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#1
05-28-2010, 07:35 AM
 Indian Member Join Date: Sep 2007 Location: Mumbai, India Posts: 1,742
Cutting a pizza into largest number of pieces

...with 4 straight cuts .

I was reading "Classic Brainteasers" by Martin Gardner (Orient Paperbacks edition ).

He gives the solution as 11 with illustrative figure.( The formula applicable is 0.5 xsquared +0.5 x +1, with x being number of cuts).

Now if the 4th cut is made horrizontally through the centre of pizza ,one can get 14 pieces ( double of 7 pieces ).

Only condition specified is 4 straight cuts, not size of pieces.

What am I missing ?
#2
05-28-2010, 07:42 AM
 zut Charter Member Join Date: Apr 2000 Location: Detroit, MI Posts: 3,589
He's assuming, without explicitly stating, that the "pizza cutting" represents a 2D problem. Else the brainteaser would be "watermelon cutting" or some such.
#3
05-28-2010, 07:47 AM
 Sparky812 Member Join Date: Aug 1999 Location: Great White North Posts: 2,732
What is your question?
If you're suggesting a horizontal cut will produce twice as many slices of pizza, please review your logic. The question specifically asks for the largest number of pieces of pizza. Your solution would result in zero.

Last edited by Sparky812; 05-28-2010 at 07:48 AM.
#4
05-28-2010, 08:21 AM
 BigT Guest Join Date: Aug 2008
Quote:
 Originally Posted by Sparky812 What is your question? If you're suggesting a horizontal cut will produce twice as many slices of pizza, please review your logic. The question specifically asks for the largest number of pieces of pizza. Your solution would result in zero.
Naw, there'd still be 7 slices of pizza and seven slices of bread. And there are covered pizzas where 14 would actually be the correct answer.

(Of course, I'm assuming the definition that a pizza is "A baked pie consisting of a shallow bread crust covered with toppings such as seasoned tomato sauce, cheese, sausage, or olives.)
#5
05-28-2010, 09:25 AM
 Sparky812 Member Join Date: Aug 1999 Location: Great White North Posts: 2,732
Seriously!?
Honestly, why do some people insist on inventing exceptions and modifying the question and/or the definitions of the terms to suit their own ridiculous answers?

The question asks for pizza, as zut said....
Quote:
 "Else the brainteaser would be "watermelon cutting" or some such"

Last edited by Sparky812; 05-28-2010 at 09:27 AM.
#6
05-28-2010, 10:31 AM
 Hampshire Member Join Date: Jan 2003 Location: Minneapolis Posts: 9,081
If you want to start breaking the rules of a basic 2-D puzzle I'm sure there are even more ways to do it than just a horizontal cut.
How about folding the the pizza in half a few times then making the cuts?
#7
05-28-2010, 10:55 AM
 Indian Member Join Date: Sep 2007 Location: Mumbai, India Posts: 1,742
Yes ! It is meant to be a 2D puzzle.

I rechecked that puzzle . It is apple pie and not pizza.
#8
05-28-2010, 11:31 AM
 Omar Little Guest Join Date: Apr 2000
Quote:
 Originally Posted by indian I rechecked that puzzle . It is apple pie and not pizza.
As if that makes a difference?
#9
05-28-2010, 11:34 AM
 Indian Member Join Date: Sep 2007 Location: Mumbai, India Posts: 1,742
Quote:
 Originally Posted by Wilbo523 As if that makes a difference?
No it won't
#10
05-28-2010, 11:36 AM
 Shagnasty Charter Member Join Date: May 2000 Posts: 21,307
Quote:
 Originally Posted by Wilbo523 As if that makes a difference?
I don't know. Is it with ice cream or without? Also, are we really talking 'slices' or did it say 'pieces'? Some apple pie crusts are really flaky.

