How many pieces can you get?

[BouncinRound]

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)

[kerowo]

I'm going to go with 2n pieces. Figuring the best you can do is cut all the existing pieces in half.

[Duke]

[ 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).

[BouncinRound]

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.

[BouncinRound]

ahh, good path gotten enough to generalize a formula for n cuts though?

[Duke]

[ 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.

[BouncinRound]

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.

[gull]

Ohh... I remember doing this once, but I forgot the solution.

[Siegmund]

2, 4, 7, 11 is the right pattern.

There is a simple argument for why the answer is what it is.

[BouncinRound]

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

[bigpooch]

I assume you mean straight lines. Then the maximum number
of pieces is just [n(n+1)/2] +1. After n lines divide the
circle into the maximum number of pieces, the next line can
cross through all n of these lines and then divide through
(n+1) regions, so there are (n+1) more regions; then, by
induction, the result follows.

Of course, this also applies not only to lines and a circle,
but also to planes and a sphere.

[jman3232]

if you have 3 lines you can make 8 pieces...

[Winenose]

[ QUOTE ]
if you have 3 lines you can make 8 pieces...

[/ QUOTE ]

Really?

[BouncinRound]

[ QUOTE ]
I assume you mean straight lines. Then the maximum number
of pieces is just [n(n+1)/2] +1. After n lines divide the
circle into the maximum number of pieces, the next line can
cross through all n of these lines and then divide through
(n+1) regions, so there are (n+1) more regions; then, by
induction, the result follows.

Of course, this also applies not only to lines and a circle,
but also to planes and a sphere.

[/ QUOTE ]

ya to the best of my knowledge 3 cuts = 7 pieces and this guy has the right generalization, well done chap

[Nicholasp27]

[ QUOTE ]
if you have 3 lines you can make 8 pieces...

[/ QUOTE ]

this is a circle, not a cake

[thylacine]

[ QUOTE ]
[ QUOTE ]
if you have 3 lines you can make 8 pieces...

[/ QUOTE ]

this is a circle, not a cake

[/ QUOTE ]

and even if it were, you would need an infinite number of lines to cut it into more than one piece. /images/graemlins/grin.gif

[bigpooch]

Actually, about planes and the sphere: the maximum number
of pieces you can cut the sphere by n planes is
C(n+1,3)+n+1 for n>=0.

[jstnrgrs]

[ QUOTE ]
[ QUOTE ]
if you have 3 lines you can make 8 pieces...

[/ QUOTE ]

Really?

[/ QUOTE ]

A sphere can be cut into 8 pieces with 3 plains.

[BouncinRound]

ya perhaps a sphere can, but this post was about cutting a circle though (a 2d object). like a pizza without cutting off the cheese or something wacky. anyways, big pooch was on the money last time I checked