What is the greatest possible value of a^b^c^d?

Question

taken from a practice for an admission test:

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

What am I doing wrong?

Answer 1

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.