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There are arithmetic puzzles with some numbers (typically four) and rules for combining them (perhaps just the four basic arithmetic operators or including things like combining digits 2 and 3 to make numbers like 23).

Write twenty-four from four numbers is an example which was recently posed on this site.

What is the general name of this type of puzzle, or is it just an "arithmetic puzzle"?

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I have not heard any special name for them before.

But here, such puzzles are called "Number Formation Puzzles," and if you google, you can find "About 26,300 results", so it is more or less a common name.

Meanwhile, if you google arithmetic puzzles you can find many different types of such a puzzles:

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