# What is the greatest possible value of a^b^c^d? [closed]

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?

• 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 $a^{(b^{(c^d)})}$. Jan 26 at 23:58
• @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. Jan 26 at 23:59
• What institution is this practice admission test for? Jan 27 at 8:03
• @user21820 actually it's a pretest for admission to the Cyberchallenge programming test. (cyberchallenge.it/training) Jan 27 at 14:41
• Oh ok thanks. At least it is not a university. Jan 27 at 17:45

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.