#1
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How many pieces can you get?
So, consider a circle. One line through the circle cuts it into two pieces.
A second line, depending on where it lies, makes one or two more pieces. What is the most number of pieces that can be made from a certain number of cuts? How many pieces can you make with n cuts? (e.g. the minimum # of pieces would be n+1) |
#2
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Re: How many pieces can you get?
I'm going to go with 2n pieces. Figuring the best you can do is cut all the existing pieces in half.
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#3
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Re: How many pieces can you get?
[ QUOTE ]
I'm going to go with 2n pieces. Figuring the best you can do is cut all the existing pieces in half. [/ QUOTE ] Think in terms of regions. How many of the regions can you cut into two? 1 line = 2 regions 2 lines = 4 regions 3 lines = 7 for me, and I don't know if that's optimal 4 lines = 11 for me, and once again I don't know if that's optimal or not See what I mean? I don't know the answer, but this is how to start looking at it. Because you have a straight line, though, you'll be precluded from splitting every single region so it won't be 2^n, though that would obviously be a loose upper bound (what you'd get if you split every region every time). |
#4
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Re: How many pieces can you get?
nah, I'm doodlin with it myself... they don't have to be equal cuts, just cuts that maximize the amount of pieces.
So for like 3 cuts you can have 7 pieces, 4 I found a way to get 10, 5 cuts can get your 14 pieces,...etc? Then I keep reverting back to symmetry and not making the correct cuts from there. I'm getting close to having enough samples to maybe come to a conclusion though. |
#5
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Re: How many pieces can you get?
ahh, good path gotten enough to generalize a formula for n cuts though?
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#6
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Re: How many pieces can you get?
[ QUOTE ]
ahh, good path gotten enough to generalize a formula for n cuts though? [/ QUOTE ] Yeah I haven't done hat. I won't be able to look at this more until later. I'm almost certain that this is a solved problem, though. |
#7
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Re: How many pieces can you get?
Holy guacamole I love the internet, I found a solution. But if you want you can keep working on it and I can tell you if you're on the right track(close to the right answer). I'll monitor the post for the next 12 hours - (6 hours for sleep somewhere in there). So, if you're interested lemme know.
Thanks for the help Duke. |
#8
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Re: How many pieces can you get?
Ohh... I remember doing this once, but I forgot the solution.
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#9
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Re: How many pieces can you get?
2, 4, 7, 11 is the right pattern.
There is a simple argument for why the answer is what it is. |
#10
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Re: How many pieces can you get?
ya; 2, 4, 7, 11, 16, 22 are the max # of pieces for 1 -> 6 cuts. Now, can you find the pattern and make an generalization for n number of cuts? I'll post the answer tomorrow before class. Holla
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