# Cutting a pizza into largest number of pieces **URL:** \ **Category:** Factual Questions\ **Created:** [May 28, 2010, 11:35am UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060 "2010-05-28T11:35:08Z")\ **Posts on this page:** 16\ **Page:** 3
**Author:** ![Ostrya](https://avatars.discourse-cdn.com/v4/letter/o/35a633/32.png) [@Ostrya](https://boards.straightdope.com/u/Ostrya)\ **Post date:** [June 1, 2010, 11:34am UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/41 "2010-06-01T11:34:41Z")
> [@astro](#): > > [Here’s a visual of the 11 slice solution](http://mathforum.org/library/drmath/view/57858.html) I needed this illustration to understand. Thank you, **astro**! ---
**Author:** ![Khaki\_Campbell](https://avatars.discourse-cdn.com/v4/letter/k/9de053/32.png) [@Khaki\_Campbell](https://boards.straightdope.com/u/Khaki_Campbell)\ **Post date:** [June 1, 2010, 1:21pm UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/42 "2010-06-01T13:21:39Z")
> [@astro](#): > > [Here’s a visual of the 11 slice solution](http://mathforum.org/library/drmath/view/57858.html) Now a more interesting problem would be how to do those four cuts so that the areas of the slices are the least unequal. ---
**Author:** ![chrisk](https://avatars.discourse-cdn.com/v4/letter/c/6de8d8/32.png) [@chrisk](https://boards.straightdope.com/u/chrisk)\ **Post date:** [June 1, 2010, 1:27pm UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/43 "2010-06-01T13:27:10Z")
> [@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. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [June 1, 2010, 2:53pm UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/44 "2010-06-01T14:53:48Z")
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. ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [June 1, 2010, 2:54pm UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/45 "2010-06-01T14:54:14Z")
[Martin Gardner memorial thread](http://boards.straightdope.com/sdmb/showthread.php?t=564214&highlight=martin+gardner). ---
**Author:** ![Sparky812](https://avatars.discourse-cdn.com/v4/letter/s/eb9ed0/32.png) [@Sparky812](https://boards.straightdope.com/u/Sparky812)\ **Post date:** [June 1, 2010, 5:43pm UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/46 "2010-06-01T17:43:38Z")
> [@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. > [@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: > [@](#): > > …with 4 straight cuts and the fifth sentence states: > [@](#): > > Only condition specified is 4 straight cuts, not size of pieces. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 1, 2010, 5:51pm UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/47 "2010-06-01T17:51:55Z")
Yeah, but my cuts are straight. What’s the problem? ---
**Author:** ![iamthewalrus\_3](https://avatars.discourse-cdn.com/v4/letter/i/258eb7/32.png) [@iamthewalrus\_3](https://boards.straightdope.com/u/iamthewalrus_3)\ **Post date:** [June 1, 2010, 6:26pm UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/48 "2010-06-01T18:26:05Z")
> [@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. ---
**Author:** ![Peter\_Morris](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/peter_morris/32/359_2.png) [@Peter\_Morris](https://boards.straightdope.com/u/Peter_Morris)\ **Post date:** [June 1, 2010, 8:31pm UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/49 "2010-06-01T20:31:39Z")
> [@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” ---
**Author:** ![Meow\_Max](https://avatars.discourse-cdn.com/v4/letter/m/59ef9b/32.png) [@Meow\_Max](https://boards.straightdope.com/u/Meow_Max)\ **Post date:** [June 2, 2010, 1:17am UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/50 "2010-06-02T01:17:46Z")
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! ---
**Author:** ![Siam\_Sam](https://avatars.discourse-cdn.com/v4/letter/s/d78d45/32.png) [@Siam\_Sam](https://boards.straightdope.com/u/Siam_Sam)\ **Post date:** [June 2, 2010, 2:41am UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/51 "2010-06-02T02:41:08Z")
I prefer cutting my pizzas into 4 slices instead of 8, because I can’t eat 8 slices. ---
