MPOTD - Math Problem Of The Day
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Suppose a triangle is completely covered by 25 circles of diameter 2. Can the same triangle be completely covered by 100 circles of diameter 1? why? * No advanced knowledge is required to solve this problem. P.S. The answer to the last problem I posted can be found here[^].[
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Suppose a triangle is completely covered by 25 circles of diameter 2. Can the same triangle be completely covered by 100 circles of diameter 1? why? * No advanced knowledge is required to solve this problem. P.S. The answer to the last problem I posted can be found here[^].[
My articles and software tools
I think it can be covered with no more than 75 circles of diameter 1. Don't ask why! ;P - - - - - - - - - - - - - - - - - - Memory leaks is the price we pay \0 01234567890123456789012345678901234
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Suppose a triangle is completely covered by 25 circles of diameter 2. Can the same triangle be completely covered by 100 circles of diameter 1? why? * No advanced knowledge is required to solve this problem. P.S. The answer to the last problem I posted can be found here[^].[
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When you stack the circles, you can easily cover the triangle with a million circles :P
A student knows little about a lot. A professor knows a lot about little. I know everything about nothing.
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I think it can be covered with no more than 75 circles of diameter 1. Don't ask why! ;P - - - - - - - - - - - - - - - - - - Memory leaks is the price we pay \0 01234567890123456789012345678901234
Kastellanos Nikos wrote: I think it can be covered with no more than 75 circles of diameter 1. Don't ask why! Ok, I won't. It can be proven mathematically that 75 circles of diameter 1 can cover 3/4 of the original triangle. :-D[
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Suppose a triangle is completely covered by 25 circles of diameter 2. Can the same triangle be completely covered by 100 circles of diameter 1? why? * No advanced knowledge is required to solve this problem. P.S. The answer to the last problem I posted can be found here[^].[
My articles and software tools
Yes. It's the (r squared) in the area formula. Therefore it would take four times as many circles because 2 squared is 4 and 1 squared is still 1. Therefore you'd need four times as many circles. That's my answer and I'm sticking with it! :-D BTW, is there a way to do super/sub scripts in the forums? Mike
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Yes. It's the (r squared) in the area formula. Therefore it would take four times as many circles because 2 squared is 4 and 1 squared is still 1. Therefore you'd need four times as many circles. That's my answer and I'm sticking with it! :-D BTW, is there a way to do super/sub scripts in the forums? Mike
MikeBeard wrote: That's my answer and I'm sticking with it! Okay, so I changed my mind. ;) The math is basically the same, but I was using the diameter rather than the radius. As it turns out, the answer is still yes, but now the d=2 r=1 circles would mean that the answers would be 1 and .25. The smaller circles are off by a factor of 4, so if the 25 2 inch diameter circles covered it, 100 1 inch should cover it. Mike
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MikeBeard wrote: That's my answer and I'm sticking with it! Okay, so I changed my mind. ;) The math is basically the same, but I was using the diameter rather than the radius. As it turns out, the answer is still yes, but now the d=2 r=1 circles would mean that the answers would be 1 and .25. The smaller circles are off by a factor of 4, so if the 25 2 inch diameter circles covered it, 100 1 inch should cover it. Mike
MikeBeard wrote: MikeBeard wrote: That's my answer and I'm sticking with it! That's the kind of confidence I expect from a developer! :-D MikeBeard wrote: so if the 25 2 inch diameter circles covered it, 100 1 inch should cover it. What you are saying is that because the 100 smaller circles have the same total area as the 25 larger circles they should be able to cover the same triangle. Unfortunately this will not be considered a correct answer by a picky math professor. Suppose you are twice as large as your kid brother, you wouldn't use two of his jackets to cover your upper body even if the total areas of the two jackets add up to exactly the area of your own jacket, would you? ;P[
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Yes. It's the (r squared) in the area formula. Therefore it would take four times as many circles because 2 squared is 4 and 1 squared is still 1. Therefore you'd need four times as many circles. That's my answer and I'm sticking with it! :-D BTW, is there a way to do super/sub scripts in the forums? Mike
How about the <sup> and <sub> tags? :)
When I can talk about 64 bit processors and attract girls with my computer not my car, I'll come out of the closet. Until that time...I'm like "What's the ENTER key?" -Hockey on being a geek
