# No fair! You got more than me!

December 11, 2009 2:30 AM Subscribe

Way to overthink a plate of ~~beans~~ pizza.

I know nothing about maths, but love the effort that went into this.

I know nothing about maths, but love the effort that went into this.

Hmm... weird... here's the same story in a different paper then.

posted by jonathanstrange at 3:39 AM on December 11, 2009

posted by jonathanstrange at 3:39 AM on December 11, 2009

Heaven help these poor souls if they ever try their hand at pizza in Chicago where they cut their round pizzas into squares.

posted by caddis at 4:12 AM on December 11, 2009 [1 favorite]

posted by caddis at 4:12 AM on December 11, 2009 [1 favorite]

*Heaven help these poor souls if they ever try their hand at pizza in Chicago where they cut their round pizzas into squares.*

If I am ever arrested for aggravated assault, chances are good that it will be in direct response to this type of pizza-slicing. How in God's name am I supposed to eat the blasted thing? Don't even talk to me about forks. Pizza is meant to be primitive hand-to-mouth food, dammit. Don't go cutting in such a way that makes keeping everything on top like trying to balance a handful of marbles on a sheet of copy paper. Where is the

*structure*, I ask you, where is the

*utility*, the

*design*-- where, when you get right down to it, is the crust?

posted by WidgetAlley at 4:21 AM on December 11, 2009 [3 favorites]

One way to cut more fairly/accurately: get a larger pizza cutter.

posted by MuffinMan at 4:25 AM on December 11, 2009

posted by MuffinMan at 4:25 AM on December 11, 2009

After the "overthink", "boffins" and pizza puns, I was expecting a really awful, anti-intellectual, lolmathematicians article. But this was actually very good. And I'm also surprised this problem is so hard. Lemme just draw it on the board here...

posted by DU at 4:25 AM on December 11, 2009

posted by DU at 4:25 AM on December 11, 2009

Here are two other boffins dealing with the same problem.

posted by mmoncur at 5:01 AM on December 11, 2009 [1 favorite]

posted by mmoncur at 5:01 AM on December 11, 2009 [1 favorite]

It seems to me (and whenever someone says 'It seems to me...' inevitably what follows is wrong) that if even numbered cuts are easily solved, then odd cuts are easily solved by ignoring the last cut.

Example- 2 cuts divide the pie into 4 pieces A,B,C,D clockwise. The even method says you alternate- one person takes A & C and the other takes B & D, and this is even. Now if you make one more cut, and it goes through B & D, for instance, then you get (clockwise) A, B1, B2, C, D1, D2. Why not just ignore the last cut and give one person A & C and the other one the Bs & Ds?

There's an ingenious proof to this, but it's too small to fit in the comment box.

posted by MtDewd at 5:01 AM on December 11, 2009 [2 favorites]

Example- 2 cuts divide the pie into 4 pieces A,B,C,D clockwise. The even method says you alternate- one person takes A & C and the other takes B & D, and this is even. Now if you make one more cut, and it goes through B & D, for instance, then you get (clockwise) A, B1, B2, C, D1, D2. Why not just ignore the last cut and give one person A & C and the other one the Bs & Ds?

There's an ingenious proof to this, but it's too small to fit in the comment box.

posted by MtDewd at 5:01 AM on December 11, 2009 [2 favorites]

SBS is the best thing free-to-air on Australian TV and my personal saviour during the Great Adolescent Depression of 1991. But their web-ness is a shamozzle.

posted by evil_esto at 5:13 AM on December 11, 2009

posted by evil_esto at 5:13 AM on December 11, 2009

Heaven help these poor souls if they ever try their hand at pizza in Chicago where they cut their round pizzas into squares.

Heaven help these poor souls if they ever try their hand at pizza in Chicago where they cut their round pizzas into squares.

ABOMINATION

If you have a rectangular pizza, it is cut into squares.

