Jamison Lahman said:
That is why they use an example with two die. The sum of the two die is not of equal probability. There is one way to make 12 (6+6) but there are six ways to make a sum of 7 (6+1, 5+2...). In that case, 6+1 is the microstate and the sum of 7 is the macrostate. Since there 6 ways make a sum of 7, that means its multiplicity is 6.
My point was that WE decided what counted as important macrostates. We could have found something else significant, eg doubles (1+1, 2+2 etc), in which case we'd get different results.
I think the Rubik's cube is a flawed example because of the differences between combinations versus permutations. This was not my strong suit and I am tired, so I'm not sure I should dare go further! :-)
The problem is it is hard to equate anything physical to entropy. The configuration of the rubik's cube is a matter of statistics but 'solving' it correctly would require mechanical energy.
This is what I mean about apples and oranges. If we're allowed to count ordered states of the Rubik's cube as low entropy as far as the 2nd law is concerned, then we're back to the problem I started with.
Think about it this way. Consider having multiple Rubik's cubes and consider them in unison. Regardless the number, they still have only a single ordered state, but each time you add a cube, the number of states sky-rockets. However, the amount of energy required to solve these cubes only goes up linearly. Solving 2 cubes only takes twice as long, and twice the energy, but the number of states has squared. As we add cubes, the energy required is quickly drawfed by the amount of order if measured in this way. But it only makes sense to talk about the energy used if we also measure order in terms of useful energy.
Although now that I've made that argument I'm not sure why it doesn't apply to any method of measuring order
Our "most-special" state is the one with the lowest multiplicity and is defined by statistics, not us.
But we decide how to measure order and what states count as significant. Again, I think this works if we have defined order in terms of useful energy, but it won't work if we allow order to be measured in any way.
This does mean though that a solved puzzle would have the same entropy as an unsolved puzzle, as far as the 2nd law is concerned, because they all have the same amount of useful energy.