [Person]₁

"Two Person 2s perforate^[15] with time Person 1."

→ "You both will pierce me." (fifth sentence in an unnumbered dataset)

Book Pahlavi

([Question]^∞ / [Language]₃) = [Pen]^∞ + (x log (-x))

"Question-transubstantiation of language 3 is pen-transubstantiation with right becoming left."

"Language 3's puzzle is a right-to-left writing system."

([Language]^-1 / ([God] / i))^{1/[Pen] + y log (-y)}

"Part of language of god of ceasing existence^[16] is written with up becoming down."

→ "The word for the god of destruction is written upside-down."

Book Pahlavi has a lot of interesting things, namely that the same symbol can be written for up to four different sounds, but the most distinctive aspect for this puzzle's purpose is that the word for the destructive spirit Ahriman is written upside-down.

([Hair] / [Head])

"Hair of head"

→ "hair" (Number 7 in the numbered list of Middle Persian words.)

Braille

([Question]^∞ / [Language]₄) = [Pen]^∞ / n [Filled Circle]

"Question-transubstantiation of language 4 is pen-transubstantiation of n filled circles."

→ "Language 4's puzzle is a writing system of dots.

([Person] - ([Eye] + [Eye]))^{[Book]^p} = [Language]₄

"Person without eye and eye can^[17] read language 4."

→ "The blind can read language 4."

It's Braille.

[Fox]₁^>t

"Fox 1^[18] is more hasty."

→ "This fox is too quick!" (The first of only three example sentences.)

EAN-13

([Question]^∞ / [Language]₅) = [Pen]^∞ / n [Straight line]

"Question-transubstantiation of language 5 is pen-transubstantiation of n straight lines."

→ "Language 4's puzzle is a writing system of vertical bars.

n = 2 [Straight line] + [Light]^∞ + 2 [Straight line] – [Light]^∞

"n is two straight lines with light-transubstantiation and two straight lines without light-transubstantiation."

→ "A number is two white^[19] bars and two black bars."

EAN-13 is a barcode standard. A single number consists of four bars (alternating black and white) adding to a constant width.

[Book]_{(a – [Brain] / ³√(xyz))}

"Book a without intelligence of location."

→ "The Lost Symbol^[20] (ISBN book)." (The third in the numbered list of barcode descriptions, and also the third barcode.)

Guazacapán Xinka

([Question]^∞ / [Language]₆) = n (a^b) + n (a^1/b)

"Question-transubstantiation of language 6 is n (a b's) and n (a is b'd)."

→ "Language 6's puzzle is a bunch of sentences with active and passive modalities."

[Language]^-1 / ([Person]₃^b + 0t) ≠ [In] + ([Question]^∞ / [Language]₆)

"Part of language of (Person 3 b's with zero time) is not in question-transubstantiation of language 6."

→ "Language 6's puzzle does not contain expressions for sentences with a third-person subject in present tense and active modality."

The puzzle for Guazacapán Xinka has first and third person, present and past tenses, and active and passive modalities. Bizarrely, the puzzle features every possible combination except the 3rd person present active.

1^[Eye] - t = 1

"Unit sees without time unit."

→ "It saw it." (Row 1 in the numbered dataset.)

Inuktitut

([Question]^∞ / [Language]₇) = n (a^b = c) + n (a^b)

"Question-transubstantiation of language 7 is n (a b's c) and n (a b's)."

→ "Language 7's puzzle is a bunch of transitive and intransitive statements."

[Language]^-1 / (c / a^b = c₁) ≈ [Language]^-1 / (a / a^b = c)

"Part of language of (c of a^b = c₁) is similar to part of language of (a of a^b = c)."

→ "The object of a sentence with a definite object is similar to the subject of a sentence with an indefinite object."

The Inuktitut problem has three default sentence structures for intransitive, transitive-definite, and transitive-indefinite. In transitive sentences with definite objects, the object inflects like the subject of the other sentence structures, while the subject gets a different ending.

([Dog] / [Person]₂)^[Eye] – t = [Person]₂

"Dog of person 2 sees without time person 2."

→ "Your_sg dog saw you_sg." (Sentence 1.)

Kilivila

([Question]^∞ / [Language]₈) = n (a^b = c) + n (a_?^b = c) + n (n_?a^b = c)

"Question-transubstantiation of language 8 is n (a b's c), n (which a b's c), and n (which n of a b's c)."

→ "Language 8's puzzle is a series of sentences—either declarative, or questions asking 'which' or 'how many'."

n_? [Man] ≠ n_? [Woman] ≈ n_? [Dog] ≠ n_? [Person]^1/n ≠ n_? 1

"'How many men is dissimilar to how many women is similar to how many dogs is dissimilar to how many children is dissimilar to how many things."