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Yes. It's the (r squared) in the area formula. Therefore it would take four times as many circles because 2 squared is 4 and 1 squared is still 1. Therefore you'd need four times as many circles. That's my answer and I'm sticking with it! :-D BTW, is there a way to do super/sub scripts in the forums? Mike
MikeBeard wrote: BTW, is there a way to do super/sub scripts in the forums? Yes. 22 is 4 11 is still 1
"When I was born I was so surprised that I didn't talk for a year and a half." - Gracie Allen
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Suppose a triangle is completely covered by 25 circles of diameter 2. Can the same triangle be completely covered by 100 circles of diameter 1? why? * No advanced knowledge is required to solve this problem. P.S. The answer to the last problem I posted can be found here[^].[
My articles and software tools
Assuming that the circles are lieing on the triangle in a grid layout, each circle is lieing on an imaginary square. If a circle with diameter 2 fits on such a square, then you can fit 4 circles with diameter 1 on it. It's simple really - a circle with diameter 2, is as high as two circles with diameter 1, and wide as two circles with diameter 1. 2 by 2 is 4. Hence, there will be four times as many circles with diameter 1. 100/25 = 4, thus the first question can be answered: Yes. (That's the simplest explanation I could come up with without fumbling with pi and radiuses. A drawing would convince anyone I believe) -- Ich bin Joachim von Hassel, und ich bin Pilot der Bundeswehr. Welle: Erdball - F104-G Starfighter
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Assuming that the circles are lieing on the triangle in a grid layout, each circle is lieing on an imaginary square. If a circle with diameter 2 fits on such a square, then you can fit 4 circles with diameter 1 on it. It's simple really - a circle with diameter 2, is as high as two circles with diameter 1, and wide as two circles with diameter 1. 2 by 2 is 4. Hence, there will be four times as many circles with diameter 1. 100/25 = 4, thus the first question can be answered: Yes. (That's the simplest explanation I could come up with without fumbling with pi and radiuses. A drawing would convince anyone I believe) -- Ich bin Joachim von Hassel, und ich bin Pilot der Bundeswehr. Welle: Erdball - F104-G Starfighter
The problem says that the triangle must be completly covered. That does not mean that the circles must fit inside the triangle - the triangle fits inside a group of 25 overlaped circles. With the pattern you describe, there would be gaps in the center of the larger circle. Steve
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The problem says that the triangle must be completly covered. That does not mean that the circles must fit inside the triangle - the triangle fits inside a group of 25 overlaped circles. With the pattern you describe, there would be gaps in the center of the larger circle. Steve
Steve Mayfield wrote: With the pattern you describe, there would be gaps in the center of the larger circle. Think about it. No triangle can ever be fully covered with circles with finite sizes. There will always be gaps! Hence the original layout had gaps, and it's ok to assume that the new layout may also have gaps. I admit that the problem description is a bit vague. -- Ich bin Joachim von Hassel, und ich bin Pilot der Bundeswehr. Welle: Erdball - F104-G Starfighter
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Suppose a triangle is completely covered by 25 circles of diameter 2. Can the same triangle be completely covered by 100 circles of diameter 1? why? * No advanced knowledge is required to solve this problem. P.S. The answer to the last problem I posted can be found here[^].[
My articles and software tools
the size of the triangle is undefined. the layout of the circles are undefined - we do not know if they stacked, everlapping or even touching? I say that they are stacked verticaly (like coins) over a triangle smaller than "diameter 1". [ Jason De Arte | Toy Maker | 1001010.com ]
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Steve Mayfield wrote: With the pattern you describe, there would be gaps in the center of the larger circle. Think about it. No triangle can ever be fully covered with circles with finite sizes. There will always be gaps! Hence the original layout had gaps, and it's ok to assume that the new layout may also have gaps. I admit that the problem description is a bit vague. -- Ich bin Joachim von Hassel, und ich bin Pilot der Bundeswehr. Welle: Erdball - F104-G Starfighter
Jörgen Sigvardsson wrote: No triangle can ever be fully covered with circles with finite sizes. What do you mean by that? One large pizza could completely cover your mouse pad, which is of rectangular shape, and a rectangle can be seen as two triangles. Since one circle can cover two triangles, there is certainly no problem for 25 circles to cover one, right? ;P Yes, the problem does not say whether the 25 circles are overlaping or not, which means there is no restriction that they must not overlap.[
