Three-Three-Three

[BigBeachBanana]

Use mathematical operations to make the following equations true. The operations are not limited to parenthesis, factorial, exponential, trig functions, etc...
3 3 3 = 0
3 3 3 = 1
3 3 3 = 2
3 3 3 = 3
3 3 3 = 4
3 3 3 = 5
3 3 3 = 6
3 3 3 = 7
3 3 3 = 8
3 3 3 = 9
3 3 3 =10

 

[Harry_the_cat]

Spoiler

3 x (3 - 3) = 0
3 ^ (3 - 3) = 1
(3 + 3) / 3 = 2
3 x 3 / 3 = 3
3 + (3 / 3) = 4
3! / 3 + 3 = 5
3 x 3 - 3 = 6
3! + 3 / 3 = 7
\(\displaystyle \lfloor {\sqrt{33}} \rfloor + 3 = 8\)
3 + 3 + 3 = 9
\(\displaystyle \lceil 3! * {\log{33} } \rceil = 10\)

 

[BigBeachBanana]

Spoiler

3 x (3 - 3) = 0
3 ^ (3 - 3) = 1
(3 + 3) / 3 = 2
3 x 3 / 3 = 3
3 + (3 / 3) = 4
3! / 3 + 3 = 5
3 x 3 - 3 = 6
3! + 3 / 3 = 7
\(\displaystyle \lfloor {\sqrt{33}} \rfloor + 3 = 8\)
3 + 3 + 3 = 9
\(\displaystyle \lceil 3! * {\log{33} } \rceil = 10\)

Well done. Haven't seen the floor and ceiling function used before.

 

[Otis]

Spoiler

\(\displaystyle \int_{3}^{3}3 \; \text{dx}\) = 0

[imath]\;[/imath]

 

[lookagain]

Spoiler

\(\displaystyle \int_{3}^{3}3 \; \text{dx}\) = 0

[imath]\;[/imath]

The 3s should be able to remain in their original horizontal placement for this type
of puzzle.

 

[lookagain]

Spoiler

3 x (3 - 3) = 0
3 ^ (3 - 3) = 1
(3 + 3) / 3 = 2
3 x 3 / 3 = 3
3 + (3 / 3) = 4
3! / 3 + 3 = 5
3 x 3 - 3 = 6
3! + 3 / 3 = 7
\(\displaystyle \lfloor {\sqrt{33}} \rfloor + 3 = 8\)
3 + 3 + 3 = 9
\(\displaystyle \lceil 3! * {\log{33} } \rceil = 10\)

Spoiler

3! - 3 - 3 = 0
(3*(3 - 3))! = 1 \(\displaystyle \ \ or \ \ \) 3!/(3 + 3) = 1
3 - 3/3 = 2 \(\displaystyle \ \ or \ \ \) 3 - (3 - 3)! = 2
3 + 3 - 3 = 3 \(\displaystyle \ \ or \ \ \) 3^((3 - 3)!) = 3 \(\displaystyle \ \ or \ \ \) 3*3 - 3! = 3
(3! + 3!)/3 = 4 \(\displaystyle \ \ or \ \ \) 3! - 3!/3 = 4
3! - 3/3 = 5 \(\displaystyle \ \ or \ \ \) 3! - (3 - 3)! = 5
3*3!/3 = 6 \(\displaystyle \ \ or \ \ \) 3! + 3 - 3 = 6
3! + (3 - 3)! = 7
(3!/3)^3 = 8 \(\displaystyle \ \ or \ \ \)3! + 3!/3 = 8
3^(3!/3) = 9 \(\displaystyle \ \ or \ \ 3! + \sqrt{3*3} \ \) = 9
\(\displaystyle 3/ \sqrt{.3*.3 \ } \ \) = 10

 

[Cubist]

Spoiler: alternative method for 10

lcm(fib(3)+3, fib(3)) = 10

where fib(n) returns the nth Fibonacci number. Wolfram (click) allows it anyway

 

[Dr.Peterson]