If you have a round pizza, it is cut into wedges.

That is how it is done, those are the rules.

posted by louche mustachio at 5:14 AM on December 11, 2009 [4 favorites]

*How in God's name am I supposed to eat the blasted thing?*

You eat it, in the company of friends, with your dominant hand. Per preference, you may carry a drink or napkin in the other hand.

*Don't even talk to me about forks.*

Why would I? Forks are for deep-dish.

*Don't go cutting in such a way that makes keeping everything on top like trying to balance a handful of marbles on a sheet of copy paper.*

Chicagoans avoid this effect by using the correct amount of sauce. When in the proper proportion, the strata fuse in a way that allows the whole to retain its integrity in smaller quantities.

*Where, when you get right down to it, is the crust?*

At the bottom of the pizza, lending bite and structure. If it's so limp that you can fold it up over the cheese, something's gone monstrously wrong.

posted by Iridic at 5:15 AM on December 11, 2009

*odd cuts are easily solved by ignoring the last cut*

Except we're talking about the situation where there is an equal angle between every cut. Not the situation where you've already cut it into four 90 degree pieces, and then you add another cut through the middle of two of them, because that leaves you with four 45 degree pieces and two 90 degree pieces.

posted by chrismear at 5:19 AM on December 11, 2009 [1 favorite]

*If it's so limp that you can fold it up over the cheese, something's gone monstrously wrong.*

I suspect this is the problem, right here. And, to be fair, I have never had genuine Chicago pizza-- my experience is limited to Tennessee's rather inferior fare when someone, for some inexplicable reason, decided it would be a Good Idea to cut it into squares. Several pizza delivery boys will never know how close they came to an extremely slow and painful death-- they're just lucky I opened the box

*after*they'd left.

posted by WidgetAlley at 5:25 AM on December 11, 2009

*Except we're talking about the situation where there is an equal angle between every cut.*

I didn't see anything about equal angles. If there are equal angles then the cuts probably go through the center, and there's no problem. I assume one is making random cross-pizza cuts. If there are 3 cuts making 6 pieces, you pick 1 of them, combine the pieces on either side of the cut, and you have a 2-cut problem.

posted by MtDewd at 5:49 AM on December 11, 2009

The answer to problems of this kind is psychological, not mathematical. When I was young and I had to divide food with a sibling, the procedure was simple: one cuts, the other takes first pick. Would it work with larger parties? Sure - just ask the waiter to bring down a pizza slice and appoint one person to do the cutting, in the knowledge that they'll get the last slice. It will then be in their interest to cut the damn pizza as evenly as possible. Self-interest trumps calculus and arithmetic and geometry, etc. every time.

posted by degreezero at 5:49 AM on December 11, 2009 [4 favorites]

posted by degreezero at 5:49 AM on December 11, 2009 [4 favorites]

*I didn't see anything about equal angles.*

Look again.

*If there are equal angles then the cuts probably go through the center...*

Ah, the venerable Method of Probably.

*I assume one is making random cross-pizza cuts.*

Math problems are always easy to solve when you can assume the problem says what you want.

*When I was young and I had to divide food with a sibling, the procedure was simple: one cuts, the other takes first pick. Would it work with larger parties?*

That's what I thought this was going to be at first. An N-person algorithm for fairly cutting a cake. And I was going to call double on these researchers.

posted by DU at 5:55 AM on December 11, 2009 [1 favorite]

*If it's so limp that you can fold it up over the cheese, something's gone monstrously wrong.*

Nah.

posted by Bernt Pancreas at 6:20 AM on December 11, 2009

DU- thanks, I knew I was missing something.

