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On June 13 2014 14:42 Simberto wrote:
Ok, now i am rather sure that no number larger then 6 exists for which this can not be done. Because that would require that number to be dividable by all primes smaller than itself. I can not rigorously proof that no larger number with that property exists, but considering i am a physicist and not a mathematician, the fact that if such a number exists, it is probably so large that you can't reasonably draw a star with that many points anyway is enough for me. 2*3*5*7*11*13*17*19 ~9.7 million, no number below 20 except for 1,2,3,4,6 has the property that we are looking for. So you can definitively draw any star with up to 9.7 million points except for those with 1,2,3,4,6 points.
I'll agree with that conclusion. A rigorous proof that there are no stars that are impossible to draw larger than a 6 pointed star I think would have to call on the theorem that shows that there is always a prime between any integer j and 2j when j>= 2. For an n pointed star, as long as a prime p exists between n and n/2 and p != n-1 then a star exists that meets our criteria. I'm just stuck on how to show that p != n-1 for all cases other than 6, perhaps I'll have to read up more on the prime theorem I referred to, though I've forgotten the name of it.
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On June 14 2014 00:15 Najda wrote:Show nested quote +On June 13 2014 14:42 Simberto wrote:
Ok, now i am rather sure that no number larger then 6 exists for which this can not be done. Because that would require that number to be dividable by all primes smaller than itself. I can not rigorously proof that no larger number with that property exists, but considering i am a physicist and not a mathematician, the fact that if such a number exists, it is probably so large that you can't reasonably draw a star with that many points anyway is enough for me. 2*3*5*7*11*13*17*19 ~9.7 million, no number below 20 except for 1,2,3,4,6 has the property that we are looking for. So you can definitively draw any star with up to 9.7 million points except for those with 1,2,3,4,6 points.
I'll agree with that conclusion. A rigorous proof that there are no stars that are impossible to draw larger than a 6 pointed star I think would have to call on the theorem that shows that there is always a prime between any integer j and 2j when j>= 2. For an n pointed star, as long as a prime p exists between n and n/2 and p != n-1 then a star exists that meets our criteria. I'm just stuck on how to show that p != n-1 for all cases other than 6, perhaps I'll have to read up more on the prime theorem I referred to, though I've forgotten the name of it.
Ok, with that theorem it's incredibly easy to prove. I kinda forgot it exists. There exists a prime between j and 2j. Thus, there exists a prime between p#1*...*p#(j) and p#1*...*p#j*p#(j+1), as p#(j+1) > 2. This means that p#1*...*p#(j+1) does not encompass all primes < it as dividers.
Furthermore, as p#(j+1) is strictly >2 for all j>2, there also exists a prime between p#1*...*p#j and p'1*...*(p2(j+1)-1), which means that (ok fuck this lets define P(x) = p#1*...*p#x), at least one of the primes that is not a divider of P(j+1), but which is < P(j+1) is also smaller than P(j+1)-1, as it(the smaller prime) is smaller than p#1*...*(p#(j+1)-1), which is strictly smaller than P(j+1)-1. This is true for every j>2, which means that 6 is the largest exception.
So conclusion: Happiness, you can draw any n-edged star with n>6 in the way previously described.
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On June 14 2014 01:06 Simberto wrote:Show nested quote +On June 14 2014 00:15 Najda wrote:On June 13 2014 14:42 Simberto wrote:
Ok, now i am rather sure that no number larger then 6 exists for which this can not be done. Because that would require that number to be dividable by all primes smaller than itself. I can not rigorously proof that no larger number with that property exists, but considering i am a physicist and not a mathematician, the fact that if such a number exists, it is probably so large that you can't reasonably draw a star with that many points anyway is enough for me. 2*3*5*7*11*13*17*19 ~9.7 million, no number below 20 except for 1,2,3,4,6 has the property that we are looking for. So you can definitively draw any star with up to 9.7 million points except for those with 1,2,3,4,6 points.