Last edited by Shagnasty; 05-28-2010 at 11:37 AM.
#11
05-28-2010, 11:37 AM
 Munch Guest Join Date: Mar 2000
Quote:
 Originally Posted by Wilbo523 As if that makes a difference?
You can't fold an apple pie.
#12
05-28-2010, 12:00 PM
 Hampshire Member Join Date: Jan 2003 Location: Minneapolis Posts: 9,081
Quote:
 Originally Posted by Munch You can't fold an apple pie.
Then how do you fit it in your mouth?
#13
05-28-2010, 12:02 PM
 Smeghead Guest Join Date: Apr 2000
Quote:
 Originally Posted by Hampshire Then how do you fit it in your mouth?
That's what she said.
#14
05-28-2010, 12:48 PM
 Happy Poster BANNED Join Date: Apr 2009 Posts: 1,192
Quote:
 Originally Posted by Hampshire Then how do you fit it in your mouth?
Open it really wide
#15
05-28-2010, 01:36 PM
 Manduck Guest Join Date: May 1999
Quote:
 Originally Posted by Happy Poster Open it really wide
That's what he said.
#16
05-28-2010, 02:14 PM
 robardin Member Join Date: Jun 2005 Location: Flushing, NY Posts: 4,216
I think the OP is worded in a way that supposes we are familiar with this puzzle already, which I'm not. Are these pieces meant to be congruent pieces? Non-congruent but equal in volume? Or just "get as many fragments of any size you can" with 8 straight cuts? And what does "straight cut" mean -- through center (diameter), edge to edge (chord of the circle), one point of the cut must touch the edge of the pie but not both, or just "no wiggles"?

Because if the intuitive meaning of "cut this pie" (meaning, into equal and identical pieces with straight line cuts that go from one edge to the other) is in play I don't see how it can't be 8 and only 8.
#17
05-28-2010, 02:24 PM
 robardin Member Join Date: Jun 2005 Location: Flushing, NY Posts: 4,216
OK if it's just "any number of fragments" I just drew on on a piece of paper with four lines edge-to-edge (chords) resulting in 11 fragments. Are you saying even more is possible?
#18
05-28-2010, 02:27 PM
 Blaster Master Guest Join Date: Feb 2006
Quote:
 Originally Posted by robardin I think the OP is worded in a way that supposes we are familiar with this puzzle already, which I'm not. Are these pieces meant to be congruent pieces? Non-congruent but equal in volume? Or just "get as many fragments of any size you can" with 8 straight cuts? And what does "straight cut" mean -- through center (diameter), edge to edge (chord of the circle), one point of the cut must touch the edge of the pie but not both, or just "no wiggles"?
you have to make as many pieces as possible without regard to size or shape of the pieces. All "straight cuts" means is a chord (why would you stop anywhere but to the edge, it will never result in more pieces than going to the edge) with "no wiggles".

And to the OP, like another said, the point of the puzzle is that it's 2D and maximizing the number of intersections; it's just more interesting to talk about a pizza or a pie than a circle and let the 2D part be an implied restriction. Besides, one could argue that it stops being a piece of pizza if you make a horrizontal cut because the bottom pieces would just be baked bread, not pizza. Also, if it were a 3D puzzle, I think you could get more than 14 pieces if you allowed diagonal cuts, but I can't be sure since I can't easily draw it out and count pieces.
#19
05-28-2010, 02:31 PM
 robardin Member Join Date: Jun 2005 Location: Flushing, NY Posts: 4,216
Well if the solution is given as "11" then I guess I've got it, or some variant of it. But it intrigues me to think of how I might PROVE that this is the largest number. Hmmmm.

Visually, I drew a circle, slashed three lines in it like an A so it kind of looks like the Anarchy logo, giving me 7 segments. I then laid a fourth line diagonally across, like taking the crossbar of the A and rotating it 30 degrees around the middle of the line. Then I counted the segments between the lines and it comes out to 11.
#20
05-28-2010, 02:35 PM
 TerpBE Guest Join Date: Apr 2004
Quote:
 Originally Posted by Wilbo523 As if that makes a difference?
Yes, it's actually very difficult to slice an apple pie so that you end up with a piece of pizza.
#21
05-28-2010, 02:40 PM
 Hampshire Member Join Date: Jan 2003 Location: Minneapolis Posts: 9,081
Quote:
 Originally Posted by robardin OK if it's just "any number of fragments" I just drew on on a piece of paper with four lines edge-to-edge (chords) resulting in 11 fragments. Are you saying even more is possible?
Yea, they were taking it into 3 dimensions which seems to refer to a similar puzzle of cutting a round cake with only 3 cuts into 8 pieces of equal volume.
SPOILER:
Two vertical cuts quatering the cake ' + ' and then a horizontal cut through the cake
#22
05-28-2010, 02:51 PM
 Indistinguishable Guest Join Date: Apr 2007
Quote:
 Originally Posted by robardin Well if the solution is given as "11" then I guess I've got it, or some variant of it. But it intrigues me to think of how I might PROVE that this is the largest number. Hmmmm.
Let's say the "linearity" of a particular configuration of cuts is the maximum number of pieces a new line could cut across. Notice that if adding a cut to a configuration gives it linearity N (i.e., after the new cut, there's some way to lay a line across it cutting across N pieces), then at most one of the relevant piece-separators is due to the new cut, so prior to the new cut, the linearity must have been at least N - 1. I.e., each cut increases the linearity by at most 1. Also, the pizza clearly starts out with linearity 1. Thus, before the Nth cut, the linearity is <= N.