The waiter is harried enough to cut the pizza off center, but can still manage to have all the edge-to-edge cuts crossing at a single point, and with the same angle between adjacent cuts.

posted by MtDewd at 6:45 AM on December 11, 2009

The waiter is harried enough to cut the pizza off center, but can still manage to have all the edge-to-edge cuts crossing at a single point, and with the same angle between adjacent cuts.

posted by MtDewd at 6:45 AM on December 11, 2009

This totally ignores the vital crust:center ratio. The center is most likely to have more sauce and cheese. Even if the pieces have the same square-inchage, some will have an imbalance of toppings, cheese, sauce and crust.

Stuffed crust is an abomination and will not be considered as a response to this problem.

The pizza of my youth has thin chewy crust and is cut in squares. I don't know if it's so good because I grew up with it and therefore assume that's what pizza should be, but I'm continuing my research with uncommon dedication.

posted by theora55 at 7:19 AM on December 11, 2009

Stuffed crust is an abomination and will not be considered as a response to this problem.

The pizza of my youth has thin chewy crust and is cut in squares. I don't know if it's so good because I grew up with it and therefore assume that's what pizza should be, but I'm continuing my research with uncommon dedication.

posted by theora55 at 7:19 AM on December 11, 2009

A tried-and-true freshman dorm guide to equitable pizza distribution. Upon reflection, it seems that the key to making it work was the introduction of another variable into the equation: weed.

posted by Balonious Assault at 7:21 AM on December 11, 2009

posted by Balonious Assault at 7:21 AM on December 11, 2009

Bah we should just call this Euclidian pizza mathematics with its obvious bias towards round pizzas....

I reject any solution that requires the pizza be round by definition... What about the equal distrabution of oblong, square/rectangular or n-dimensional pizzas?

posted by Nanukthedog at 7:42 AM on December 11, 2009

I reject any solution that requires the pizza be round by definition... What about the equal distrabution of oblong, square/rectangular or n-dimensional pizzas?

posted by Nanukthedog at 7:42 AM on December 11, 2009

Right. Well if they're so damned clever they can figure this one out next.

posted by contessa at 7:43 AM on December 11, 2009

posted by contessa at 7:43 AM on December 11, 2009

Some people got a lot of time on their hands. Poor souls.... they'll never get laid.

oh, wait - I sleep with a mathematician! What is

posted by Neekee at 7:52 AM on December 11, 2009

oh, wait - I sleep with a mathematician! What is

*wrong*with me???posted by Neekee at 7:52 AM on December 11, 2009

*I sleep with a mathematician! What is wrong with me???*

You've got nerd cooties now. That stuff doesn't wash off.

posted by contessa at 7:59 AM on December 11, 2009

*Right. Well if they're so damned clever they can figure this one out next.*

Group B should ask that Pizza #2 be made with the meat confined to one half of the pie.

posted by Iridic at 8:02 AM on December 11, 2009

Caddis. I've had dozens of round Chicago-style pizzas, and none of those were cut into squares, as I remember.

When I grew up in Indiana, many of our round pizzas were cut into squares, including Pizza King (as linked by sciurus). When I was in college the first time around, we often ordered pizza from that very Pizza King, especially freshman year in my dorm.

Our distribution went as follows: I bought the pizza and ate as much as I could. Roommate Dan would get my leftovers. Rock and Dave across the hall would get his leftovers. RA Dave down the hall would get their leftovers. It was pretty thin, though, so it didn't tend to get that far.

Back to the square cutting business: I think it's fairly well known that

Sorry if I beanplated this a little much. I'm a pizza geek.

posted by The Potate at 8:07 AM on December 11, 2009

When I grew up in Indiana, many of our round pizzas were cut into squares, including Pizza King (as linked by sciurus). When I was in college the first time around, we often ordered pizza from that very Pizza King, especially freshman year in my dorm.

Our distribution went as follows: I bought the pizza and ate as much as I could. Roommate Dan would get my leftovers. Rock and Dave across the hall would get his leftovers. RA Dave down the hall would get their leftovers. It was pretty thin, though, so it didn't tend to get that far.