I'll agree with that conclusion. A rigorous proof that there are no stars that are impossible to draw larger than a 6 pointed star I think would have to call on the theorem that shows that there is always a prime between any integer j and 2j when j>= 2. For an n pointed star, as long as a prime p exists between n and n/2 and p != n-1 then a star exists that meets our criteria. I'm just stuck on how to show that p != n-1 for all cases other than 6, perhaps I'll have to read up more on the prime theorem I referred to, though I've forgotten the name of it. Ok, with that theorem it's incredibly easy to prove. I kinda forgot it exists. There exists a prime between j and 2j. Thus, there exists a prime between p#1*...*p#(j) and p#1*...*p#j*p#(j+1), as p#(j+1) > 2. This means that p#1*...*p#(j+1) does not encompass all primes < it as dividers. Furthermore, as p#(j+1) is strictly >2 for all j>2, there also exists a prime between p#1*...*p#j and p'1*...*(p2(j+1)-1), which means that (ok fuck this lets define P(x) = p#1*...*p#x), at least one of the primes that is not a divider of P(j+1), but which is < P(j+1) is also smaller than P(j+1)-1, as it(the smaller prime) is smaller than p#1*...*(p#(j+1)-1), which is strictly smaller than P(j+1)-1. This is true for every j>2, which means that 6 is the largest exception. So conclusion: Happiness, you can draw any n-edged star with n>6 in the way previously described.
This thread is supposed to handle stupid questions. Not math!
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On June 14 2014 01:29 boxerfred wrote:Show nested quote +On June 14 2014 01:06 Simberto wrote:On June 14 2014 00:15 Najda wrote:On June 13 2014 14:42 Simberto wrote:
Ok, now i am rather sure that no number larger then 6 exists for which this can not be done. Because that would require that number to be dividable by all primes smaller than itself. I can not rigorously proof that no larger number with that property exists, but considering i am a physicist and not a mathematician, the fact that if such a number exists, it is probably so large that you can't reasonably draw a star with that many points anyway is enough for me. 2*3*5*7*11*13*17*19 ~9.7 million, no number below 20 except for 1,2,3,4,6 has the property that we are looking for. So you can definitively draw any star with up to 9.7 million points except for those with 1,2,3,4,6 points.
I'll agree with that conclusion. A rigorous proof that there are no stars that are impossible to draw larger than a 6 pointed star I think would have to call on the theorem that shows that there is always a prime between any integer j and 2j when j>= 2. For an n pointed star, as long as a prime p exists between n and n/2 and p != n-1 then a star exists that meets our criteria. I'm just stuck on how to show that p != n-1 for all cases other than 6, perhaps I'll have to read up more on the prime theorem I referred to, though I've forgotten the name of it. Ok, with that theorem it's incredibly easy to prove. I kinda forgot it exists. There exists a prime between j and 2j. Thus, there exists a prime between p#1*...*p#(j) and p#1*...*p#j*p#(j+1), as p#(j+1) > 2. This means that p#1*...*p#(j+1) does not encompass all primes < it as dividers. Furthermore, as p#(j+1) is strictly >2 for all j>2, there also exists a prime between p#1*...*p#j and p'1*...*(p2(j+1)-1), which means that (ok fuck this lets define P(x) = p#1*...*p#x), at least one of the primes that is not a divider of P(j+1), but which is < P(j+1) is also smaller than P(j+1)-1, as it(the smaller prime) is smaller than p#1*...*(p#(j+1)-1), which is strictly smaller than P(j+1)-1. This is true for every j>2, which means that 6 is the largest exception. So conclusion: Happiness, you can draw any n-edged star with n>6 in the way previously described. This thread is supposed to handle stupid questions. Not math!