Furthermore, the number of new pieces a cut creates (by splitting old pieces) is clearly at most the linearity of the configuration prior to the cut. Combining this with the previous observation, we find that the Nth cut creates <= N new pieces. In other words, after N cuts, a total of <= 1 + 2 + ... + N = N * (N + 1)/2 many pieces is created. Adding the 1 piece the pizza starts with, we see that after N cuts, the pizza is left in <= 1 + N * (N + 1)/2 many pieces.

In the particular case where N = 4, this gives us an upper bound of 1 + 4 * 5/2 = 11 many pieces.

Last edited by Indistinguishable; 05-28-2010 at 02:54 PM.
#23
05-28-2010, 03:52 PM
 astro Member Join Date: Jul 1999 Location: Taint of creation Posts: 29,360
#24
05-28-2010, 03:53 PM
 Chronos Charter Member Join Date: Jan 2000 Location: The Land of Cleves Posts: 50,758
I can get 16 congruent pieces with 4 cuts. First, cut along the diameter. Then, take one half and stack it on top of the other half, and make the second cut along the midline. For the third and fourth cuts, stack and cut on the midline again.
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#25
05-28-2010, 04:01 PM
 Indistinguishable Guest Join Date: Apr 2007
Also clearly the maximum possible, for the problem allowing stacking but no folding.
#26
05-28-2010, 04:09 PM
 Rigamarole Guest Join Date: Dec 2005
Quote:
 Originally Posted by Chronos I can get 16 congruent pieces with 4 cuts. First, cut along the diameter. Then, take one half and stack it on top of the other half, and make the second cut along the midline. For the third and fourth cuts, stack and cut on the midline again.
Again, you're making a 2D exercise into a 3D one. There's no such thing as "stacking" things on top of each other in a two dimensional plane.
#27
05-28-2010, 04:15 PM
 robardin Member Join Date: Jun 2005 Location: Flushing, NY Posts: 4,216
Quote:
 Originally Posted by Chronos I can get 16 congruent pieces with 4 cuts. First, cut along the diameter. Then, take one half and stack it on top of the other half, and make the second cut along the midline. For the third and fourth cuts, stack and cut on the midline again.
But you got cheese on the bottom of the other pieces. That is a fail.
#28
05-28-2010, 05:43 PM
 Smeghead Guest Join Date: Apr 2000
Quote:
 Originally Posted by robardin But you got cheese on the bottom of the other pieces. That is a fail.
Ah, but what if the pizza comes with an unlimited number of those little plastic picnic tables that go in the middle to keep the box of the cheese, huh? WHAT THEN??!
#29
05-28-2010, 05:53 PM
 gazpacho Charter Member Join Date: Oct 1999 Posts: 4,969
Quote:
 Originally Posted by robardin But you got cheese on the bottom of the other pieces. That is a fail.
You flip one piece over so that the cheese side faces the cheese side. After that crust is on the outside and stacking is no longer an issue.
#30
05-28-2010, 08:38 PM
 Patty O'Furniture Guest Join Date: May 1999
Quote:
 Originally Posted by Chronos I can get 16 congruent pieces with 4 cuts. First, cut along the diameter. Then, take one half and stack it on top of the other half, and make the second cut along the midline. For the third and fourth cuts, stack and cut on the midline again.
Ha ha. If you can cut this piece of paper into four equal sections, I'll give you a quarter.
#31
05-28-2010, 09:13 PM
 Patty O'Furniture Guest Join Date: May 1999
Let me try to run post #22 through the degeekulator.