→ "The word for 'how many' differs depending on the thing being counted."

Kilivila has a number of noun classes, each with a certain prefix, which is affixed to "vila" to get "how many…". There are separate classes for men, women and animals, children, canoes (not mentioned), and objects.

([Woman]₃ + [Brain])^[Eye] + t = 1¹

"Woman 3 with intelligence sees with time something."

→ "That clever woman will see something." (Sentence 7.)

Kimbundu

([Question]^∞ / [Language]₉) = n (a^b – t = c) + n (? = a^b – t = c) + n (a¹ + 0t)

"Question-transubstantiation of language 9 is n (a b's c), n (which a b's c), and n (which n of a b's c)."

→ "Language 9's puzzle is a series of sentences—either declarative transitive in the past, interrogative transitive in the past, or existential present."

([Sound]y)_-1 / ([Language]^-1)_n = (([Sound]y)¹ / ([Language]^-1)_n+1)

"Sound height^[21] -1 of part of language n is sound height 1 of part of language n+1."

→ "The last tone of a word is equal to the first tone of the next word."

In Kimbundu, the last tone of any word is entirely dependent on the first tone of the next word. If it happens to be the last word, it's high tone for questions, low tone for statements.

[Person]₁^[Eye] – t = (n [Man] = [In] + [Square])

"Person 1 sees without time n men in square."

→ "I saw the men in the square^[22]." (Sentence 7.)

Lakhota

([Question]^∞ / [Language]₁₀) = n (< a + a + > a)

"Question-transubstantiation of language 10 is n (less a and a and more a)."

→ "Language 10's puzzle is several words of varying magnitude."

[Sound]^-1 / (< a) = [Sound]^-1 / (> a) – z

"Part of sound of less a is part of sound of more a without depth."

→ "The phoneme of lesser magnitudes are in front of the phoneme of greater magnitudes."

The 2004 Lakhota dataset presents sets of words similar in meaning but differing in magnitude. The words differ only in their fricatives, which are articulated further back in the mouth the greater the magnitude of the word.

{1 = ([Straight line] + [Curved line] – [Straight line]); 1^{(1 / ([Curved line] log [Straight line])) ^ p}}

"Unit is (straight and curved without straight); unit can be curve becoming straight."

→ "It is curved and can be straightened." (The fifth entry in the table.)

Mongolian

([Question]^∞ / [Language]₁₁) = [Language][Book]

"Question-transubstantiation of language 11 is language book."

→ "Language 11's puzzle is a dictionary."

[Book]^[Language] = (([Heavy] + [Light] - [Heavy]) = – ([Heavy] + [Light] - [Light]))

"Book says heavy and light without heavy is not heavy and light without light."

→ "The book defines light as "the opposite of heavy".

Unfortunately the solution doesn't say much about Mongolian, and I can only really define it with one of its entries.

[Food]⁰ = [Earth]⁰ / [Fruit] + [Person][Food] + [Light]^∞

"Food-material is earth-material of fruit with person-food with light-transubstantiation."

→ "Sugar is a white nutritious rock^[23] of fruit." (Entry 7 in the dictionary; the full sentence reads "White coloured and tasty/sweet nutritious crystal object obtained from various specific beets".)

Mountain Arapesh (Bukiyip)

([Question]^∞ / [Language]₁₂) = nⁿ

"Question-transubstantiation of language 12 is function of numbers."

→ "Language 12's puzzle is counting."

26 [Egg] = (6 × (4^{–1/[Language]}) + 2) [Egg], 26 [Fruit]⁰ [In] = (24 + 2) [Fruit]⁰ [In]

"26 eggs is 6 times 4 not spoken with 2 eggs, 26 fruit-material insides is 24 with 2 fruit-material insides."

→ "26 eggs is written 6 × (implied 4) + 2 eggs, but 26 betel nuts^[24] is written 24 + 2 betel nuts.

Mountain Arapesh has two different counting systems depending on what's being counted. Betel nuts and sago bark are counted in the language's normal base of 6, but eggs and sugar cane are counted in sets of 4 (which are themselves counted in base 6). The word for 4 is occasionally omitted.

13 [Square] / [Tree]⁰ [Out]

"13 squares of tree-material outside"

→ "13 sheets of sago bark" (Entry 7 in the ordered list.)

Muna

([Question]^∞ / [Language]₁₃) = n (a^b = c) + n (a^b)

"Question-transubstantiation of language 13 is n (a b's c) and n (a b's)."

→ "Language 13's puzzle is a series of transitive and intransitive sentences."

[Monkey] ≠ [Monkey]₁ / [Book]^∞

"Monkey is not similar to monkey 1 of book-transubstantiation."

