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taken from a practice for an admission test: enter image description here

the most obvious answer is $3^4^2^1=6561$, but apparently, there isn't.

What am I doing wrong?

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    $\begingroup$ It seems, from the parenthetical part of the question, that they intend to ask about $((a^b)^c)^d$, even though the usual interpretation of their formula would be $a^{(b^{(c^d)})}$. $\endgroup$ Jan 26 at 23:58
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    $\begingroup$ @AndreasBlass Well, that's the first problem with the question. The second is that $((a^b)^c)^d$ is maximized by $((3^4)^2)^1 = 6561$ (we can put $4,2,1$ in any order), which is still not one of the options. $\endgroup$ Jan 26 at 23:59
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    $\begingroup$ What institution is this practice admission test for? $\endgroup$
    – user21820
    Jan 27 at 8:03
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    $\begingroup$ @user21820 actually it's a pretest for admission to the Cyberchallenge programming test. (cyberchallenge.it/training) $\endgroup$ Jan 27 at 14:41
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    $\begingroup$ Oh ok thanks. At least it is not a university. $\endgroup$
    – user21820
    Jan 27 at 17:45

1 Answer 1

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You're not doing anything wrong; the question is incorrect.

If a^b^c^d is interpreted in the usual way as $a^{(b^{(c^d)})}$, the maximum possible value is $2^{3^{4^1}} = 2417851639229258349412352$.

If it is interpreted as $((a^b)^c)^d$, then it simplifies to $a^{b\cdot c\cdot d}$, which makes the problem easier because we only have to check the four cases for what $a$ is. The maximum possible value is the one you found, with $a=3$, giving $3^{12} = 6561$.

Out of the four given answers, only the first one is even possible to get: under the first interpretation, $3^{4^{1^2}} = 81$. None of the answers are possible under the second interpretation.

My best guess is that they solved the problem under the first interpretation, but accidentally took $a,b,c,d$ to be a permutation of $1,2,3,2$ instead. In that case, $81 = 3^{2^{2^1}}$, $256 = 2^{2^{3^1}}$, and $512 = 2^{3^{2^1}}$ are possible results (but $729$ is not), and $512$ is the largest possible out of any permutation.

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