I think there's plenty people who would call us stupid for trying to define the art of star drawing to this level, but I think it's entertaining. Good work Simberto, you've solved my classroom doodles problem 
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So my parents bought me one of those electric toothbrushes last week. It’s nice and all, but the thing is it has a timer that beeps every 30 seconds it’s on and keeps track of how long you brush, displaying a happy face after 2 minutes on the timer. This timer also links to a weekly calendar online that shows how long I’ve brushed each day. Normally this would be a cool feature, but the problem is that my parents can look at this and they keep reminding me to brush my teeth more because, “I didn’t get the recommended 2 minutes of brushing last night” etc. Well, after a week or so of hearing them complain about it to me, I had a brilliant idea. I can just turn it on and let it run for those 2 minutes and not even use it. Now I can skip brushing my teeth completely and nobody will know the difference!
Along with the toothbrush, I recently got a new car a few months ago. I don’t drive very often but I’ve noticed that the car will often start beeping at me until I click in my seatbelt. As annoying as this is, I’ve found a solution. If I buckle in my seatbelt before I step into the car, I don’t have to wear it and the car knows better than to beep at me! Why don’t more people know about this? In any case thought I’d share this secret with you guys.
What are some other tricks to beating the system that you guys might have?
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in before you forget and leave it on all night
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On June 14 2014 05:51 brian wrote:
in before you forget and leave it on all night
If it's anything like mine, it stops automatically after 2 minutes.
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Did you expect a 5-1 for the Netherlands against Spain?
WHOOOHOOOOO!!!!!!!
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Why don't you just put on the damn seatbelt, Epishade?
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i would recommend both good oral hygiene and road saftey
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On June 13 2014 15:54 Mataza wrote:Show nested quote +On June 13 2014 15:42 Thieving Magpie wrote:On June 13 2014 13:55 Najda wrote:On June 13 2014 13:49 Simberto wrote:
6-sided is possible if you do the pointy outside first, then the hexagon in the middle. 9 Should be possible in some similar way, too. True, but I should have been specific in saying you can only draw lines from point to point. Also each point has to have the same angle. I can draw a lopsided 6 pointed star like this: ![[image loading]](http://i.imgur.com/uFM2s7H.png) but it doesn't meet the angle requirement. Even with those parameters its easy to draw a 6 sided star that ends point to point with each point having the same angles as each other. ![[image loading]](http://i.imgur.com/UgUrBe6.png) Isn't this basically cheating(besides being ugly, how about you copy/paste the star and then add a new line to the copy)? You added 2 inner lines that do not appear as edges of the star and you lack the third inner line to make those rotational symmteric. I do know that achieving the third line is impossible without breaking any (unspoken) rule, like not drawing one line twice.
Well he first asked if it was possible to draw a 6 sided star that looked like his. Two people made two different ways to do it. He then added that points have to reach that the angles of each point should be even--I did it with mine by retracing two lines.
What he really means to say was
"Can you draw these stars exactly like how I prefer drawing them"
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On June 14 2014 07:27 Thieving Magpie wrote:Show nested quote +On June 13 2014 15:54 Mataza wrote:On June 13 2014 15:42 Thieving Magpie wrote:On June 13 2014 13:55 Najda wrote:On June 13 2014 13:49 Simberto wrote:
6-sided is possible if you do the pointy outside first, then the hexagon in the middle. 9 Should be possible in some similar way, too. True, but I should have been specific in saying you can only draw lines from point to point. Also each point has to have the same angle. I can draw a lopsided 6 pointed star like this: but it doesn't meet the angle requirement. Even with those parameters its easy to draw a 6 sided star that ends point to point with each point having the same angles as each other. Isn't this basically cheating(besides being ugly, how about you copy/paste the star and then add a new line to the copy)? You added 2 inner lines that do not appear as edges of the star and you lack the third inner line to make those rotational symmteric. I do know that achieving the third line is impossible without breaking any (unspoken) rule, like not drawing one line twice. Well he first asked if it was possible to draw a 6 sided star that looked like his. Two people made two different ways to do it. He then added that points have to reach that the angles of each point should be even--I did it with mine by retracing two lines. What he really means to say was "Can you draw these stars exactly like how I prefer drawing them"
I couldn't put into words the requirements of the star drawing so I just posted the example instead and decided it would be best to hone the definition through discussion. Your drawing does meet the initial requirements but the requirements changes as the discussion continued.
