Science problems thread: 2/8/2013 19:39:16 |
Moros
Level 50
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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.
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Science problems thread: 2/8/2013 19:41:02 |
Moros
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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)
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Science problems thread: 2/9/2013 11:14:27 |
nich
Level 60
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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.
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Science problems thread: 2/9/2013 13:21:48 |
Darth Mylor {Warlighter}
Level 13
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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?
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Science problems thread: 2/9/2013 15:29:37 |
Moros
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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.
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Science problems thread: 2/9/2013 16:07:52 |
RvW
Level 54
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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...!?
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Science problems thread: 2/9/2013 16:10:25 |
professor dead piggy
Level 59
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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.
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Science problems thread: 2/9/2013 16:32:30 |
Seahawks
Level 54
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if x is even then you live, if x is odd then you die
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Science problems thread: 2/9/2013 17:26:50 |
Moros
Level 50
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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?
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Science problems thread: 2/9/2013 18:06:32 |
RvW
Level 54
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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
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Science problems thread: 2/9/2013 19:20:26 |
Moros
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Nope, I honestly have no idea, but I expect the number of combinations to be around a billion.
I think you could start with the amount of possible perfect tenths on the piano's 88 keys. There are 52 white keys, so that's 42 possible perfect tenths. The number of diminished tenths (an interval of 9,5 notes, e.g. F to G#(with an octave in between)) 60. But you could play a perfect and a diminished tenth at the same time (if you had flexible fingers), as long as they fall into each other.
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Science problems thread: 2/9/2013 21:25:09 |
Art Vandelay
Level 54
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Yang-Mills and Mass Gap, solve
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Science problems thread: 2/9/2013 23:14:49 |
Moros
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Huh?
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Science problems thread: 2/9/2013 23:55:44 |
RvW
Level 54
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Come again? When I search for "Yang-Mills" I get this: http://en.wikipedia.org/wiki/Yang-Mills which does not seem overly related (maybe I'm stupid, but what does quantum chromodynamics have to do with either music or counting problems...!?). For "Mass gap" I get http://en.wikipedia.org/wiki/Mass_gap and have the same question.
On a (presumably related??) note, I might as well yell: "SAT problem; let a solver construct the BDD and count the number of possibilities", but I'm old-fashioned and figure puzzles should be solved with understanding, not throwing overpowered tools at them and getting an answer which is correct, but which you don't understand.
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Science problems thread: 2/10/2013 02:24:24 |
Art Vandelay
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The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the "mass gap:" the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap and will require the introduction of fundamental new ideas both in physics and in mathematics.
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Science problems thread: 2/10/2013 02:24:41 |
Art Vandelay
Level 54
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if you solve that you are one of the smartest humans in history
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Science problems thread: 2/10/2013 04:23:04 |
nich
Level 60
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You should also post your solution here before taking any other action.
Since I didn't get to post a problem before and in order to get us back on track here's a fairly straightforward one:
A man owns 7 different coloured pairs of gloves, 7 different coloured pairs of socks to match his gloves, and 7 different coloured ties to match these. He keeps them all in the same drawer without the gloves and socks paired up. One morning he wakes up hungover and late for an important meeting, in his confused and panicked state he is unable to distinguish between the three types of garment and starts grabbing randomly. How many items must he grab before he is assured of a matching set of gloves socks and tie?
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Science problems thread: 2/10/2013 05:02:30 |
Hennns
Level 58
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assuming he`s unlucky and picks in order, he can pick 7 pairs of socks and 7 pairs of gloves without having 3 pairs of everything, the next pick which then is a tie will give him a complete set. So he`ll have to pick 7x2+7x2+1 =29 iteams before he`s guaranteed a set.
so the answer is 29.
(Note if a pair of socks/gloves= 1 item the answer is 7+7+1=15.
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Science problems thread: 2/10/2013 05:04:58 |
nich
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Splendid work, you are correct.
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Science problems thread: 2/10/2013 14:54:56 |
Hennns
Level 58
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Okey, my turn to post a problem :D
I`m a 60kg astronaut,who have gone outside my spaceship to fix the cable which is used to secure people on the outside of the spaceship. Unfortunately do I lose control and start slipping away from the spaceship, when I`m 50m away I realize I`ve to do something, but the only thing I\ve is my 5kg toolbox :(
What must I do to get back to the spaceship, and will my 20min left of Air be enough to survive back to the spaceship?
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