Without relying on a visual I, I think the best way to explain it verbally is:

The first cut (1) can only hope to produce two sections, a and b. Hopefully that doesn't need its own proof.

The second cut (2) can be made in one of two ways.
• Either entirely within section a or section b, the result is identical to that of cut #1 (splitting a single section into two sections). This results in a single additional section c, for a total of three sections.
• Make a cut that intersects cut #1. This results in two additional sections, for a total of four sections.

From this last step we observe that, if the choice is either intersecting the previous cut or not intersecting it, the choice that yields more sections is intersecting the previous cut. Let's apply this newly acquired practical knowledge when making cut #3

The third cut (3) can be made in one of four ways:
• Entirely within any single section (one additional section produced)
• Intersecting a single previous cut (two additional sections produced)
• Intersecting the previous two cuts' intersection point (two additional sections produced)
• Intersecting both previous cuts (splitting three existing sections in two, and resulting in three additional sections)

We can see that the fourth option yields the largest number of sections.

The fourth and final cut should be made in such a way that it intersects all three previous cuts. If we continue on with additional cuts, and always wish to generate the maximum number of sections, we must always cut in such a way that we intersect all previous cuts.
#32
05-28-2010, 09:34 PM
 Smeghead Guest Join Date: Apr 2000
Wait, wait - what if the four lines represent the four blades of a blender? You'd approach an infinite number of slices then!

I don't know why I'm screwing with this thread. I'll stop now.
#33
05-28-2010, 10:40 PM
 Notassmartasithought Guest Join Date: Apr 2009
Martin Gardner died just last Saturday.

RIP

http://www.nytimes.com/2010/05/24/us...ref=obituaries
#34
05-29-2010, 05:01 PM
 Chronos Charter Member Join Date: Jan 2000 Location: The Land of Cleves Posts: 50,758
Oh, man, he was a great popularizer. Are there any compendia of his work?
#35
05-30-2010, 10:44 PM
 Notassmartasithought Guest Join Date: Apr 2009
Quote:
 Originally Posted by Chronos Oh, man, he was a great popularizer. Are there any compendia of his work?
You can always check Amazon.com, for instance, to see what's available new and used.

