What verb should be used with "puzzles"? We "raise" them, "cast" them, ... or we simply "say" them?

What is the right verb for when puzzles are shared with another person? Do we "raise" puzzles? Do we "give" puzzles? Do we "cast" puzzles? ... or do we simply "say" puzzles?

What, for example, can be put in the following blank space?

In this reality show, they ____ puzzles and participants try to solve them.

I want the puzzle to be in the sentence.

The interesting thing is that the main dictionaries on the web Cambridge Marriam Webster Oxford do not give proper examples to solve this puzzle!

The word that first came into my mind when reading your question was pose, defined by the online M-W as (among other things):

a : to set forth or offer for attention or consideration let me pose a question

I would say that we pose puzzles just like we pose questions. However, it depends on context. For example, I would say that

The mysterious event presented the police with a puzzle.

However, if we're referring to one person actively showing another something as a puzzle to be solved, then I would use pose:

I posed a puzzle for my friends to solve.

And if this is somehow obligatory, as in your example of a game show host, if it is a task that must be done, I would use set:

The host sets puzzles for the contestants to solve.

So, for me, set is for cases where the puzzle is somehow work to do. Not really for describing something which is puzzling (where present would be more common) nor for the action of telling a friend a riddle to solve (where pose is more common).

This Google NGram seems to support this interpretation (for what it's worth, this is far from conclusive) with presents a puzzle being more common than poses a puzzle (the latter is likely mostly used for active cases like my second example above) but pose a puzzle being far more common than present a puzzle (which has no hits at all) and set/sets a puzzle being by far the rarest.

The dubious evidence above suggests that set a puzzle is far less common than either present or pose. In the specific example you cite, that of a game show, set does indeed work and sounds perfectly natural, but I feel that setting puzzles is not as common outside that specific context. Perhaps because set carries a connotation of homework. So I would pose a puzzle for my friend to solve, but I would set a puzzle for a game contestant (who has to solve it in order to proceed) and I would write that a murder presented the police with a puzzle.

The word present is frequently used with puzzles

A few ideas:

• perhaps "present puzzles", or
• "present puzzling questions to" (although "puzzling question" is less optimal without additional context as a "puzzling question" frequently means an ill-formed question, but perhaps less so if words like "solve" are used along with it (see below)),
• "present puzzle challenges"

or maybe

• "challenge participants with puzzles to solve" (challenge could be an alternative to present if used this way.

Usually you would couple "present" with "to" or "with":

• "present puzzles to participants for them to solve" or

• "present participants with puzzles"

We set puzzles.

In this reality show, they set puzzles and participants try to solve them.

This refers strictly to the puzzles. The puzzles are set. Set here has the meaning of 'being constructed or devised', some work has gone in to preparing the puzzle. This is correct using the exact form of the OP's test sentence.

That the participants are expected to solve those puzzles is spelt out explicitly. We do not 'set puzzles to the contestants', though they might be given or presented to them. But that would not use the form of words in the OP.

Amongst the many subtly different meanings of the word 'set' in the OED are 'to prepare', 'to set a trap', and 'to set down on paper', most or all of which are appropriate for a puzzle.