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Topic: How many corners (Read 8822 times) |
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Rivaa
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How many corners
« on: Dec 19th, 2012, 10:29pm » |
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There is a rectangle, if i cut a corner than how much corners will remain..? 
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cartoonle
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Re: How many corners
« Reply #1 on: Dec 19th, 2012, 11:46pm » |
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Grimbal
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Re: How many corners
« Reply #2 on: Dec 20th, 2012, 8:42am » |
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3
It depends how much of the corner you cut. You could end up with a triangle.
So the answer "3" is not necessarily wrong.
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| « Last Edit: Dec 20th, 2012, 8:44am by Grimbal » |
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towr
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Re: How many corners
« Reply #3 on: Dec 20th, 2012, 9:02am » |
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* It also depends on whether you use a straight cut
* The piece you cut off also has corners; so a straight cut close to the corner you cut off would leave you with 8 corners, 5 on the large piece, 3 on the small
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| « Last Edit: Dec 20th, 2012, 9:02am by towr » |
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Death
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Re: How many corners
« Reply #4 on: Dec 22nd, 2012, 4:55am » |
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Just for completeness:
* If you cut from one corner to the opposite corner you get 6 corners (2 triangles)
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rmsgrey
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Re: How many corners
« Reply #5 on: Dec 31st, 2012, 8:18am » |
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4 corners - if I cut a corner in the College Quad, then the Quad still has the same number of corners, even though I haven't walked around any
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marlonmark
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Re: How many corners
« Reply #6 on: Jan 2nd, 2013, 1:37am » |
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I think you are talking about the one that remains in the rectangle hence
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MpAdvisor
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Re: How many corners
« Reply #7 on: Feb 20th, 2013, 8:07pm » |
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it depends on the cut if it is straight u will get5 corners and if it is zig zag then the answer is undefined
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Jacob Black
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Re: How many corners
« Reply #8 on: Mar 12th, 2013, 3:35am » |
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Rectangles has four corner and if you cut one the there will be five corners left.
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Apala
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Re: How many corners
« Reply #9 on: Mar 13th, 2013, 12:43am » |
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answer is 3 because i m going to cut the corner in round shape and then 3 will remain 
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ravibhole_1
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Re: How many corners
« Reply #10 on: Apr 6th, 2013, 11:51pm » |
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IF cut is straight 5 corners
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allinonetech01
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Re: How many corners
« Reply #11 on: Apr 13th, 2013, 3:10am » |
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If cut from middle then answer is:3
If cut less than half then answer is:5 
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Nursejim
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Re: How many corners
« Reply #12 on: Sep 21st, 2013, 11:34pm » |
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To me the answer might be different. If you cut of a corner then you are left with a 5 sided figure but you also have a triangle now and as we know has 3 sides. Sides equals corners so you end up with 8 corners in my book.
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djewels
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Re: How many corners
« Reply #13 on: Sep 22nd, 2013, 3:01am » |
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5 Corners
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webtasarim
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Re: How many corners
« Reply #14 on: Oct 6th, 2013, 10:41am » |
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i think 5
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UgoLocal02
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Re: How many corners
« Reply #15 on: Jun 12th, 2014, 2:28am » |
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5
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medo90zezo
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Re: How many corners
« Reply #16 on: Jun 22nd, 2014, 7:05am » |
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i think its 5 corners
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| « Last Edit: Nov 5th, 2014, 6:38am by medo90zezo » |
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rloginunix
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Re: How many corners
« Reply #17 on: Jun 22nd, 2014, 2:14pm » |
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Let us turn that old page.
Fold, cut - and then engage.
For the purposes of this mathematical thought experiment imagine having a very very very very long and narrow rectangular piece of paper with all four corners being 90 degrees.
"A cut" = a single 45-degree non-diagonal straight line scissors cut of a small (triangular) corner area.
"A fold" = folding of the current piece of paper formation in exactly half - align two opposite edges and make a crease (with your thumb and index fingers).
"Angular line" = slanted line of the cut (45 degrees relative to the short side of the rectangle).
If the strip of paper is folded N times how many angular lines will there be? How many vertexes along the cut side will there be?
P.S.
I just came up with this problem trying to make the original a bit more interesting.
I suggest starting the sequence at zero: 0 folds = 1 slanted line, 1 fold = 2 lines, etc.
