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Previous puzzle

Take this puzzle of mine that I created around a week ago:

Take these 3 functions: $f(x):=x+8,g(x):=x^2-3,h(x):=\sqrt x$

Starting from $x=0$,$$\color{black}{\text{How many times will you need to apply }f,g,\text{ and }h\text{ to get }78}?$$or is the puzzle simply impossible?

My attempt:

$$\begin{align}g(0)=-3\quad&(1\text{ function})\\g(g(0))=6\quad&(2\text{ functions})\\g(g(g(0)))=33\quad&(3\text{ functions})\\f(g(g(g(0))))=41\quad&(4\text{ functions})\\f(f(g(g(g(0)))))=49\quad&(5\text{ functions})\\h(f(f(g(g(g(0))))))=7\quad&(6\text{ functions})\\g(h(f(f(g(g(g(0)))))))=46\quad&(7\text{ functions})\\f(f(f(f(g(h(f(f(g(g(g(0)))))))))))=78\quad&(11\text{ functions})\end{align}\\\therefore11\text{ functions are needed to solve this puzzle}$$

However, my question is

Is this the most optimized solution there is, or can you find a more efficient solution?

Hint:

The title hints at a way of finding the most efficient solution. (if I have hinted at it correctly)

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1 Answer 1

2
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It can be done in

6

$g(g(h(g(h(g(0)))))) \\ = g(g(h(g(h(-3))))) \\ = g(g(h(g(\sqrt 3 i)))) \\ = g(g(h(-6))) \\ = g(g(\sqrt 6 i)) = g(-9) = 78$

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  • $\begingroup$ Oh my, good job! My most optimized solution was with 9 functions but I did not think of the complex realm! good job! $\endgroup$
    – CrSb0001
    Commented Nov 29, 2023 at 1:54
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    $\begingroup$ Actually, there is a 7 that doesn't use complex numbers: gfgghfg and another one that doesn't even use negatives: gfghhff $\endgroup$ Commented Nov 29, 2023 at 2:01
  • $\begingroup$ Notably, since every h is directly inside a g, the choice of which complex square root you use is irrelevant. $\endgroup$ Commented Dec 1, 2023 at 12:50

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