## Monday, June 30, 2008

### Mega Cubes

Finally!!! After 3 years of waiting since Mr. Verdes announced that he found a way to mass produce them, the 6x6x6 and 7x7x7 cubes are out for sale! You may not share in my excitement, but to make the 7x was quite a feat of engineering.

In order to show the solution complexity of these cubes, take a look at the number of permutations (the number of possible ways to mix-up the cube, and therefore the number of possible solutions). Making some assumptions about the symmetricality of the pieces of the puzzle, I'm pretty confident that I correctly calculate the number of possible permutations as:
8! x 37 x 24!4 x 48!3 x 12! x 210 = 2.14 x 10237
4!24 x 8!6

To put the size of this number into perspective, it's generally accepted that there are 'only' an estimated 4 x 1081 atoms in the entire universe. That's more than a tripleplex order of magnitude. Wow. Can you feel the excitement? Or, in another fantastically incomprehensible comparison between two ridiculously large numbers, our big number is within 1000 times of A SHANNON NUMBER of Shannon Numbers. (1 x 10240). Ha. Chess has got nothing on this big momma of cubes. Bear in mind that the original 3x3x3 Rubik's cube (that you nevertheless still peeled the stickers off of), has only a puny 4.3 x 1019 possible permutations. Pshaw. A THREE YEAR OLD can solve that one, and teenagers can solve it in LESS THAN 10 SECONDS. That's no competition for the 7x7x7.

And because I just know you're still with me, and so eager for more gratuitous quantifying, I'll go on.

So, there are two hundred fourteen septseptuagintillion possible ways to the mix up the 7x7x7. This is written out as:
214,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

And the whole point is to find the one correct solution.

Incidentally, if anyone feels so inclined to chip in, I would really, really like one of each, but my wife would kill me if I spent \$80 on another cube. But, if you'd like something in return other than a daily dose of punishing complaints on your nth favorite blog, I'd be willing to trade: would you be interested in my drawing a PORTRAIT of you or your kid in exchange for a cube? That's a pretty good deal for a rubik's cube... (Does that sound desperate enough?)