Back to the square cutting business: I think it's fairly well known that

*Midwestern*-style pizza is commonly cut into squares. This is, of course, excluding Chicago Style pizza which tends to be very very thick.Sorry if I beanplated this a little much. I'm a pizza geek.

posted by The Potate at 8:07 AM on December 11, 2009

On second though, I rescind my beanplating apologies above. I forgot what this thread is about.

posted by The Potate at 8:09 AM on December 11, 2009

posted by The Potate at 8:09 AM on December 11, 2009

Oh God! I and a couple of friends who are also MeFites are meeting for pizza and beer tonight. We'll spend all night drawing diagrams and talking about this. We may never get around to watching Primer.

posted by Kattullus at 8:21 AM on December 11, 2009

posted by Kattullus at 8:21 AM on December 11, 2009

Many boffins died to bring us this information.

posted by albrecht at 8:27 AM on December 11, 2009 [2 favorites]

posted by albrecht at 8:27 AM on December 11, 2009 [2 favorites]

i don't know about anyone else, but the cross cut pizza (i've heard it referred to as 'chicago style') raises a new questiion: do you prefer innies or outies?

posted by lester's sock puppet at 8:31 AM on December 11, 2009

posted by lester's sock puppet at 8:31 AM on December 11, 2009

You wanna know how to split a pizza? They pull a knife, you pull a pizza slicer. He takes one of your big slices from the tray, you take one of his from his plate.

posted by kirkaracha at 8:36 AM on December 11, 2009 [2 favorites]

*That's*the*Chicago*way! And that's how you split a pizza. Now do you want to do that? Are you ready to do that?posted by kirkaracha at 8:36 AM on December 11, 2009 [2 favorites]

My wife is from Illinois, therefore I am an expert on the matter: When you want a round pizza cut into squares, it is ordered as "party pizza" or "party cut". This is because the party pizza is meant for small children's parties. If you are ordering party pizza as adults, you might as well be wearing a mickey mouse letterman jacket and ""Cars"" shoes that light up.

posted by boo_radley at 9:30 AM on December 11, 2009

posted by boo_radley at 9:30 AM on December 11, 2009

The only really good pizza that I had in NYC was Chicago-style pizza. If you want something that you can fold lengthwise into a little point, make a fucking paper airplane for Christ's sake.

posted by Halloween Jack at 9:40 AM on December 11, 2009

posted by Halloween Jack at 9:40 AM on December 11, 2009

Wait, what?

That's obviously false. Draw a line top to bottom starting an inch to the right of the pizza's center. Make all your cuts parallel to that line and on its right. Ta-da.

posted by kenko at 9:40 AM on December 11, 2009

*if you use 5, 9, 13, 17... cuts, the person who gets the centre ends up with less.*That's obviously false. Draw a line top to bottom starting an inch to the right of the pizza's center. Make all your cuts parallel to that line and on its right. Ta-da.

posted by kenko at 9:40 AM on December 11, 2009

*The only really good pizza that I had in NYC was Chicago-style pizza. If you want something that you can fold lengthwise into a little point, make a fucking paper airplane for Christ's sake.*

BURN THE WITCH!

posted by contessa at 9:54 AM on December 11, 2009

How many of us are going to have pizza for dinner tonight?

posted by sciurus at 10:06 AM on December 11, 2009

posted by sciurus at 10:06 AM on December 11, 2009

*This is because the party pizza is meant for small children's parties.*

And children are famous for really loving to have hot cheese and sauce escape the surface of the pizza onto their fingers.

*Make all your cuts parallel to that line and on its right.*

INTERSECTION FAIL

posted by DU at 10:17 AM on December 11, 2009

*That's obviously false. Draw a line top to bottom starting an inch to the right of the pizza's center. Make all your cuts parallel to that line and on its right. Ta-da.*

This is outside the scope of the problem definition. All cuts must pass through a single point.

posted by mhum at 10:30 AM on December 11, 2009

jonathanstrange's link leads to this helpful image.

