@Alex #38

> So Puzzle 12 would be \\(\lfloor\frac{n}{2}\rfloor\\) and Puzzle 13 would be \\(\lfloor\frac{n+1}{2}\rfloor\\). Am I crazy?

I get a different form of the answer to Puzzle 13: \\(g(n) = \lceil \frac{n}{2} \rceil\\). But \\( \lfloor \frac{n+1}{2} \rfloor\\) is the same as \\(\lceil \frac{n}{2} \rceil \\): if n is even, we get \\(\frac{n}{2}\\); if n is odd, we get \\(\frac{(n+1)}{2}\\). So I think we agree (obviously still assuming \\(f, g : \mathbb{N} \to \mathbb{N} \\)).

Is there a name for this kind of "gauge"-like transformation between functions? Our forms are different but in the given domain and range, they behave identically.

> So Puzzle 12 would be \\(\lfloor\frac{n}{2}\rfloor\\) and Puzzle 13 would be \\(\lfloor\frac{n+1}{2}\rfloor\\). Am I crazy?

I get a different form of the answer to Puzzle 13: \\(g(n) = \lceil \frac{n}{2} \rceil\\). But \\( \lfloor \frac{n+1}{2} \rfloor\\) is the same as \\(\lceil \frac{n}{2} \rceil \\): if n is even, we get \\(\frac{n}{2}\\); if n is odd, we get \\(\frac{(n+1)}{2}\\). So I think we agree (obviously still assuming \\(f, g : \mathbb{N} \to \mathbb{N} \\)).

Is there a name for this kind of "gauge"-like transformation between functions? Our forms are different but in the given domain and range, they behave identically.