Spoiler

3*(3-3) = 0
3!/(3+3) = 1
3-3/3 = 2
3+3-3 = 3
3+3/3 = 4
3!-3/3 = 5
3*3-3 = 6
3!+3-3 = 7
(3!/3)^3 = 8
3+3+3 = 9
(3!-3)/.3 =10 or [imath]3.\bar{3}\times 3=10[/imath]

 

[BigBeachBanana]

Spoiler: alternative method for 10

lcm(fib(3)+3, fib(3)) = 10

where fib(n) returns the nth Fibonacci number. Wolfram (click) allows it anyway

The challenge for 10 is the most interesting one. Here are some other solutions that I've seen:

Spoiler: Solutions for 10

[math]\sqrt{\frac{\sqrt{3\cdot3}}{3\%}}\\[/math][math]\log_{\sqrt{\sqrt{3}}}3+3!\\ 3\cdot3+\left(\frac{d}{dx}3\right)![/math]

 

[Cubist]

The challenge for 10 is the most interesting one. Here are some other solutions that I've seen:

Spoiler: Solutions for 10

[math]\sqrt{\frac{\sqrt{3\cdot3}}{3\%}}\\[/math][math]\log_{\sqrt{\sqrt{3}}}3+3!\\ 3\cdot3+\left(\frac{d}{dx}3\right)![/math]

I particularly like the last one ?

 

[lookagain]

The challenge for 10 is the most interesting one. Here are some other solutions that I've seen:

Spoiler: Solutions for 10

[math]\sqrt{\frac{\sqrt{3\cdot3}}{3\%}}\\[/math][math]\log_{\sqrt{\sqrt{3}}}3+3!\\ 3\cdot3+\left(\frac{d}{dx}3\right)![/math]

The first and the second one do not count, for the same reason as I told Otis, because it changes the puzzle. The idea is to keep the digits all on the same level and same size. That means you don't make any a subscript, a superscript, or a numerator or denominator by a horizontal fraction bar, for instance.

If you want to be free to do all that stuff as in the 1st and 2nd offerings, it will have to be a different puzzle/different thread with different/more open rules.

Sent on my cell phone.

 

[Dr.Peterson]

The poser of a puzzle gets to set the terms, and the OP is pretty clear that this is meant to be wide open.

I myself do prefer to narrow the rules enough to make it interesting; my experience is that allowing floor and ceiling trivializes it, so I personally consider that to be cheating (but am not going to criticize anyone chooses to do so).

Ultimately, the way this one was stated, we need to decide what counts as an "operation". I consider the use of decimals as potentially illegal, and avoided that until I saw no other way; and the use of a vinculum in a decimal is truly questionable, which is why I didn't post an answer until I had an alternative to that -- but I did post it, because I like it! I question whether the derivative should count, because it operates on a function, not on a number. On the other hand, it can be fun to see how creative people can get!

And if we can't have fun in a puzzle, why do it at all?

 

[lookagain]

The poser of a puzzle gets to set the terms, and the OP is pretty clear that this is meant to be wide open.

The poser of the problem is just passing along an already established puzzle that
has 0 through 9 for the goals, but it has been meddled with by adding in a goal of
10 that does not belong in the original puzzle.
This has the set format as I already explained, else you would just announce you
have three digits/numbers (here there are three 3s), and you would get to place
them in any orientation with any "operations."

Ultimately, the way this one was stated, we need to decide what counts as an "operation". I consider the use of decimals as potentially illegal, and avoided that until I saw no other way; and the use of a vinculum in a decimal is truly questionable, which is why I didn't post an answer until I had an alternative to that -- but I did post it, because I like it! I question whether the derivative should count, because it operates on a function, not on a number.

And if we can't have fun in a puzzle, why do it at all?

If it were a good/challenging puzzle, it would have stopped at the goal of 9. Then,
it could have excluded the decimal points. I also would have not allowed the use
of the derivative. However, 3 is a constant function, so its derivative is 0.

And I'll throw back to you, if we can't have some consistent guidelines and not too
open rules for these numerical puzzles, why do them at all!?

 

[Cubist]

I know why a semi-disagreement has begun

333 = 666 / 2

 

[Otis]

The 3s should be able to remain in their original horizontal placement

Very well.

Spoiler

int(3,x,x=3..3)

[imath]\;[/imath]