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What would happen if all fossil fuels on earth were suddenly used up over night?
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Bad things.
Specifically, the economy would break down. And not in that weird "collapse" way that happens every few years and basically means slightly negative growth ratio. I mean really break down. Anything involving Energy or plastics would basically stop being usable.
Next step is probably starvation and lots and lots of revolutions and fighting. Then a lot of people die. At some point things are going to kind of stabilize at a level someplace between the middle ages and the industrial revolution, without the possibility of another industrial revolution due to a lack of coal, but a lot of stuff lying around from before. So my guess is end result is some kind of postapocalyptic medieval society.
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Well, don't forget that nuclear energy would still be available. With such big worldwide demand and no conventional fuel sources whatsoever, mass switch into nuclear power plants would probably be the only option. Obviously it wouldn't happen overnight and there would probably be a bigass economical crysis, but I'm not entirely convinced that we would eventually emerge as far back as in the middle ages.
Intially, it would be horrible, though. There would be no car transportation until fully electric or solar cars were implemented. No airplanes and ships would be operational, which means global production couldn't be outsorced anymore. No fuel means also no food supplies in shops, no employees at their desks, no police or ambulances or firefighters available. At least for a while, the cities would suddenly lose their point of exisitence and I guess there would be a mass migration back to rural areas, where direct food farming for your own demands would happen all over again. Political regimes would probably dissolve into millions of little, autonomic countries, ruled by those who own the local sources of nuclear, geothermal or solar energy. Bicycle gangsters would roam the world, spreading terror... All in all, nice premise for a science-fiction book.
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On June 14 2014 10:51 Najda wrote:Show nested quote +On June 14 2014 07:27 Thieving Magpie wrote:On June 13 2014 15:54 Mataza wrote:On June 13 2014 15:42 Thieving Magpie wrote:On June 13 2014 13:55 Najda wrote:On June 13 2014 13:49 Simberto wrote:
6-sided is possible if you do the pointy outside first, then the hexagon in the middle. 9 Should be possible in some similar way, too. True, but I should have been specific in saying you can only draw lines from point to point. Also each point has to have the same angle. I can draw a lopsided 6 pointed star like this: but it doesn't meet the angle requirement. Even with those parameters its easy to draw a 6 sided star that ends point to point with each point having the same angles as each other. Isn't this basically cheating(besides being ugly, how about you copy/paste the star and then add a new line to the copy)? You added 2 inner lines that do not appear as edges of the star and you lack the third inner line to make those rotational symmteric. I do know that achieving the third line is impossible without breaking any (unspoken) rule, like not drawing one line twice. Well he first asked if it was possible to draw a 6 sided star that looked like his. Two people made two different ways to do it. He then added that points have to reach that the angles of each point should be even--I did it with mine by retracing two lines. What he really means to say was "Can you draw these stars exactly like how I prefer drawing them" I couldn't put into words the requirements of the star drawing so I just posted the example instead and decided it would be best to hone the definition through discussion. Your drawing does meet the initial requirements but the requirements changes as the discussion continued.
Which is where the split happened. You initially talked about how to make stars, but making stars was not the discussion you wanted. The fact that its a star has become irrelevant, the discussion is actually about "assuming X drawing limitations, what is possible for us to draw?"
Which is far less interesting than "How many ways can we draw this star?"
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I don't know, i actually found the second question more interesting, because it actually has some mathematic value to it. Of course you can draw anything without taking the pen off the paper if you can redraw lines and don't have to any other limitations, unless there are jumps in the shape you wish to draw. That is trivial.
And the problem of actually clearly formulating the boundaries you care about is pretty common. In this case, they slowly developed through dialog. And yes, the discussion was about stars, but under certain limitations. Which makes sense, because without limitations the discussion is rather pointless because the answer is trivially obvious.