Lord help these people should they ever decide to split fries.

posted by mazola at 11:00 AM on December 11, 2009 [1 favorite]

Lord help these people should they ever decide to split fries.

posted by mazola at 11:00 AM on December 11, 2009 [1 favorite]

I've been sitting on a similar geometry problem for years. Whenever I'm bored I try to solve it:

Between two people there is one slice of pizza with angle theta and radius r. At what distance from the centre h can you make a straight width-wise incision (edge-to-edge, not crust-to-center) such that both people get an equal share?

posted by spamguy at 11:48 AM on December 11, 2009

Between two people there is one slice of pizza with angle theta and radius r. At what distance from the centre h can you make a straight width-wise incision (edge-to-edge, not crust-to-center) such that both people get an equal share?

posted by spamguy at 11:48 AM on December 11, 2009

*Between two people there is one slice of pizza with angle theta and radius r. At what distance from the centre h can you make a straight width-wise incision (edge-to-edge, not crust-to-center) such that both people get an equal share?*

Seems like you're just asking for the dimensions of the isosceles triangle with half the area of the circular sector.

Area of isosceles triangle with side length h and angle theta = 1/2 * h^2 * sin(theta)

Area of sector (theta in radians) = r^2 * theta

So set one equal to half the other, and h = sqrt(r^2 * theta/2sin(theta)). Did I miss something?

posted by albrecht at 11:56 AM on December 11, 2009

MetaTalk:

posted by l33tpolicywonk at 11:56 AM on December 11, 2009

*Upon reflection, it seems that the key to making it work was the introduction of another variable into the equation: weed.*posted by l33tpolicywonk at 11:56 AM on December 11, 2009

*Area of sector (theta in radians) = r^2 * theta*

Sorry, that's r^2 * theta/2, of course. But that's what I used anyway.

posted by albrecht at 11:58 AM on December 11, 2009

*Between two people there is one slice of pizza with angle theta and radius r. At what distance from the centre h can you make a straight width-wise incision (edge-to-edge, not crust-to-center) such that both people get an equal share?*

Assuming theta < π (because you said "one slice"):

The area of the slice is πr

^{2}* θ/2π = r

^{2}θ/2.

The base of the triangle you cut off from the slice is b = 2h*tan(θ/2). The area of that triangle is bh/2 = h

^{2}*tan(θ/2).

You want this to be half the area of the entire slice, so h

^{2}*tan(θ/2) = r

^{2}θ/4.

Is there a neat way to reduce that? I don't know my trig identities well enough to say.

posted by DU at 12:05 PM on December 11, 2009 [1 favorite]

*This is outside the scope of the problem definition. All cuts must pass through a single point.*

And the point can't be on the edge, I take it?

posted by kenko at 12:08 PM on December 11, 2009

Regularly in college, 5 of us (girls) would order a large pizza and have them cut it in 10 slices (instead of the traditional 8) so we didn't have to deal with divvying it up later. Luckily the pizza places in my college town were very forgiving of our silly request.

Also - sometimes I like a pizza slice to be a little skinnier than the others. Like, hey I want another slice but not a full slice? Oh yes, this tiny slice will do very nicely, thanks inaccurate pizza slicer person!

posted by sararah at 12:55 PM on December 11, 2009 [1 favorite]

Also - sometimes I like a pizza slice to be a little skinnier than the others. Like, hey I want another slice but not a full slice? Oh yes, this tiny slice will do very nicely, thanks inaccurate pizza slicer person!

posted by sararah at 12:55 PM on December 11, 2009 [1 favorite]

Conclusion: they have seriously bad pizza slicers in Shreveport.

