[SOLVED] Let's see some creativity!
Without using "+" make the number 9 using only three 3's, and no other digits using any mathematical symbol you want. So "3+3+3=9" and similar expressions are off limits.
Here are a couple of the examples I've found so far: \[\frac{3^3}{3}\]\[\sqrt{3^3\cdot3}\]I know of several more possibilities (not including various possible applications of negation). Which ones can you get?
PS: \(-(-\sqrt{3^3\cdot3})\), \(-(-3-3-3)\) and similar don't count.

- KingGeorge

- schrodinger

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- anonymous

\[\sqrt {3 \times 3} \times 3\]

- anonymous

\[3^0 \times 3 \times 3\]

- anonymous

\[\sqrt{3!3!3!}\]

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## More answers

- KingGeorge

@abb50 That has a 0 in it. Let me correct the problem to specify against that.
@Cortegu10 \(\sqrt{3!3!3!}=\sqrt{6^3}=6\sqrt{6}\neq9\)

- anonymous

oh damn i tried :)

- UnkleRhaukus

this probably dosent count as there are four 3's
best i can do \[3 \log_3(3)^3=9\]

- KingGeorge

Too many 3's there :(
Nice try though. I know of at least 3 more expressions using increasingly convoluted nested functions.

- KingGeorge

Hint (for a couple): Keep thinking with factorials and exponents. And remember, "-" isn't completely ruled out. Just don't abuse it.

- experimentX

\[ \sqrt{3!3!} + 3\]

- KingGeorge

Excellent. I hadn't thought that one.
Also, feel free to abuse the floor and ceiling function.

- experimentX

\[ 3^{\frac{3!} 3}\]

- KingGeorge

Excellent once again!

- experimentX

LOL .. not sure if it works
\[ \left \lfloor {3*3 + \sin 3} \right \rfloor \]
\[ \left \lceil {3*3 + \cos 3} \right \rceil \]

- apoorvk

greatt!!!! sin3 works!!

- experimentX

perhaps log 3 too :D

- KingGeorge

If you could do those without the "+" sign, those would be accepted. I'm pretty sure you can get rid of it however.

- apoorvk

sin3 is something between 0 and 1, so it will. cos3 unfortunately is negative.

- apoorvk

hmm.. the plus sign..

- KingGeorge

Just to be clear,
(Partial) list of lesser known functions I will accept:
\[\lfloor9.5=9\rfloor\]\[\lceil8.5=9\rceil\]\[^33=3^{3^3}\]
Also, I will accept \(\ln\) for \(\log_e\) and \(\log\) for \(\log_{10}\) as allowable functions.

- experimentX

Ah great this works
ceil 9^(-cos(3))

- experimentX

\[ \lceil (3*3)^{-\cos 3} \rceil \]

- KingGeorge

This is probably the most convoluted solution I've come up with \[\left\lceil \sqrt[3]{\left(\left(\lfloor\sqrt3\rfloor3\right)!\right)!}\right\rceil\]

- lgbasallote

@experimentX \[\sqrt{3!3!} \times 3\]..youarent allowed to use + lol

- lgbasallote

i mean \[\sqrt{3!3!} + 3\]

- lgbasallote

you cant use plus

- KingGeorge

Should've caught that :/

- KingGeorge

I have at least 4 more solutions no one has posted so far =D

- experimentX

\[ \left\lceil \sqrt[3]{\left( \sqrt{\left(3!3!\right)}\right)!}\right\rceil\]

- anonymous

\[y=\left(\frac{x^3}{3}\right)\]dy/dx at x=3 lol but it's identical to (3^3)/3. Not sure if it counts.

- experimentX

\[ (3*3)^{\lfloor \sqrt 3\rfloor }\]

- experimentX

i bet sqrt 3 can be replaced with ln
log using ceil ..

- KingGeorge

Probably, but let's try and get things that look new, and not just replacing one part with another.

- KingGeorge

I still have 3 more solutions that look different from any posted above.

- experimentX

http://www.wolframalpha.com/input/?i=3%5Eceil%28log%5B10%2C+3%5E3%5D%29

- experimentX

lol ... this ceil function is so useful
http://www.wolframalpha.com/input/?i=3%5Eceil%28+sqrt%283*ln+3%29%5D%29

- anonymous

e^(3ln(3))/3

- KingGeorge

How about \[\large 3^{\lceil\sqrt3\rceil\cdot\lfloor\sqrt3\rfloor}\]

- experimentX

lol ... i think we should ban usage of ceil
http://www.wolframalpha.com/input/?i=3*+ceil%28log+%283*3%29%29

- KingGeorge

Alright. Let's ban the ceiling function for now. What else have we got?
btw, I still have 2 more different solutions

- experimentX

is e allowed??

- KingGeorge

Let's restrict it so we don't have \(e\), \(\pi\), \(\phi\), or other constants like that for now.

- KingGeorge

Also, let's stay out of integrals and derivatives for now as well. Maybe I'll do this again with those allowed.

- experimentX

this seem to have interesting result
http://www.wolframalpha.com/input/?i=floor%28ln%28%283*3%29%21%29+-+3%29

- anonymous

33 (mod 3)

- KingGeorge

\(33\equiv0\pmod3\), although you have a case in that \(9\equiv0\pmod3\) as well.

- KingGeorge

I've got to go to bed now. Keep posting solutions, and I'll post the ones I have left tomorrow.

- UnkleRhaukus

\[\frac{3\times3!}{\Gamma(3)}=9\]

- experimentX

Nice idea

- KingGeorge

Here are the other ideas I've had that look different (mostly) from previous answers\[3^{3!}-(3!)!\]\[\lfloor \log(^33))\rfloor-3\]Recall that \(^33=3^{3^3}=3^{27}\)

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