I was offered a problem in which you were given four 9's like this:

9 9 9 9

And by adding mathematical functions (+, -, /, *, floor/ceiling, log, ln, trig, factorial) you could make it sum to the numbers 1-100.

Which made me think: Given a positive integer \(n\), how many integers can you achieve from using some sequence of these mathematical functions -- can you get all of them? Only positive integers? I have a feeling that you can get at least every positive integer, but I don't have the evidence to prove it.

What are your thoughts?

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## Comments

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TopNewestSo, what exactly are we allowed to use? Just one \(n\), or \(n\) 9's, or something else? Also, do we have to use all the digits we are given?

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Well the introductory problem was just stating motivation for the question. I mean for any positive integer \(n\), greater than \(1\) I guess (since factorial is the only way to grow). You're allowed to use I guess the functions I stated (since I'm sure there are some crazy functions out there that can easily allow you to get any number).

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Define \(\spadesuit(n) = n+1\). Then, \(\spadesuit(1) = 2\), \(\spadesuit(\spadesuit(1)) = 3\), \(\spadesuit(\spadesuit(\spadesuit(1))) = 4\), etc.

So, you don't really need a "crazy" function in order to make every positive integer. This is why providing a list of well defined functions is necessary for this to be a well posed question.

Regardless, what digits and how many of them are we allowed to use? Without any restrictions, I can "make" any number by starting with that number (no functions needed).

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And \(n\) is an arbitrary integer, not every integer. I think that should be pretty obvious, to be frank.

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My question was what digits are we allowed to use. The example you have has four 9's. Is that what we are starting with here?

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