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The word conundrum "sounds" very Latin (or at least, it does not sound English enough to me). Yet, it seems its origin is unclear. Wiktionary states:

A word of unknown origin with several variants, gaining popularity for its burlesque imitation of scholastic Latin, as "hocus-pocus" or "panjandrum". If there is more to its origin than a nonce coinage, Anatoly Liberman suggests the best theory is that connecting it with the Conimbricenses, 16C scholastic commentaries on Aristotle by the Jesuits of Coimbra which indulge heavily in arguments relying on multiple significations of words.

In effect, this blog, sponsored by the Oxford University Press, proposes seven alternative origins. Three are directly related to Latin (quoting in extenso from above):

  1. Since conundrum means “pun” and presupposes an imaginary or fanciful agreement between some two objects, the etymon may be Greek koinon duoin (Latin commune duorum); substitute Latin duorum for Greek duoin, and you will get a good approximation of conundrum.

  2. Perhaps conundrum is a modified and disguised form of Latin conventum “agreement.” For v the letter u often turns up in books. Conuentum could have been misunderstood and mispronounced as conundrum.

  3. One of the citations in the OED runs as follows: “These conimbrums, whether Reall or Nominall, went down with Erasmus like chopt hay.” (1651; the first citation of conundrum goes back to 1586.) Here is the etymology, published in The Nation 57, 1893, No. 1481, p. 370 and signed by the initials C.S.P.: “There surely can be no doubt what this word [that is, conimbrums] is. The reference to realists and nominalists shows that something in the scholastic philosophy is referred to; and ‘conimbrum’ is easily recognized as meaning argumentum Conimbrienum. The doctors of Coimbra, in their celebrated commentaries published in the sixteenth century, have in all cases a great deal to say of the ‘multiplex significatio’ of one word and another. Indeed, such remarks are their great weapon. They used it for all it was worth, and a little more. Accordingly, a dealer in verbal quibbles might naturally have been called by Oxford students a Conimbricus, and his quillet Conimbrienum argumentum. The original c, which this hypothesis requires, is preserved in another old form of the word ‘conuncrum’. Conimbrica was in the sixteenth century the most usual Latin form of the name Coimbra, though Conimbria is also common.

Certainly fascinating stuff. Now, as experienced Latinists, you might be able to give some lights on this issue. For instance, another instance of word transformation going through one of the above channels? Or perhaps evidence that commune duorum is a rare phrase in (mediaeval?) Latin? I am curious on your thoughts on this. Surely a definitive answer will not be given, but any light on the issue would, imo, be a good answer.

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Sadly, I think you are right, and myself doubt very much that any useful insights can be provided. Indeed, I strongly suspect that the writer of your quoted OED citation may well, in frustration, have been cunningly turning the question into a conundrum of his own. No matter how many etymologies may be put forward, I think that we should accept that there's never going to be one that's really satisfactory.

[On a technical ppoint :a conundrum is not a pun precisely, but a riddle or puzzle needing a bit of lateral thinking to solve it. It is often answerable by punning. Punning itself is exemplified by the (good, but untrue) story that General Sir Charles Napier sent the one-word dispatch ‘Peccavi’ after conquering the Indian province of Sind in 1843. ‘Peccavi,’ I have sinned, is spoken just as I have Sind. Any Americans reading may recognize another, in a Presidential electioneering effort from the nineteenth century in 'We Polked You in ’44, We Shall Pierce You in ‘52'. Neither is the answer to a conundrum!]

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