Why is the Rubik's cube so hard? (episode 1)

Have you ever wondered why a Rubik's cube is so hard to solve? The reason the Rubik's cube is so hard to solve is because a Rubik's cube can be twisted into many different positions, yet there is only one position that is solved.

So, how many positions can one twist a 3x3x3 Rubik's cube? It turns out the answer is more than 43·10¹⁸. That sounds big, but just how big is it?

Well, here's one way we can think about it. Let's imagine I wanted to see every position of a Rubik's cube. So what I did was I drew each position on a separate piece of paper. Then I stacked all 43·10¹⁸ papers. Let's see how high the stack would be.

So I started on the kitchen table. It didn't take long before I needed a ladder. Soon, the house wasn't big enough. I had to take it outside. The ladder wouldn't take me high enough so I found a crane. Pretty soon, even that wasn't high enough. With the help of a hot-air balloon I continued to stack. Now my stack was higher than the tallest building in the world.

I really began to enjoy this project. Soon it took me where no one else had ever been. Now I was further than the Moon. I really got to know our solar system. There's the Sun, Mercury, Venus, Earth, Mars, the asteroid belt, Jupiter, Saturn, Uranus, Neptune and Pluto. Pluto's no longer a planet, but let's just assume I didn't get the memo.

I continued the stack and took myself on a grand tour of our solar system. I went straight toward Pluto. Once at Pluto, I realized I was low on fuel and food, and should probably start heading back. So I started a stack on Pluto back toward Earth. The trip back seemed to go by quickly, and eventually I returned to Earth.

It was depressing to know I wasn't even half way done. After fuelling up the tank, it was time for another trip. Resolved to finish this epic quest, I set out on multiple trips. I finally finished all 43·10¹⁸ positions. It only took 728 stacks.

Hidden out in the mass of papers is one paper that is special. Each paper has a unique picture of a Rubik's cube, but only one paper has the picture of a solved Rubik's cube. It's your job to find that paper. Good luck!

Taken and edited from Kenneth Brandon's video.

Used math:
paper thickness: 0.01 cm
total stack size: 4.3·10¹⁵ m = 4 300 000 000 000 km
1 AU: 149 598 000 km
distance to Pluto: 5 906 376 272 km = 39 500 AU
length of stack in AU: 28 743.7
number of stacks: 727.69

The distance from Pluto to Sun was used, because over the course of a couple years, that would be the average distance from the Earth to Pluto as well.

It's 728 stacks where the last stack isn't a full stack.