posted by mazola at 1:32 PM on December 11, 2009

posted by mazola at 1:32 PM on December 11, 2009

Actually, I still have trouble with this five-cut thing. Suppose you have the circle x

posted by kenko at 5:02 PM on December 11, 2009

^{2}+y^{2}= 16, and you make your first two cuts the lines x = √12 and y = √12, so that the point through which all the cuts travel is (√12, √12). Now make three more cuts in the upper right of the circle—lines with negative slope that pass through (√12, √12). Surely you end up with a bigass piece that includes the origin. So there must be more to the problem statement than I'm getting.posted by kenko at 5:02 PM on December 11, 2009

Wait, what? You're supposed to slice up and

posted by Zinger at 5:06 PM on December 11, 2009

*share*pizza?posted by Zinger at 5:06 PM on December 11, 2009

*Now make three more cuts in the upper right of the circle—lines with negative slope that pass through (√12, √12).*

Are these all equi-angular?

posted by DU at 5:50 PM on December 11, 2009

albrecht:

Many boffins

posted by chrismear at 6:22 PM on December 11, 2009 [2 favorites]

*Many boffins died to bring us this information.*Many boffins

**dined**to bring us this information.posted by chrismear at 6:22 PM on December 11, 2009 [2 favorites]

kenko:

The problem statement also requires that the cuts are equi-angular around their common point:

posted by mhum at 6:49 PM on December 11, 2009

*Actually, I still have trouble with this five-cut thing. ... So there must be more to the problem statement than I'm getting.*The problem statement also requires that the cuts are equi-angular around their common point:

For a positive integer N, divide a pizza into 2N slices by choosing an arbitrary point P in the pizza and making N straight cuts through P, the cuts meeting to form 2N equal angles.Without this stipulation, the problem becomes less interesting, as in your example. If the pizza cutter isn't even trying to make the cuts fair, then pretty much anything can happen. If the pizza cutter is at least trying to make the angle of each slice equal, then a giant slice containing the origin will also result in two pretty big slices on either side of the giant slice.

posted by mhum at 6:49 PM on December 11, 2009

DU:

I don't see an obvious closed-form solution to this equaition. However, we can go a little bit further. For the sake of simplicity, let r = 1 so that h can be interpreted as the fraction of the radius where you want to make the cut. Plotting out sqrt(θ/4tan(θ/2)) we get this graph, which shows that h ranges from just over 0.7 to just over 0.6 as θ ranges from 0 to Pi/4 (h actually ranges from 1/sqrt(2) to sqrt(Pi/8)).

We can get numerical solutions for the most common slice sizes. If the pizza was originally sliced into 8 pieces, h = 0.688.... If it was 10 slices, h = 0.695.... And if the pizza was sliced into 6 pieces, h = 0.673...

All of this indicates that, from a practical standpoint, spamguy's problem is solved if you cut the slice about two-thirds of the way away from the pointy end, more or less.

posted by mhum at 7:09 PM on December 11, 2009

*You want this to be half the area of the entire slice, so h*

Is there a neat way to reduce that? I don't know my trig identities well enough to say.^{2}*tan(θ/2) = r^{2}θ/4.Is there a neat way to reduce that? I don't know my trig identities well enough to say.

I don't see an obvious closed-form solution to this equaition. However, we can go a little bit further. For the sake of simplicity, let r = 1 so that h can be interpreted as the fraction of the radius where you want to make the cut. Plotting out sqrt(θ/4tan(θ/2)) we get this graph, which shows that h ranges from just over 0.7 to just over 0.6 as θ ranges from 0 to Pi/4 (h actually ranges from 1/sqrt(2) to sqrt(Pi/8)).

We can get numerical solutions for the most common slice sizes. If the pizza was originally sliced into 8 pieces, h = 0.688.... If it was 10 slices, h = 0.695.... And if the pizza was sliced into 6 pieces, h = 0.673...

All of this indicates that, from a practical standpoint, spamguy's problem is solved if you cut the slice about two-thirds of the way away from the pointy end, more or less.

posted by mhum at 7:09 PM on December 11, 2009

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