**Author:** ![Superhal](https://avatars.discourse-cdn.com/v4/letter/s/b4bc9f/32.png) [@Superhal](https://boards.straightdope.com/u/Superhal)\ **Post date:** [June 2, 2010, 3:40am UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/52 "2010-06-02T03:40:45Z")
This thread still going? Imho, the answer as given by Gardner is correct. Pizza, cut horizontally, would not be pizza anymore. On the other hand, other things can be cut horizontally and still be what it is, e.g. cake. ---
**Author:** ![Sparky812](https://avatars.discourse-cdn.com/v4/letter/s/eb9ed0/32.png) [@Sparky812](https://boards.straightdope.com/u/Sparky812)\ **Post date:** [June 2, 2010, 11:52am UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/53 "2010-06-02T11:52:51Z")
> [@Chronos](#): > > Yeah, but my cuts are straight. What’s the problem? I misread your solution, my apologies.:smack: > [@iamthewalrus\_3](#): > > **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. This does makes sense, you could get 16 pieces and satisfy all the conditions.🙂 ---
**Author:** ![Indian](https://avatars.discourse-cdn.com/v4/letter/i/c0e974/32.png) [@Indian](https://boards.straightdope.com/u/Indian)\ **Post date:** [June 2, 2010, 12:59pm UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/54 "2010-06-02T12:59:11Z")
Most likely, Martin Gardner meant it as a problem to be solved without rearranging the pieces . ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [June 3, 2010, 2:02am UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/55 "2010-06-03T02:02:22Z")
> [@chrisk](#): > > 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. I think you mean the same thing as my first suggestion, max area - min area, unless I don’t quite understand what you’re saying. But I’m still interested in the problem how to make the pieces as nearly equal as possible and can zero difference be achieved. Anyone have any detailed thoughts on that? ---
**Author:** ![koryphos](https://avatars.discourse-cdn.com/v4/letter/k/a183cd/32.png) [@koryphos](https://boards.straightdope.com/u/koryphos)\ **Post date:** [June 7, 2010, 1:50am UTC](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060/56 "2010-06-07T01:50:54Z")
If it were possible to cut the pie into eleven equal pieces, each line (slice) would divide the pie into integer-sized pieces. (There can be no fractions because it is impossible to recombine the fractions into a whole piece.) So let’s divide an imaginary pie having an area of 11 into four pieces, using two of our four slices. We have two slices remaining. Any single region may be sliced into four pieces by having those two slices intersect within that region. So the largest piece that we can have at this stage has area 4, if we want to have eleven equal pieces at the end. However, our two slices can only slice a single region into 4 pieces – there can’t be two regions with area 4 at this stage. (Slicing into 4 requires that the two remaining slices intersect, and in flat 2-d geometry, two non-collinear lines can only intersect at a single point.) So the groups of four integers that total 11, having a maximum value of 4 for a single entry, are: 4, 3, 3, 1 4, 2, 2, 3 and 3,3,3,2 These are the possible areas of our pie pieces at this point. Looking at the last possibility - 3,3,3,2 - it should be obvious that this can’t be sliced into 11 using only two slices. One of the (straight line) slices would have to intersect all four pieces, bisecting the smallest (area 2) and removing exactly 1 from each of the other 3 areas. This is impossible. The first option - 4,3,3,1 - is similarly flawed. Our single (straight) slices would have to intersect each of the three largest pieces twice, and would have to intersect in the largest piece, dividing it into 4 equal sized pieces. Geometrically, this won’t work. Straight lines cannot accomplish this subdivision. Similarly, the final option - 4, 2, 2, 3 - requires that the lines which split one area into 3 equal parts somehow intersect to split the region on the opposite side of the circle into 4 equal parts, while each at the same time transsecting an area of 2 exactly in half. It’s not rigorous … but when you sketch the little diagrams for each of them, it should be obvious that division into 11 equal parts with 4 straight lines is impossible. As to the best-case division, I’m not certain how to approach it. [Previous page](https://boards.straightdope.com/t/cutting-a-pizza-into-largest-number-of-pieces/541060.md?page=2)