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Jörgen Sigvardsson wrote: No triangle can ever be fully covered with circles with finite sizes. What do you mean by that? One large pizza could completely cover your mouse pad, which is of rectangular shape, and a rectangle can be seen as two triangles. Since one circle can cover two triangles, there is certainly no problem for 25 circles to cover one, right? ;P Yes, the problem does not say whether the 25 circles are overlaping or not, which means there is no restriction that they must not overlap.[
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Well, I made assumptions, so bite me. ;P :-D -- Ich bin Joachim von Hassel, und ich bin Pilot der Bundeswehr. Welle: Erdball - F104-G Starfighter
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the size of the triangle is undefined. the layout of the circles are undefined - we do not know if they stacked, everlapping or even touching? I say that they are stacked verticaly (like coins) over a triangle smaller than "diameter 1". [ Jason De Arte | Toy Maker | 1001010.com ]
Xiangyang does this periodically to screw with our brains. I think he enjoys it. :-D -- Ich bin Joachim von Hassel, und ich bin Pilot der Bundeswehr. Welle: Erdball - F104-G Starfighter
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Suppose a triangle is completely covered by 25 circles of diameter 2. Can the same triangle be completely covered by 100 circles of diameter 1? why? * No advanced knowledge is required to solve this problem. P.S. The answer to the last problem I posted can be found here[^].[
My articles and software tools
Ah, I didn't get a chance to think about it until I got home from work. The circles can cover a similar triangle of the same proportion and 1/4 the area. Just assemble 4 of those triangles to cover the original and you have the entire original triangle covered, hence your 75 circle comment. Just a tesselation problem.:doh:
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
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Suppose a triangle is completely covered by 25 circles of diameter 2. Can the same triangle be completely covered by 100 circles of diameter 1? why? * No advanced knowledge is required to solve this problem. P.S. The answer to the last problem I posted can be found here[^].[
My articles and software tools
Xiangyang Liu wrote: Suppose a triangle is completely covered by 25 circles of diameter 2. Can the same triangle be completely covered by 100 circles of diameter 1? why? Yes. Assume the given triangle has vertices A, B, and C. Now define points D, E, and F as the midpoints between each of the vertices. These six points can be used as the vertices of 4 smaller triangles (ADF, BDE, CFE, and DEF) that are congruent to and 1/4 the area of the original triangle ABC. These 4 triangles represent a partition of the original triangle. The idea is to cover each of the smaller triangles with 25 circles of diameter 1, and this is possible because this is just applying the known covering at a 1/2 length scale. Thus, we have covered the entire triangle with 100 circles of diameter 1.
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Ah, I didn't get a chance to think about it until I got home from work. The circles can cover a similar triangle of the same proportion and 1/4 the area. Just assemble 4 of those triangles to cover the original and you have the entire original triangle covered, hence your 75 circle comment. Just a tesselation problem.:doh:
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
andy brummer wrote: Ah, I didn't get a chance to think about it until I got home from work. The circles can cover a similar triangle of the same proportion and 1/4 the area. Just assemble 4 of those triangles to cover the original and you have the entire original triangle covered Beautiful! :rose: John Carson "I believe in an America where the separation of church and state is absolute--where no Catholic prelate would tell the President (should he be Catholic) how to act, and no Protestant minister would tell his parishoners for whom to vote ... and where no man is denied public office merely because his religion differs from the President who might appoint him or the people who might elect him. - John F. Kennedy
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Ah, I didn't get a chance to think about it until I got home from work. The circles can cover a similar triangle of the same proportion and 1/4 the area. Just assemble 4 of those triangles to cover the original and you have the entire original triangle covered, hence your 75 circle comment. Just a tesselation problem.:doh:
I can imagine the sinking feeling one would have after ordering my book, only to find a laughably ridiculous theory with demented logic once the book arrives - Mark McCutcheon
You are absolutely correct! andy brummer wrote: Just a tesselation problem. If only software development is this simple: Any big project can be divided into smaller ones of equal complexity and all developers have the same capability. All management has to do is hire more developers for bigger projects. :-D[
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