Sunday, October 12, 2014

Solution rewrite

"You're a card player?"

"Yes. I mean, I play online."

"So if you take a deck of cards that are shuffled into a particular order, you never know what order the deck of cards is in unless you look. By simple combinatorial analysis, a deck of cards has 52 factorial[i] distinct orderings. That's such a big number[ii], it might exceed the number of atoms in the universe[iii].

"Imagine a deck that is in initial order, inside a sealed box. Let's call that deck number one. We label some hydrogen atom deep inside the Sun with the number 1 to keep track of it. Then we swap the top card, the ace of spaces, with the next card, the deuce of spades.

"We'll call that deck number two, and likewise we label another atom somewhere inside the Sun as number 2 and keep track of that deck there. Keep in mind the hydrogen atoms inside the sun are not really atoms because the Sun is made of plasma[iv], but you get the idea."

"I follow."

"Now, keep rearranging the deck and numbering them as I've described in my example until you've run out of atoms in the Sun. Then keep doing that for each star in the Milky Way galaxy. You'd still have plenty of decks to arrange and there are billions of galaxies, but let's stop there."

"Uh huh."

"Imagine you hold a single deck of cards in a single arrangement such that you could point to a specific atom of the Sun and say, 'This is deck number whatever and it corresponds to this atom.' Now imagine that you peel off several of the top cards and see what order the first few cards are arranged."


"Now, of the remaining order of cards you could say that all of the other atoms, or nuclei, really, in the Sun around and next to your atom are all the remaining possibilities of decks that have the same starting arrangement as yours."

"I don't follow."

"It's like the street address of a building and the floor number and office number. The first few cards determin which galaxy in the universe to choose. The next few cards determine which star to look in. The rest of the deck could point to any single nucleus in the plasma soup inside the Sun.

"Imagine each star in the Milky Way being named after a unique group of a deck of cards whose first few cards are in the same order. Now the Sun has billions and billions of decks mapped to each atom, or nuclius, and your deck is just the one deck of cards you've shuffled and hold in your hand."

"Yeah," said Mark grinning at the stupefying image.

"So this unique deck that you've shuffled and hold in your hand is as unique deck among all shuffled decks as the Sun is unique from all other stars in the Milky Way, indeed as unique as any atom in our universe."

"I like it."

"But I'll repeat again that this deck of cards is set in a particular sequence. It doesn't change. It is not 52 factorial number of decks simultaneously held in one hand, like a universe of atoms in your palm. It is only one deck and no other deck. This particular deck is set in one unique ordering and it is not a trillion trillion trillion decks at once.

"If I look inside the deck to identify it, as I peel off each card and show you which one is next, the deck is revealed card by card. The deck order does not change in any way, it is fixed in advance. You don't know what order it's in, but that doesn't mean it could be all possible orders at the same time. Do you see the difference?"

"Sort of," said Mark. "I really like the idea of the galaxy in my palm, like the galaxy on Orion the cat's belt in Men in Black."

Samantha rolled her eyes. "I don't even know what you're talking about," she said. "I assume it's the Will Ferrel movie."

"No, Will Smith."

"Oh," said Sam.

"No, but I get it," said Mark. "There are no multiverses. Only one universe with the present state. The superposition is possibility but only because we haven't observed all the information available yet."

[i] "In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,

"5! = 5 * 4 * 3 * 2 * 1 = 120.

"The value of 0! is 1, according to the convention for an empty product.

"The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic occurrence is the fact that there are n! ways to arrange n distinct objects into a sequence (i.e., permutations of the set of objects)."
[ii] 50! ~ 3.041409320×1064
[iii] "Assuming the mass of ordinary matter is about 1.45×1053 kg ... and assuming all atoms are hydrogen atoms (which in reality make up about 74% of all atoms in our galaxy by mass, see Abundance of the chemical elements), calculating the estimated total number of atoms in the universe is straightforward. Divide the mass of ordinary matter by the mass of a hydrogen atom (1.45×1053 kg divided by 1.67×10−27 kg). The result is approximately 1080 hydrogen atoms."

(Samantha is off by perhaps 20% with her statement.)
[iv] "Plasma ..., according to natural science, is one of the four fundamental states of matter (the others being solid, liquid, and gas). ... Plasma comprises the major state of matter of the Sun."

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