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KingGeorge
[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.
\[\sqrt {3 \times 3} \times 3\]
\[3^0 \times 3 \times 3\]
@abb50 That has a 0 in it. Let me correct the problem to specify against that. @Cortegu10 \(\sqrt{3!3!3!}=\sqrt{6^3}=6\sqrt{6}\neq9\)
this probably dosent count as there are four 3's best i can do \[3 \log_3(3)^3=9\]
Too many 3's there :( Nice try though. I know of at least 3 more expressions using increasingly convoluted nested functions.
Hint (for a couple): Keep thinking with factorials and exponents. And remember, "-" isn't completely ruled out. Just don't abuse it.
\[ \sqrt{3!3!} + 3\]
Excellent. I hadn't thought that one. Also, feel free to abuse the floor and ceiling function.
\[ 3^{\frac{3!} 3}\]
Excellent once again!
LOL .. not sure if it works \[ \left \lfloor {3*3 + \sin 3} \right \rfloor \] \[ \left \lceil {3*3 + \cos 3} \right \rceil \]
greatt!!!! sin3 works!!
perhaps log 3 too :D
If you could do those without the "+" sign, those would be accepted. I'm pretty sure you can get rid of it however.
sin3 is something between 0 and 1, so it will. cos3 unfortunately is negative.
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.
Ah great this works ceil 9^(-cos(3))
\[ \lceil (3*3)^{-\cos 3} \rceil \]
This is probably the most convoluted solution I've come up with \[\left\lceil \sqrt[3]{\left(\left(\lfloor\sqrt3\rfloor3\right)!\right)!}\right\rceil\]
@experimentX \[\sqrt{3!3!} \times 3\]..youarent allowed to use + lol
i mean \[\sqrt{3!3!} + 3\]
Should've caught that :/
I have at least 4 more solutions no one has posted so far =D
\[ \left\lceil \sqrt[3]{\left( \sqrt{\left(3!3!\right)}\right)!}\right\rceil\]
\[y=\left(\frac{x^3}{3}\right)\]dy/dx at x=3 lol but it's identical to (3^3)/3. Not sure if it counts.
\[ (3*3)^{\lfloor \sqrt 3\rfloor }\]
i bet sqrt 3 can be replaced with ln log using ceil ..
Probably, but let's try and get things that look new, and not just replacing one part with another.
I still have 3 more solutions that look different from any posted above.
http://www.wolframalpha.com/input/?i=3%5Eceil%28log%5B10%2C+3%5E3%5D%29
lol ... this ceil function is so useful http://www.wolframalpha.com/input/?i=3%5Eceil%28+sqrt%283*ln+3%29%5D%29
How about \[\large 3^{\lceil\sqrt3\rceil\cdot\lfloor\sqrt3\rfloor}\]
lol ... i think we should ban usage of ceil http://www.wolframalpha.com/input/?i=3*+ceil%28log+%283*3%29%29
Alright. Let's ban the ceiling function for now. What else have we got? btw, I still have 2 more different solutions
Let's restrict it so we don't have \(e\), \(\pi\), \(\phi\), or other constants like that for now.
Also, let's stay out of integrals and derivatives for now as well. Maybe I'll do this again with those allowed.
this seem to have interesting result http://www.wolframalpha.com/input/?i=floor%28ln%28%283*3%29%21%29+-+3%29
\(33\equiv0\pmod3\), although you have a case in that \(9\equiv0\pmod3\) as well.
I've got to go to bed now. Keep posting solutions, and I'll post the ones I have left tomorrow.
\[\frac{3\times3!}{\Gamma(3)}=9\]
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 \(^33=3^{3^3}=3^{27}\)