I guess the question of "How many ways can we draw this star" (which is a different question) is kind of interesting, but once again you need limitations, otherwise it is nonsensical. If you can redraw the same lines as often as you wish, the answer is obviously "There are infinite ways of drawing this star". If you can add random additional lines wherever you want, once again the answer is "There are infinite ways of drawing this, but the result is probably not a star anymore". If you can only trace each line once, and may not redraw them, the question might get interesting.
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I need some help to understand some accountancy stuff. I'm looking at some "consolidated financial information" trying to figure out what these things mean. Montreal's 2013 consolidated statement of financial position reports a net debt of 5.3 billion, an accumulated surplus of 6.1 billion and yearly surpluses over the last few years ranging from 80 to 1200 millions.
What does accumulated surplus mean? Are those assets able to be moved toward paying off the debt or are they non financial somehow? If so why not use the accumulated surplus to pay off the debt and avoid paying interests? Is Montreal investing the surplus hoping to get a better return on capital than what they're paying on the debt?
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On June 13 2014 14:35 Simberto wrote:
An n-pointed star is not really rigidly defined, though.
After some thinking, i have come up with this solution: Basically, what you need to draw a good star is an integer x that fulfills the following conditions:
x<n-1 (If x=n-1, you are drawing a polygon, which i assume do not count as stars because otherwise this would be trivial)
x>1 (Same as above, x=1 leads to polygon)
x and n do not share any prime factors.
If at least one number like that exists, you can draw a star without taking your pen of the paper by always drawing a line to the point x further from where you are, and since there are no common prime factors, you will have to fill up all points and thus finish a star. I am pretty sure that if no x exists that fulfills those conditions, you can not draw a star that has n degrees of rotational symmetry. If more then one x exist, you can draw multiple different stars.
This means that:
1,2,3,4,6 do not work.
No idea if there are other larger numbers that do not allow continuous stardrawing, but i doubt it. Right now i can not proof that that is the case, though.
Just to add a few remarks related to this for the people interested.
Number of x that satisfy the three conditions is equal to phi(n) - 2, if n>= 3 and where phi is the well known Euler's totient function. Hence you need to prove that phi(n) > 2 when n > 6, but that can be seen from the formula here (which is easily proved in a combinatorial manner, compared to the difficult proof of the fact that there's always a prime between n and 2n):
formula and proof
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On June 14 2014 19:28 Simberto wrote:
I don't know, i actually found the second question more interesting, because it actually has some mathematic value to it. Of course you can draw anything without taking the pen off the paper if you can redraw lines and don't have to any other limitations, unless there are jumps in the shape you wish to draw. That is trivial.
And the problem of actually clearly formulating the boundaries you care about is pretty common. In this case, they slowly developed through dialog. And yes, the discussion was about stars, but under certain limitations. Which makes sense, because without limitations the discussion is rather pointless because the answer is trivially obvious.
I guess the question of "How many ways can we draw this star" (which is a different question) is kind of interesting, but once again you need limitations, otherwise it is nonsensical. If you can redraw the same lines as often as you wish, the answer is obviously "There are infinite ways of drawing this star". If you can add random additional lines wherever you want, once again the answer is "There are infinite ways of drawing this, but the result is probably not a star anymore". If you can only trace each line once, and may not redraw them, the question might get interesting.
Well, my first drawing drew each line once as well, its the first goal post he moved of "you can only make lines from point to point" where its also very possible to make 6 point star (which he showed) and then he added the "each point must share the same angle, and can only be done point to point.
The current rule set is that you can only trace from star point to star point, you are not allowed to trace over lines, the points must be at specific angles, and there can be no extra lines other than the lines I arbitrarily dictate can be there.
Do you see where I'm going with this? As a discourse it is only possible by the telling of others that their exploration of the creation of stars is invalid and that you're only allowed to discuss one version of star construction.
Let me put it this way. If I wanted to have a discussion about "fun games" is it more interesting to give others free reign in what to bring up and talk about or should I say "I only like BW if you don't talk about BW gtfo."
Personally, I'm finding the discussion talking about what the discussion was talking about more interesting than either stars or abstract math theories.
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EPT
