Why is the Rubik's cube so hard? (episode 2)

Last time we looked at how many possible positions/combinations a 3x3x3 Rubik's cube could be twisted. We found out that the answer is more than 43·10¹⁸. We also found that 43·10¹⁸ is quite large.

Many people asked about the 7x7x7 Rubik's cube. They wanted to know how many more positions are in it. Well, let's compare the two.

First we'll start with a 3x3x3 Rubik's cube. Last time we used a sheet of paper to represent a Rubik's cube. This time we'll go smaller.

A standard Rubik's cube is 57 millimetres on each side. Currently the smallest 3x3x3 in the world is the Elemental cube, it's 8 millimetres on each side (at the present time there even smaller cubes). But we will need smaller for our calculations.

Let's imagine we have a machine that can shrink a Rubik's cube down to the size of a carbon atom. It can be shown what 43·10¹⁸ carbon-atom sized Rubik's cubes would look like. Using high school chemistry we can calculate the mass of 43·10¹⁸ carbon atoms. Remember, even though 43·10¹⁸ is a big number, atoms are small.

43·10¹⁸ carbon atoms weights in an only 0.85 milligrams. That's about 1/7000^th of a Elemental cube, or some shavings off of a pencil.

Now let's look at a 7x7x7 Rubik's cube. Let's do the same thing, only this time we'll use any atom, not just carbon. The 7x7x7 cube has more than 19·10¹⁵⁹ possible positions/combinations. So, what does 19·10¹⁵⁹ atoms look like?

Well, it's more atoms than you would find in a pencil. It's more atoms than you would find in your house. More atoms than you would find in the tallest building in the world. More atoms than all the atoms in all the oceans of the Earth. More atoms than the Earth and the Moon combined. More atoms than our solar system. More than our galaxy. Even more atoms than all the atoms in the visible universe!!

If I were to stop here, I would be doing you a disservice. If I stop now, I would have failed at making an accurate comparison. In fact, to say there are more possible positions in a 7x7x7 cube than atoms in our universe would be a vast understatement. You wouldn't need 2 or 3 universes full of atoms, not even 1 000 or a million would be enough. You would need 10⁸⁰ universes full of atoms. That's a universe for each atom in this universe. Remember that in all of these universes full of atoms, there is only 1 atom in 1 universe that represents a solved 7x7x7 cube. It's your job to find that atom.

So... What does this tell us? This tells us that the difficulty of a puzzle is not determined solely on the number of possible combinations. If it were so, the 7x7x7 Rubik's cube would be impossible to solve.

Taken and edited from Kenneth Brandon's video.

Used math for the 3x3x3:
4.3·10¹⁹ carbon atoms = 0.85 mg (see at 1:15 in the video)

Used math for the 7x7x7:
there are 1.95·10¹⁶⁰ combinations

According to www.universetoday.com/36302/atoms-in-the-universe there are between 10⁷⁸ and 10⁸² atoms in the universe. 10⁸⁰ was used for the sake of simplicity (because it worked well with the calculations).

So if you had 10⁸⁰ universes full of 10⁸⁰ atoms each, that would give you a total of 10¹⁶⁰ atoms. That is roughly the number of combinations in a 7x7x7 Rubik's cube.