→ "Monkey is not grammatically similar to a story's Monkey."

The problem for Muna makes distinctions between story characters (underlined in the problem text) and common nouns.

([House]/[Person]₁)¹ + t = z⁰ + n

"House of person 1 exists with time depth-material with n."

→ "My house will be far^[25] (from here^[26])." (Sentence 2.)

Nahuatl and Arammba

([Question]^∞ / ([Language]_14.1 + [Language]_14.2)) = n ([Pen][Book] / n)_-1

"Question-transubstantiation of language 14.1 with language 14.2 is n parts of write-reads of n."

→ "The puzzle about languages 14.1 and 14.2 is a series of arithmetic^[27] equations."

2 × 20² + 3 × 20 + (15 + 4) = 4 × 6³ + 2 × 6 + 3.

Nahuatl is written in base 20 with a partial sub-base of 5. Arammba is written in base 6. I just picked an arbitrary number here to showcase both.

1 + 1 = 1 × 2. (Equation 7, which actually happens to be in Arammba, but we index into the problem's title.)

Proto-Chamic, Phan Rang Cham and Tsat

([Question]^∞ / ([Language]_15.1 + [Language]_15.2 + [Language]_15.3)) = [Language]_15.1 log ([Language]_15.2 + [Language]_15.3)

"Question-transubstantiation of language 15.1 with language 15.2 with language 15.3 is language 15.1 becoming language 15.2 with language 15.3."

→ "The puzzle about languages 15.1, 15.2, and 15.3 is the evolution of language 15.1 into languages 15.2 and 15.3."

(([Sound]^-1)_-1 / [Language]_15.1)^[Hand] + t = (([Sound]y) / ([Language])_15.3)

"Part of sound last of language 15.1 creates with time sound height of language 15.3."

→ "The final sound of language 15.1 will create the tone of language 15.3."

Proto-Chamic words underwent tonogenesis at some point during evolution into Tsat; which tone it became depended on the final consonant of the word.

5. (By coincidence, the fifth entry in the table.)

Sulka

([Question]^∞ / [Language]₁₆) = nⁿ

"Question-transubstantiation of language 16 is function of numbers."

→ "Language 16's puzzle is counting."

16 [Palm tree][Fruit] = (4 × (4 [Palm tree][Fruit]))

"16 palm-tree fruit is 4 times 4 palm-tree fruit."

→ "16 coconuts is 4 foursomes of coconuts."

Sulka counts coconuts in sets of four, among other groupings.

2 [Fruit]⁰ [In]

"2 fruit-material inside"

→ "2 betel nuts" (The third entry in an unenumerated list.)

Tzotzil

([Question]^∞ / [Language]₁₇) = n (a^b = c) + n ([Question] = a^b = c) + …

"Question-transubstantiation of language 17 is n (a b's c) and n (does a b c?) and …"

→ "Language 17's puzzle is a series of sentences, among which are transitive statements and questions."

[Language]^-1 / – t = [Language]^-1 / + t

"Part of language of without-time is part of language of with-time."

→ "The past tense morpheme is the same as the future tense morpheme."

Tzotzil uses the same particle to denote past and future tense, and requires time words to disambiguate.

([Woman]_J)¹ – ³√(xyz)

"Woman J^[28] exists without location."

→ "Juana is nowhere." (Sentence 7.)

Udihe

([Question]^∞ / [Language]₁₈) = n (a^b = c) + n ([Question] = a^b = c) + …

"Question-transubstantiation of language 18 is n (something of person n)."

→ "Language 18's puzzle is a series of possessive phrases."

[Stool leg] / ([Stool] / [Person]₂) ≠ ([Stool leg] / [Stool]) / [Person]₂

"Stool leg of (stool of person 2) is not similar to (stool leg of stool) of person 2."

→ "'Your_sg stool's leg' is not grammatically similar to 'your_sg stool leg'."

Udihe distinguishes alienability. The problem also can nest possession two layers deep.

([Shoulder] / [Person]₂)

"Shoulder of person 2."

→ "Yoursg shoulder." (The second phrase in an unnumbered list.)

Umbu-Ungu

([Question]^∞ / [Language]₁₉) = n (n)

"Question-transubstantiation of language 19 is n (n)."

→ "Language 19's puzzle is a bunch of numbers."

76 = 24 × 2 + 28.

Umbu-Ungu counts in base 24 with a sub-base of 4. However, if the number is up to 11 larger than a multiple of 24, it counts one less multiple of 24 and instead adds 28 or 32 as appropriate for the number.

35. (The seventh table entry total in the puzzle, though it is the first entry in its table.)

Authors' Notes