You know the little fun facts that Snapple prints on the inside of its bottle caps. Recently I’ve seen some that seem rather incredible, such as “A cubic mile of fog contains less than a gallon of water” and “Any Rubik’s Cube configuration can be solved in 17 moves or less”. It almost makes me wonder if they’ve started just making shit up and waiting to see how long it takes before someone calls them on it. (I could perhaps be convinced about the Rubik’s Cube one, but I just don’t know about the cubic mile of fog deal.)

The latest Rubik’s cube proof puts the maximum number of moves for any cube at 22. No idea where they got 17 from, although obviously there are a large number of configurations that *can* be solved in 17 moves, nobody has proven that all of them can be.

According to this: http://www.rexresearch.com/airwells/airwells.htm

1 cubic mile = 4.16818183 × 10^9 cubic meters

so 1 cubic mile of fog should contain ~210,000,000 - 12,000,000,000 grams of water, or

about 500,000 - 25,000,000 lb.

Water is about 8.35 lb/gallon, so it should be about 60,000 - 3,000,000 gallons of water per square mile of fog.

So assuming I did these calculations correctly, Snapple is only off by a factor of around a million.

Here’s a correction posted by a newspaper who posted the “fact” about fog:

http://legacy.signonsandiego.com/uniontrib/20061026/news_lz1c26eureka.html

If you google “A cubic mile of fog contains less than a gallon of water”, it’s shocking how many sites state this statistic as a fact.

There are 20 parts of the cube that can be out of place; wouldn’t the theoretically lowest possible minimum number of moves be 19?

That line of reasoning would assumes that each piece could be moved to its correct location in a single move, without affecting the position of the other pieces.

Probabably all the same sites that claim “a duck’s quack doesn’t echo.” :rolleyes:

Back of the envelope calculation leads me to believe a gallon of water would make roughly 10 cubic yards of fog. A square mile of fog would have to be less than a millimeter thick to be only a gallon of water.

The line of reasoning is intended to establish a lower bound, not an upper one. If it has a flaw, it is that more than one piece is moved with each move…if you just rotate one side of a solved cube by 90 deg, then you can restore 8 pieces from wrong positions to correct positions with only a single move.

But the real question is … can you get fog off a duck’s back and not have it echo when you do?

Only on a treadmill.

“But it’s a dry fog.”

The standard Rubik’s cube has 4.325E19 reachable configurations. If a “move” is defined as one of the twelve face quarter-turns, then the number of configurations reachable in 17 or fewer moves is at most 12[sup]17[/sup]+12[sup]16[/sup]+…+1=2.42E18 (actually it is fewer, since some of these moves commute), so 17 moves cannot be enough. They may be using a more expansive notion of a single “move”, though.

In discussions of the Rubic’s Cube, I’ve usually seen a 180 degree turn of a face counted as a “move”, too. That’d bring us up to 18[sup]17[/sup] + 18[sup]16[/sup] + …, or over 2E21 moves. However, then there’s the commutations you mention: At the very least, you should never make two moves to the same face in a row, so for your first move you have 18 possibilities, but only 15 for each subsequent move. So you’d actually be looking at 18*15[sup]16[/sup] + 18*15[sup]15[/sup] + …, which gives us around 1.26E20 reachable configurations. Meanwhile, with only 16 moves, you get 8.44E18 configurations.

So given that definition of a “move”, a simple back-of-the-envelope calculation proves that the perfect solution must sometimes take more than 16 moves, but is consistent with 17 moves possibly being enough. From this, it’s reasonable to guess that 17 might be the correct number, and this is probably the source of the Snapple statistic.

I agree with your analysis. (I think – I’m not too great at this stuff.) 17 seems a reasonably likely target, but nobody has yet proven that it’s possible.

The Wikipedia page mentions two configurations which require 26 quarter turns and 20 face turns, though. So 17 is out again unless the definition of “move” is broadened even further.

did you know that 63% of statistics are made up on the spot?

Only 34.5% of the time.