## KingGeorge 3 years ago [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.

1. abb50

$\sqrt {3 \times 3} \times 3$

2. abb50

$3^0 \times 3 \times 3$

3. Cortegu10

$\sqrt{3!3!3!}$

4. KingGeorge

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

5. Cortegu10

oh damn i tried :)

6. UnkleRhaukus

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

7. KingGeorge

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

8. KingGeorge

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

9. experimentX

$\sqrt{3!3!} + 3$

10. KingGeorge

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

11. experimentX

$3^{\frac{3!} 3}$

12. KingGeorge

Excellent once again!

13. experimentX

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

14. apoorvk

greatt!!!! sin3 works!!

15. experimentX

perhaps log 3 too :D

16. KingGeorge

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

17. apoorvk

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

18. apoorvk

hmm.. the plus sign..

19. 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.

20. experimentX

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

21. experimentX

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

22. KingGeorge

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

23. lgbasallote

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

24. lgbasallote

i mean $\sqrt{3!3!} + 3$

25. lgbasallote

you cant use plus

26. KingGeorge

Should've caught that :/

27. KingGeorge

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

28. experimentX

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

29. Ishaan94

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

30. experimentX

$(3*3)^{\lfloor \sqrt 3\rfloor }$

31. experimentX

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

32. KingGeorge

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

33. KingGeorge

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

34. experimentX
35. experimentX

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

36. anonymoustwo44

e^(3ln(3))/3

37. KingGeorge

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

38. experimentX

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

39. KingGeorge

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

40. experimentX

is e allowed??

41. KingGeorge

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

42. KingGeorge

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

43. experimentX

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

44. anonymoustwo44

33 (mod 3)

45. KingGeorge

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

46. KingGeorge

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

47. UnkleRhaukus

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

48. experimentX

Nice idea

49. 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}$$