Last edited by Notassmartasithought; 05-30-2010 at 10:45 PM.
#36
05-31-2010, 08:05 AM
 MikeS Charter Member Join Date: Oct 2001 Location: Williamstown, MA Posts: 3,155
Over the last few years he was still at work writing updates for new editions of his Mathematical Games columns. They're worth a gander if you don't have any of his published works already.
#37
05-31-2010, 09:44 AM
 naita Guest Join Date: Jun 2002
Now the next problem is: what orientation of these four cuts give pieces with the smallest difference in area? And what statistical measure of difference is the best to use?
#38
05-31-2010, 10:31 AM
 Indian Member Join Date: Sep 2007 Location: Mumbai, India Posts: 1,742
Quote:
 Originally Posted by naita Now the next problem is: what orientation of these four cuts give pieces with the smallest difference in area? And what statistical measure of difference is the best to use?
where are the geeks of this board ?
#39
06-01-2010, 12:12 AM
 OldGuy Charter Member Join Date: Dec 2002 Location: Very east of Foggybog, WI Posts: 2,412
Quote:
 Originally Posted by naita Now the next problem is: what orientation of these four cuts give pieces with the smallest difference in area? And what statistical measure of difference is the best to use?
My gut feeling is you could make the pieces identical in area so it would not matter what measure you used. If this is not possible, there is no correct answer for the "best measure." Three obvious choices are: max area - min area, variance (or std dev), and sum of the 55 absolute differences.
#40
06-01-2010, 07:27 AM
 naita Guest Join Date: Jun 2002
Quote:
 Originally Posted by OldGuy My gut feeling is you could make the pieces identical in area so it would not matter what measure you used. If this is not possible, there is no correct answer for the "best measure." Three obvious choices are: max area - min area, variance (or std dev), and sum of the 55 absolute differences.
I don't think it's possible to make them identical in area. Looking at the figure and attempting to move the cuts mentally, it seems to me that moving the lines to shrink one of the larger pieces inevitably affects at least one of the smaller pieces negatively.
#41
06-01-2010, 07:34 AM
 Ostrya Guest Join Date: Dec 2003
Quote:
 Originally Posted by astro Here's a visual of the 11 slice solution
I needed this illustration to understand. Thank you, astro!
#42
06-01-2010, 09:21 AM
 Khaki Campbell BANNED Join Date: Jan 2010 Posts: 607
Quote:
 Originally Posted by astro Here's a visual of the 11 slice solution
Now a more interesting problem would be how to do those four cuts so that the areas of the slices are the least unequal.
#43
06-01-2010, 09:27 AM
 chrisk Charter Member Join Date: Nov 2003 Location: Southern ontario Posts: 5,784
Quote:
 Originally Posted by OldGuy My gut feeling is you could make the pieces identical in area so it would not matter what measure you used. If this is not possible, there is no correct answer for the "best measure." Three obvious choices are: max area - min area, variance (or std dev), and sum of the 55 absolute differences.
I can suggest one measure that has the advantage of simplicity, even if it's not the 'best'. Difference between the area of the largest piece and the area of the smallest piece.
#44
06-01-2010, 10:53 AM
 Wendell Wagner Charter Member Join Date: Jul 1999 Location: Greenbelt, Maryland Posts: 11,190
The best anthologies of Martin Gardner's works that are relevant to this thread are The Colossal Book of Short Puzzles and Problems and The Colossal Book of Mathematics. There are a lot of other anthologies of his works, but they are shorter and not quite as consistently good. The two I list collect the best of the Mathematical Games columns.
#45
06-01-2010, 10:54 AM
 Exapno Mapcase Charter Member Join Date: Mar 2002 Location: NY but not NYC Posts: 22,118
#46
06-01-2010, 01:43 PM
 Sparky812 Member Join Date: Aug 1999 Location: Great White North Posts: 2,732
Quote:
 Originally Posted by Chronos I can get 16 congruent pieces with 4 cuts. First, cut along the diameter. Then, take one half and stack it on top of the other half, and make the second cut along the midline. For the third and fourth cuts, stack and cut on the midline again.
Quote:
 Originally Posted by Rigamarole Again, you're making a 2D exercise into a 3D one. There's no such thing as "stacking" things on top of each other in a two dimensional plane.

Yeah that and the first sentence of the OP states:

Quote:
 ...with 4 straight cuts
and the fifth sentence states:

Quote:
 Only condition specified is 4 straight cuts, not size of pieces.

Last edited by Sparky812; 06-01-2010 at 01:44 PM.
#47
06-01-2010, 01:51 PM
 Chronos Charter Member Join Date: Jan 2000 Location: The Land of Cleves Posts: 50,758
Yeah, but my cuts are straight. What's the problem?
#48
06-01-2010, 02:26 PM
 iamthewalrus(:3= Guest Join Date: Jul 2000
Quote:
 Originally Posted by Rigamarole Again, you're making a 2D exercise into a 3D one. There's no such thing as "stacking" things on top of each other in a two dimensional plane.
Chronos' solution works in the 2d plane, as well. Just rearrange the pieces in the plane so that one straight cut goes through all of them on each cut.
#49
06-01-2010, 04:31 PM
 Peter Morris Charter Member Join Date: Apr 2003 Location: ___\o/___(\___ Posts: 8,761
Quote:
 Originally Posted by Sparky812 Seriously!? Honestly, why do some people insist on inventing exceptions and modifying the question and/or the definitions of the terms to suit their own ridiculous answers?
Because that's literally "thinking outside the box"
#50
06-01-2010, 09:17 PM
 Meow Max Guest Join Date: Mar 2010
Ahah! You are all missing out on another interpretation of the puzzle. "Cuts" can be a verb, as well as a noun. If I interpret "4 straight cuts" as a verb, I can slice the pizza (without folding it, or making cuts in 3D) into as many arbitrary pieces as I would like...
... by using a multi-bladed knife with parallel blades!

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