How precisely are we defining "same orientation"? Within 1 degree? Within 1 minute of arc? Exactly?


0.00000001.

1. Assuming a 3-axis representation, and a 6-state die, the chance that the side with three dots would appear in the same orientation is 1/576.
2. 10^-9.

Those puzzles were too easy XD

How many permutations are on a 3x3x3 Rubik's Cube?
Give me math, not an answer :P I know you just looked it up on Google

EDIT: Damn. I didn't see that page. Never mind.

same as above. I have a riddle, but I didn;t anser the last one, so if you want to see it, select the tezt from the I to the "?" at the end.
If you know this, don't answer.
Imagine a hotel with an infinite number of rooms, numbered 1, 2, 3, and so on forever. All the rooms are full.

A man comes in and asks the desk manager whether any rooms are available, and the desk manager replies, “All our rooms are full, but I’d be happy to accommodate you.” How does he do it?

and

An infinite amount of people check in. How does the manager accommodate THEM?
Although... (8! 12! 3⁸ 2¹²)/(2*3*2)=43 252 003 274 489 856 000
Wolfram Alpha rules!

I've done this before, but let's spoil it.
What he does is he moves Guest 1 to Room 2, Guest 2 to Room 3, Guest 3 to Room 4, etc. until all of the guests are moved. He then checks in the guy into Room 1.

For the next problem, he moves each person to the room double the previous.
Guest 1 goes to Room 2. Guest 2 goes to Room 4. Guest 3 goes to Room 6. etc.
Since there are an infinite amount of odd numbers, all the guests go into those rooms.

My problem now.
Prove that 1+1=1. (Hint: 1 may not be 1)

I'll do this the Calvin and Hobbes way:
assume a planet a
assume a planet b
the planets are still all molten lava.
planet a collides with planet b
the resulting planet is but one planet.
1planet+1planet=1planet|÷planet
1+1=1

I don't know if this counts.

That works, although I came up with:
let 1 be a variable
1=0
0+0=0

Anyway, puzzle solved! Here's another one:

For the function f(x), find the tangent line at (a, f(a)).

y = f'(a)*x - f(a)

puzzle, gws?

<troll>Derive the Cubic Formula</troll>

<actualpuzzle>Derive the Quadratic Formula</actualpuzzle>

Start with y = ax² + bx + c and show me the steps to turn it into x = (-b (+/-)√(b²-4ac))/2a
If you've already done this, leave it for someone else : )

Nobody's going to do it, since everybody on this website has done it one time or another.
...
Okay, I'll bring out Math Input Panel.

puzzle, beary?

What's √2?
(Easy? Maybe.)

Aleph_something. You didn't put a dx in there.

answer is here

You are one of ten people to be beamed aboard an alien vessel. Each one of you is responsible for one tenth of the human race. If you die, they die and their land gets transfered over to the aliens. You will be put to a test in 24 hours. Until then, you can talk it over and make a plan to have the greatest possible percentage of the human race alive afterwards.
THE TEST:
you will all be led to a dark room, and there each of you will have either a green or purple hat put on his head. There can be all green hats, all purple hats, or any combination of the two. You will stand in a line, such that each one of you can clearly see all the hats in front of him, and hear anything anyone behind him says. Then the lights will be turned on. From now on the only words you can say are "purple" and "green". Each one of you, starting from the back, will be asked what color his hat is. If he guesses correctly, he will survive. Otherwise, he dies.

This was actually one of Omni's PotW's a while back (except it was an evil king and 10 dwarves). I'll refrain from answering.

Whoever's at the front of the line should listen for the people behind them to say what color their hat is. The question didn't say nobody else can talk during the questions, so it's only the last guy in line with a 50% chance.

EDIT: And if people who pass the test stay in the room, they can tell the last guy what color his hat is.

This seems too easy.

EDIT EDIT: That didn't look like a sqrt(2); it looked like integral(2) with no bounds or differential. Hence my answer. Oops.

Oops. I guess I should give you credit, but I did forget to add an important part to the question: you can only speak once. Also, you are right in the second part of the statement. As you didn't post up a new puzzle yet, I guess someone could try and answer mine with this little bug fixed.

You cheat.

Can someone who already answered, point at someone else who already answered to indicate 'THE SAME COLOR HAT AS THIS GUY OVER HERE'? That'll take care of the last person in line, and ensure the complete survival of the human race. : p

You could, but then Earth would be destroyed.

No no, just pointing. Not saying.

you can't see the people behind you.

Revival, I guess.

Three different coins, called 1, 2, and 3, are placed on a table in an arbitrary order.

The rules of this game are: you can only stack lower-numbered coins onto higher-numbered coins, and you can only move coins over by one space (for example, an order 3 1 2, you cannot move the 2 onto the 3.)
You could also think of this as a version of the Tower of Hanoi.

Find the minimum number of moves required to move these following orders into the sequence 1 2 3, and if you cannot get to 1 2 3 using these rules, answer with "Cannot do".
3 2 1
2 1 3
1 3 2

(For all participants: Yes, this is the 5^th CCC problem.)