c.1300, gessen "to estimate, appraise," originally "take aim," probably from Scandinavian (cf. Middle Danish gitse, getze "to guess," Old Norse geta "guess, get"), possibly influenced by Middle Dutch gessen, Middle Low German gissen "to guess," all from Proto-Germanic *getiskanan "to get" (see get). Sense evolution is from "to get," to "to take aim at," to "to estimate." Meaning "to hit upon the right answer" is from 1540s. U.S. sense of "calculate, recon" is true to the oldest English meaning. Spelling with gu- is late 16c., sometimes attributed to Caxton and his early experience as a printer in Bruges. Related: Guessed; guessing. Guessing game attested from 1650s.
c.1300, from guess (v.). Verbal shrug phrase your guess is as good as mine attested from 1902.