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I came across a type of function/notation on a test that I'd never seen before, and for formatting reasons, I haven't been able to find any answers online. Can anyone help me with what this means/how to solve it? Thanks in advance.

the question:

What is $^{3}3$? Hint: this does not mean $3^3$, nor is it a typo.

Matthew Leingang
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Jasmine Wang
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    This looks like a tetration. If that was on a test I'd hope they explained it in lectures. – Tob Ernack Jan 02 '18 at 17:59
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    In general the answer is “it depends.” Perhaps it’s tetration – Dan Robertson Jan 02 '18 at 17:59
  • It looks weird , a question about notation (which is a convention) in a test... It might be a tetration https://en.wikipedia.org/wiki/Tetration – leonbloy Jan 02 '18 at 18:12
  • If a question need to say that the question is not a typo then it is not a good question... At least in my opinion, and like the other said it is probably Rudy Rucker notation for tetration – ℋolo Jan 02 '18 at 18:27
  • I also think that it is tetration. However, I prefer the notation $$a\uparrow \uparrow b$$ (Knuth's up-arrow-notation) or $$a\rightarrow b\rightarrow 2$$ (Conway's-chained-array-notation) or even {$a,b,2$} (Bowers's array-notation).This operator easily creates numbers surpassing all numbers having any physical meaning. – Peter Jan 02 '18 at 18:28
  • Would be nice to know from where you got this test. – kingW3 Jan 02 '18 at 22:39
  • original poster here: the test was for an event called Fermi Questions in a competition called Science Olympiad: basically involves mental calculations to find the order of magnitude for various problems. There's no limit to what kind of problems we can have, and this was one of them. – Jasmine Wang Jan 05 '18 at 04:17

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The notation $^n a$ is usually used to denoted tetration. Tetration is the next hyperoperation after exponentiation.

So to build up through the various lower level hyperoperations from successor, addition, multiplication, and exponentiation to your particular example of tetration we have: \begin{align*} 3 + 3 &= 3 + \underbrace{(1 + 1 + 1)}_{\text{3 copies of 1 added to 3}} = 6\\ 3 \times 3 &= \underbrace{3 + 3 + 3}_{\text{3 copies of 3 combined by addition}} = 9\\ 3^3 &= \underbrace{3 \times 3 \times 3}_{\text{3 copies of 3 combined by multiplication}} = 27\\ ^3 3 &= \underbrace{3^{3^3}}_{\text{3 copies of 3 combined by exponentiation}} = 7 \, 625 \, 597 \, 484 \, 987 \end{align*}

omegadot
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