Let me now blow your mind: all of these usages of “check” (including “Exchequer”) come from exactly the same root. Moreover, the root is the game of chess. Yes, the game is chess is primary, and verification, obstruction, the Treasury, and negotiable financial instruments all get their names from a metaphor derived from the game of chess.

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First, the game of chess itself: the king, in Persian, is shāh, and shāh māt means “the King is dead”. In Russian, the game of chess is itself called shákhmaty, and “check” is said shakh. This became scaccus in Latin, and from there you get échec in French, chess in English, and lots of other forms in different languages. [EDIT: Forgive my carelessness with pronouns. By “This" in the previous sentence, I mean of course the Persian, not the Russian.] (Note that échec is also the French word for “failure” — and this also comes directly from the chess concept.)

The English treasury in early Norman times was arranged so innumerate people could settle their accounts with the king. The calculations were done by moving counters on a big table with a checkerboard-style tablecloth. If you want all the details, you can read the Dialogue of the Exchequer — or Dialogus de Scaccario in Latin, written around the 1170s or 1180s by Anglo-Norman bureaucrat Richard FitzNeal. So the concept of “Exchequer” derives very directly from chess, not much metaphor there.

For the other usages, it’s a matter of looking through the various definitions in the Oxford English Dictionary. As the OED says: “From its use in chess the word has been widely transferred in French and English. In the sense-extension the noun and verb have acted and reacted on each other, so that it is difficult to trace and exhibit the order in which special senses arose.”

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From the chess sense, we quickly get “to come into collision with, strike, hit”; the OED has a couple of 16th- and 17th-century usages. Chaucer used “check” in the sense of “to arrest, stop, or retard the onward motion or course of (a person or thing)” in The House of Fame, written around 1379-80: “When they metten in that place, They wer a-cheked bothe two.” Also, the verb was used in the 15th century in the sense of “to challenge (a sentinel).”

There are also the hunting and hawking senses of “to stop short” or “to recoil”, as well as the financial sense of “to stop (a person) from receiving a part of his wages as a fine or penalty” (attested from the 16th to the 19th centuries).

Once we’ve gotten this far, it’s not far at all to the figurative sense of “to stop (action, growth, exhibition of feeling, and the like); to stay the course of; to repress, restrain” (see Shakespeare’s Sonnet 15 (“Men as plants increase, Cheared and checkt euen by the selfe-same skie.”), or Milton’s Paradise Lost (“Half his strength he put not forth, but check’d His Thunder in mid Volie.”).), or “to curb or control”. That’s enough to get us the political sense of “checks and balances”.

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But what about the financial sense of negotiable instruments? Another meaning, also found from the 17th century onward, is “to control (a statement, account, etc.) by some method of comparison; to compare one account, observation, entry, etc., with another, or with certified data, with the object of ensuring accuracy and authenticity.” I pay you by a note drawn on my bank, you take the check to your bank and the bank takes it to my bank (or you can just skip the middleman and go directly to my bank), and then the bank verifies that it’s O.K. by comparing the note with the stub that it retained. And from there, “to check up on” or “to check on” something, meaning “to examine carefully or in detail, to ascertain the truth about.” So the “verification” sense is derivative of the financial sense. And the check-mark is also the mark that results from such verification, though that usage seems to be more recent: the OED doesn’t have it earlier than the 20th century.