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allay (v.)[allay 词源字典]: Old English alecgan "to put down, remit, give up," a Germanic compound (cognates: Gothic uslagjan, Old High German irleccan, German erlegen), from a- "down, aside" + lecgan "to lay" (see lay).

Early Middle English pronunciations of -y- and -g- were not always distinct, and the word was confused in Middle English with various senses of Romanic-derived alloy and allege, especially the latter in an obsolete sense of "to lighten," from Latin ad- "to" + levis (see lever).
Amid the overlapping of meanings that thus arose, there was developed a perplexing network of uses of allay and allege, that belong entirely to no one of the original vbs., but combine the senses of two or more of them. [OED]
The double -l- is 17c., a mistaken Latinism. Related: Allayed; allaying.[allay etymology, allay origin, 英语词源]
perplex (v.): late 14c. as an adjective, "perplexed, puzzled, bewildered," from Latin perplexus "involved, confused, intricate;" but Latin had no corresponding verb *perplectere. The Latin compound would be per "through" (see per) + plexus "entangled," past participle of plectere "to twine, braid, fold" (see complex (adj.)).

The form of the English adjective shifted to perplexed by late 15c., probably to conform to other past participle adjectives. The verb is latest attested of the group, in 1590s, evidently a back-formation from the adjective. Related: Perplexing, which well describes the history of the word.
puzzle (n.): c. 1600, "state of being puzzled," from puzzle (v.); meaning "perplexing question" is from 1650s; that of "a toy contrived to test one's ingenuity" is from 1814.