I was wondering how many possible configurations there are of a Rubik's cube of size greater than $3\times3\times3$ (e.g. for $4\times4\times4$, $5\times5\times5$, $\dots 10\times10\times10)$? We know that for a $3\times3\times3$-cube there are about $4.3 \times 10^{19}$ configurations, what about the bigger cubes?
[Math] How many configurations of a 10x10x10 Rubik’s cube
combinatoricsgroup-theoryrubiks-cube
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Best Answer
I'll give an upper bound.
We are considering a 10 by 10 cube here. A cube has 6 faces. Assuming that the cube has six colors, there should six options for each square on a face. There are a 100 squares per face. That means there are $6^{100}$ permutations per face. Multiplied by the six faces of the cube gives an upper bound of $6^{600}$ possibilities.
In an attempt to lower the bound, assume that after the first face, there are only 5 options left for each remaining square. After forming the permutations on the second face there would only be 4 options, etc. multiplying that out we get, $(720)^{100}$ as an estimate