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dudiobugtron
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Re: How many corners
« Reply #18 on: Jun 22nd, 2014, 2:42pm » |
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on Jun 22nd, 2014, 2:35pm, rloginunix wrote:| the strip of paper is so long that the new cuts never interfere with the old ones. |
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It's not the length of the paper that affects this so much as the length of the cut. For any paper length and N, you can make a cut small enough that it doesn't interfere with any other cuts. All you need to do is make sure that you 'cut off' at most one corner. Which is arguably the requirement given in the definition for it to be a cut anyway.
One thing that may need more clarification, though, is whether you are allowed to fold width-wise; this doesn't appear to be disallowed in the problem statement.
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rloginunix
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Re: How many corners
« Reply #19 on: Jun 22nd, 2014, 2:46pm » |
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Pictorial description of the cuts:
Clarification:
the strip of paper is so long that no matter how many times we fold it we can always make the "less than diagonal" cut (see the image).
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rloginunix
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Re: How many corners
« Reply #20 on: Jun 22nd, 2014, 2:55pm » |
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Sorry for the poor comment (I removed it now) - it was the left over of me cutting the real paper with real scissors.
The process of folding is repetitive: fold the strip in half, then fold that half in half, then fold that quarter in half again and so on. No folding width-wise, only length-wise. Also we make only one cut in only one corner.
Is that more clear now? Let me know if I should explain some more.
[edit]
Just to be sure: fold the strip of paper in half N times, make one cut in one corner (as shown in the diagram above).
[/edit]
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| « Last Edit: Jun 22nd, 2014, 4:30pm by rloginunix » |
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dudiobugtron
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Re: How many corners
« Reply #21 on: Jun 22nd, 2014, 2:59pm » |
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Very clear, thanks. Now I will start thinking about the solution. 
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dudiobugtron
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Re: How many corners
« Reply #22 on: Jun 22nd, 2014, 6:39pm » |
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Actually, there is one thing I need a bit of clarification on: can an angular line go in either direction? eg: would cutting off the bottom-left corner (instead of the bottom right) also generate an angular line?
If so, then here is my take on it:
Each time you fold it, you double the number of sheets that have been folded over. With no folds, there is 1 sheet, so one angular line. With 2 folds, there are 2 sheets, so 2 lines. 3 folds gives 4 lines, 4 gives 8, etc...
The general formula is 2^N.
For the number of corners: every pair of two angular lines forms a triangular shape cut out of the bottom side. That is except for if the cut chops off one or both of the original bottom corners.
Assuming it doesn't (so, you always cut the side which doesn't have the 'free' ends), then you halve the number of lines to get the number of triangles, then multiply that by 3 to get the number of extra corners. So the number of extra vertices added along the cut side is:
3 * 2^(N-1)
Making the total 2 + 3 * 2^(N-1) for that side, or
4 + 3 * 2^(N-1) for the whole shape.
If you cut the 'free' ends off, then you end up one less 'full triangle', but two more 'half triangles'. (Each 'half-triangle' is obviously a triangle in its own right, but it's only half the size of the others.)
Each 'half triangle' adds two vertices, but at the cost of cutting one of the original vertices off. So the formula for how many vertices are in the whole shape is 4 (from the original rectangle) - 2 (the two cut off), + 4 (the new ones from the two half triangles), + 3 * (2^(N-1) - 1) (the extra sides added from the full triangles, of which there is one less. You can reorganise this into:
3 + 3 * 2^(N-1)
For the whole shape. So one less than if you had cut on the other side.
edit: fixed 'cuts' to 'angular lines' where appropriate.
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| « Last Edit: Jun 22nd, 2014, 6:40pm by dudiobugtron » |
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rloginunix
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Re: How many corners
« Reply #23 on: Jun 23rd, 2014, 8:31am » |
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Good stuff. That's the numbers I got.
My only comment is that if you count the vertexes induced by the cut only you get (what do you know) two apparently well known integer sequences 3*2^(N-1) and 3*2^(N-1) + 1. I checked those against oeis.org.
I see now where (and why) my problem statement remains ambiguous. Where = location of the cut. Why = if it's in my head it doesn't mean the rest of the world knows about it. Also my English is no good - the words "angular line" should likely be replaced with "the line of the cut".
So hopefully we straightened it out now and came up with an entertaining easy problem:
Fold a long narrow rectangular strip of paper in half N times. Cut any one of the corners off with a single straight line cut.
How many cases are possible? How many lines of the cut will there be? How many vertexes will there be?
P.S.
Feel free to simplify/disambiguate it more.
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pembelajaran.sd
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Re: How many corners
« Reply #24 on: Jun 25th, 2014, 7:10pm » |
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I answer 4 corners...
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