User blog:Edwin Shade/Just How Many Combinations Are There In A Rubik's Cube ?

Being inspired by Bubby3's blog post and a video highlighting just how large a number 43,252,003,274,489,856,000 is, or the number of possible scrambles for a 3x3x3 puzzle cube, I decided to make a blog post with the purpose of describing to you just how large these combinatoric numbers are, especially when we get into higher level cubes.

First, there is the 1x1x1, pictured below.



And yes, this is a joke. The 1x1x1 cube has only one possible scramble, which is leaving it alone. If you had one 1x1x1 cube you could easily fit it into your pocket.

Next is where things get interesting. The 2x2x2 cube has 3,674,160 combinations. Assuming each puzzle cube is 1 and 3/4 inches in length, you could stack them almost 101 and a half miles high ! This over 18 times the height of Mt.Everest, and well into the Earth's Mesosphere, as pictured below.



Just staring up at this stack of cubes would no doubt give you vertigo. But what if you grouped them as compactly as possible into a big pile ? Well, your pile would be about 11,300 cubic feet, or about 5 and a half trailers worth of cubes. You would need a medium-sized house to house all those cubes, and even then you would not have floor space left over to put in furniture.

The 3x3x3 has 43,252,003,274,489,856,000 possible scrambles, which if stacked up would stretch from Earth to 261 light years away, over half the distance to the Pleiades starcluster. This stack could also go from Earth to Alpha Centauri 59 times, or from Earth to to the Moon almost 6 and a half billion times ! (Assuming each cube has a side length of 2 and 1/4 inches.)

If you were to group together all these cubes, they would occupy a space equal to about the volume of Uranus, or a cube with a side length of 28,000 miles. Here is the planet Uranus next to Earth, so as to get an idea of how massive this pile would be.



This is only the beginning however. A 4x4x4 cube has 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 possible scrambles, which if we assume each cube to have a side length of 2 and a half inches, a stack of 4x4x4's would not only reach the length of the observable universe, but would also be larger than this diameter by over 17 orders of magnitude !

It is difficult to imagine just how long this would be, so difficult in fact that I will finish the rest of this blog post tomorrow with an analogy prepared for this.

Expect this done with cubes up to size 22x22x22, currently the highest order puzzle cube that has been made in real life.

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