question 5: prove 3.333recurring=1/3

*ahem* question 5: prove 3.3recurring=10/3 and I will need to see you LaTeX for an answer to be considered valid.

Well, since 3 fits three times in 10, with a 1 left.
And 0,3 fits three times in 1 as well, with 0,1 left.
And because the laws of maths say that if you divide something the comma (or whatever sign people use to separate decimals and whole numbers) moves one spot to the right, this continues forever. Thus 10/3=3,3333...
Good?

Nope. Prove the "laws of maths". And do it quick before someone just googles the answer. This is actually an interesting fundamentally important tenet of mathematics, and noone ever knows it. It also creates arguments =D

Sorry, I forgot this: http://www.latex-project.org/

http://www.latex-project.org/

Noone wants to see that, you sicko. If you arent going to play the game then I wont give you any more hints.

Haha, found it:

Actually, that link will only lead you to a site describing LaTeX ;) You're being the sick one here.

Haha, you think i didn't click a link called latex project? Great work, you are correct.

Okay, here's a new problem:
What's the number of possible ways you can put notes and rests in a single 4/4 measure of sheet music? Not counting different clefs, and no notes or rests shorter than hemidemisemiquavers. And you can't just put lots of rests in different orders, but right next to each other, because when combined they last the same time. (I mean you can't have two quarter rests next to each other for example, because they would sound the same as a halve rest)

Good luck with that!

And I forgot one thing: because d flat sounds the same as c sharp, they are identical and count as a single combination, despite being notated differently.

That counts for all sharps and flats by the way. And e sharp equals f, f flat equals e, b sharp equals c and c flat equals b, so they are regarded as one.
(sorry for the triple post)

Moros that seems fiendishly difficult, there are some other considerations unspecified as well:
Using just one octave? Can you use triplets? (or other tuplets for that matter) Can you have more than one note at a time?

Still I think I'll probably pass.

hmm i wonder if anyone knows game theory here. I will put this question and see if you can solve it (this is an easy one).

You are stuck in of an island. Escape is impossible, so you start exploring the island. Suddenly, group of x cannibals appear, and they seem hungry. They are all experts at game theory, and will eat each other after they eat you. However, only one of them can eat someone (so 2 cannibals can't eat 1 guy) and eating someone leaves them defensless and can be eaten themselves. How can you predict whether you will live or not?

Mylor, it isn't your turn, you can post that question if you solve one from someone else.
And nich, you can't use tuplets, but you can use all 88 keys of the piano, and you can use up to five of them at a time (because a human has 5 fingers) But the lowest and highest cannot be more than a perfect tenth apart.

Moros, sounds like a not-too-difficult combinatorics problem. But I can't solve it because I lack sufficient background (domain) knowledge. Could you help me "translate" the problem description for people who aren't all that music-savvy?

• Total length of our "composition" should be one full note.
  
• Shortest note we can use is 1/16 (??).
  
• Each note can have 88 different values; you can play at most five of those at the same time (subject to certain restrictions, but I have no idea what a "perfect tenth" is).
  

Questions:

• Do all notes have to start at their "natural" boundaries? For instance, is it allowed to have a 1/16th note followed by a 1/2 note (and then another 7 1/16th notes)? My intuition says this would probably sound horrible, so we might want to disallow it, but from your description so far it seems okay.
  
• Should all notes played at the same time have the same length? Is it okay to start playing a 1/2 note C and in parallel play first a 1/4 note D, then 1/4 note E? How about even weirder combinations: start playing 1/2 note C and 1/4 note D. After that 1/4 note D play a 1/2 note E (so that for the length of 1/4 note we're playing C and E), then after the 1/2 note C ends, play a 1/4 note F (so that we end with the second half of the 1/2 note E and the 1/4 note F, having played a composition of 3/4 notes in length).
  
• Is there anything special about rests? In other words: we're allowed to play at most five notes at a time (if they're "close enough"); is it okay to model a rest as playing zero notes?
  
• You said 1/4 note rest immediately followed by another 1/4 note rest is indistinguishable from 1/2 note rest. Just to be sure, 1/4 note C immediately followed by another 1/4 note C is different from 1/2 note C, right?
  
• My five notes at a time, most people have two hands...!?
  

humans have 5 finger? seriously? I'm going to answer mylors question.

Mylor I think they will probably eat you out of self preservation. How can I predict that? I wish you could phrase the question differently, I can predict anything. Once there are only 2 cannibals not eating left, then once one starts eating they will get eaten 100% of the time. So when there are 3 left the third one is free to chow down knowing the other 2 cant touch him. If x=2 then you live would be an exception. It makes me think i must be wrong because usually logic questions have neat answers with no exceptions and i know nothing about game theory, so I am trying to empathise with cannibals who I have nothing in common with.

if x is even then you live, if x is odd then you die

Shortest note we can use is 1/16 (??).

Nope, a hemidemisemiquaver is an 1/64st note.

but I have no idea what a "perfect tenth" is).

Something I quickly put together.
A perfect tenth is any interval of two note exactly ten notes apart, so there's nine notes between them. And a perfect tenth is about the largest chord a human hand can make.

Do all notes have to start at their "natural" boundaries? For instance, is it allowed to have a 1/16th note followed by a 1/2 note (and then another 7 1/16th notes)? My intuition says this would probably sound horrible, so we might want to disallow it, but from your description so far it seems okay.

They're allowed.

Should all notes played at the same time have the same length? Is it okay to start playing a 1/2 note C and in parallel play first a 1/4 note D, then 1/4 note E? How about even weirder combinations: start playing 1/2 note C and 1/4 note D. After that 1/4 note D play a 1/2 note E (so that for the length of 1/4 note we're playing C and E), then after the 1/2 note C ends, play a 1/4 note F (so that we end with the second half of the 1/2 note E and the 1/4 note F, having played a composition of 3/4 notes in length).

Following the official rules of sheet music, there can be (but don't have to be) up to two "voices", or tracks. So what you're saying is possible, but there can only be notes of two different timings at a time.

Is there anything special about rests? In other words: we're allowed to play at most five notes at a time (if they're "close enough"); is it okay to model a rest as playing zero notes?

Sure, that's what a rest essentially is.

Just to be sure, 1/4 note C immediately followed by another 1/4 note C is different from 1/2 note C, right?

Yup, unless you use ties, but I'll leave them out for now ;)

My five notes at a time, most people have two hands...!?

Completely true, and I understand your confusion. But we use two different bars for two hands. And I'm focussing on a single hand right now, if we wanted to add a second hand we'd just need the final answer squared, right?

This is more difficult than I thought... The half-notes (more specifically, the lack of half notes between B, C and E, F) combined with that perfect-tenth makes counting possible combinations messy.

Nope, two hands does not equal "one hand squared". Since you can only hit each note once at a time any combination where one hand wants to play a half-note C at the same time the other hand wants to play a quarter-note C is disallowed. And if one hand plays a half-note C we cannot distinguish between the other hand playing that same note or a rest of the same length. And then there's two identical notes played at different (but overlapping) times; that's not possible either. (And possibly: do both hands get to use two of their own voices, or do they have to share? If so, that would severely limit the number of possibilities for two hands.)

Do you know the answer, or are you trying to get other people to solve this for you? :p