Etymology
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complex (adj.)

1650s, "composed of interconnected parts, formed by a combination of simple things or elements," from French complexe "complicated, complex, intricate" (17c.), from Latin complexus "surrounding, encompassing," past participle of complecti "to encircle, embrace," in transferred use, "to hold fast, master, comprehend," from com "with, together" (see com-) + plectere "to weave, braid, twine, entwine," from PIE *plek-to-, suffixed form of root *plek- "to plait."

The meaning "involved, intricate, complicated, not easily analyzed" is first recorded 1715. Complex sentence, for one containing one or more subordinate clauses in addition to the principal clause, is attested from 1776.

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implex (adj.)
"intricate, complicated," 1710, from Latin implexus "interwoven, entwined," past participle of implectere, from assimilated form of in- "in" (from PIE root *en "in") + plectere "to plait, twine, braid" (from suffixed form of PIE root *plek- "to plait"). Used by 18c. critics in reference to plots.
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complication (n.)

early 15c., "complex combination or intricate intermingling," from Latin complicationem (nominative complicatio), noun of action from past participle stem of complicare "to fold together, fold up, roll up," from com "with, together" (see com-) + plicare "to fold, weave" (from PIE root *plek- "to plait").

From 1690s as "an additional disorder which develops during the course of an existing one," hence, generally, "that which renders (an existing situation) complex, involved, or intricate."

Complication commonly implies entanglement resulting either in difficulty of comprehension or in embarrassment; complexity, the multiplicity and not easily recognized relation of parts; as business complications; the complexity of a machine; the complexity of a question of duty. [Century Dictionary]
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perplex (v.)

1590s, "embarrass, puzzle, bewilder, fill (someone) with uncertainty," evidently a back-formation from perplexed, a variant of the adjective perplex (late 14c.), "perplexed, puzzled, bewildered," from Latin perplexus "involved, confused, intricate;" but Latin had no corresponding verb *perplectere. The Latin compound would be per "through" (from PIE root *per- (1) "forward," hence "through") + plexus "entangled," past participle of plectere "to twine, braid, fold" (from suffixed form of PIE root *plek- "to plait").

The form of the English adjective began to shift to perplexed by late 15c., probably to conform to other past-participle adjectives, and the adjective perplex became obsolete from 17c. The verb is the latest attested of the group. The sense of "make intricate, involve, entangle, make difficult to be understood" is from 1610s. Related: Perplexing, which well describes the history of the word; perplexingly.

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imbroglio (n.)

1750, "a jumble;" 1818 as "complicated misunderstanding, intricate entanglement" (of persons, nations, etc.), from Italian imbroglio, from imbrogliare "confuse, tangle," from assimilated form of in- "into, in, on, upon" (from PIE root *en "in") + brogliare "embroil," probably from French brouiller "confuse" (see broil (v.2); also compare embroil).

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inexplicable (adj.)

early 15c., from Latin inexplicabilis "that cannot be unfolded or disentangled, very intricate," figuratively, "inexplicable," from in- "not, opposite of" (see in- (1)) + explicabilis "that may be explained" (see explicable).

As a noun, 1745, "something that cannot be explained;" jocular inexplicables "trousers" is from 1829. Related: Inexplicably; inexplicability.

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fast and loose 
described as "a cheating game played with a stick and a belt or string, so arranged that a spectator would think he could make the latter fast by placing a stick through its intricate folds, whereas the operator could detach it at once." [James O. Halliwell, "Dictionary of Archaic and Provincial Words," 1847]. The figurative sense (1550s) is recorded earlier than the literal (1570s).
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perplexity (n.)

mid-14c., perplexite, "bewilderment, doubt, uncertainty," from Old French perplexite "confusion, perplexity," from Late Latin perplexitatem (nominative perplexitas), from Latin perplexus "confused, involved, interwoven," from per- "completely" + plexus "entangled," past participle of plectere "to twine" (from suffixed form of PIE root *plek- "to plait"). From 1590s as "something that causes perplexity, an intricate or involved state or confusion."

The intellect of man is forced to choose
perfection of the life, or of the work,
And if it take the second must refuse
A heavenly mansion, raging in the dark.
When all that story's finished, what's the news?
In luck or out the toil has left its mark:
That old perplexity an empty purse,
Or the day's vanity, the night's remorse. 

[William Butler Yeats, "The Choice," 1933]
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Minerva 

in ancient Roman mythology, one of the three chief divinities (with Jupiter and Juno), a virgin goddess of arts, crafts, and sciences; wisdom, sense, and reflection (later identified with Greek Athene), late 14c., Mynerfe, minerve, from Latin Minerva, from Old Latin Menerva, from Proto-Italic *menes-wo- "intelligent, understanding," from PIE root *men- (1) "to think, remember," with derivatives referring to qualities of mind or states of thought (see mind (n.)). Compare Sanskrit Manasvini, name of the mother of the Moon, manasvin "full of mind or sense." Related: Minerval.

Minerva Press, a printing-press formerly in Leadenhall Street, London; also a class of ultra-sentimental novels, remarkable for their intricate plots, published from about 1790 to 1810 at this press, and other productions of similar character. [Century Dictionary]
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labyrinth (n.)
c. 1400, laberynthe (late 14c. in Latinate form laborintus) "labyrinth, maze, great building with many corridors and turns," figuratively "bewildering arguments," from Latin labyrinthus, from Greek labyrinthos "maze, large building with intricate passages," especially the structure built by Daedelus to hold the Minotaur, near Knossos in Crete, a word of unknown origin.

Apparently from a pre-Greek language; traditionally connected to Lydian labrys "double-edged axe," symbol of royal power, which fits with the theory that the original labyrinth was the royal Minoan palace on Crete. It thus would mean "palace of the double-axe." But Beekes finds this "speculative" and compares laura "narrow street, narrow passage, alley, quarter," also identified as a pre-Greek word. Used in English for "maze" early 15c., and in figurative sense of "confusing state of affairs" (1540s). As the name of a structure of the inner ear, the essential organ of hearing, from 1690s.
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