It never occurred to me that those were pineapple slices, although now that you say it, they clearly are. I just saw them as yellow backgrounds so the numbers would be more visible.
Truly a mathematician's sort of pizza; reducing it to a bare minimum set of ingredients that have been known to make up a pizza (even if they're in an unconventional form), calling it a previously solved problem, and not even bothering to bake it :p
Original pizza didn't have tomato. Tomato was not known in Italy (or anywhere outside America) until the early 1500's, after the Spanish discovery / conquest. Pizza has existed in Italy since before there was a Italy, or a Roman empire.
Now that we know that there is a cut pattern that makes 1+n(n+1)/2 pieces out of n cuts, we could ask about fairness in several ways: - What is the pattern that makes the smallest piece the biggest? - What is the pattern that makes the biggest piece the smallest? - What is the pattern that minimizes the variance of the area of the pieces? - Are some of those questions equivalent?
The thing to remember is that you don't want to come even close to parallel lines, because each line must intersect each other line. So change the angle the smallest possible amount on each cut, staying just shy of the limit an infinitely small (but not zero) amount. Once you pass 180 degrees you get a parallel line that cannot possibly cut all lines, so your range of motion is half a circle. So between (exclude the limits) 0 and 180 degrees you'll be making cuts with the two outermost cuts as close to perpendicular as you can get. The closer the first cut for both of these is to the far edge of the circle, the less long their adjoining pieces are. And since we're talking about cuts with an angle, we're talking triangular pieces for these, two so the longer they get, the bigger they get. If you plot your intersections you should get a half circle with anywhere between zero or infinite surface area. of intersections. So the question becomes: When I draw half a circle inside a circle, how close away can the start and beginning of that half circle be from the edge? The closer they are, the smaller their pieces. Then again, that's only true if the half circle of intersections is for a smaller circle than the pizza. If the intersection half-circle is infinitely bigger, the intersection half-circle is effectively a straight line, so the closer the line is to the middle, the fairer the cuts. So depending on the difference between the pizza's radius and the intersection half-circle's radius, the best option is either as close to the edge or as far from the edge. I wonder what the middle point would be where the pizza and intersection radius are the same. This is probably basic, but math was a long time ago for me.
@@bartolhrg7609After you make all n cuts you see what's the size of the smallest piece. Now you consider if you can cut differently such that the smallest piece is larger.
Love how Brady jumped right in for -1/12, implying that if you take this algorithm to infinity you wind up with a pizza cut into 11/12 regions. Get the Sixty Symbols squad on this, I smell a Nobel prize for topological defects in higher dimensional pizzas!
Well that proofs then, if you cut through the middle, that infinity is the same as 6 cuts, since that leaves you 12 regions. With the lazy cutting technique infinity using the -1/12 answer is impossible since you either have 11 regions or 16, but no 12. 8-)
@@Nemelis0 it's not that you get 11 of 12 regions, it's that you get 11/12ths of "the concept of a region". Think of it in terms of countries, if I cut a country in half I wind up with two countries, I can move the line wherever but I still get out 2 whole units. If I cut an infinite number of times though, I get 11/12ths of a country, 11/12ths of whole indivisible unit.
I'm afraid you need a blade or pizza roller that can cut over 1.00 efficiency. In other words, it needs to cut into non-parallel non-curved spacial direction while being a curved-space object.
@@Yezpahr @Yezpahr I mean, moving this from the imperfect doughy world of pizza and into the mathematically pure realm of infinitely thin cuts and infinitely flat planes the algorithm still holds. The algorithm is just a greedy method of partitioning a flat region (doesn't have to be a circle) into as many sub-regions as possible by bisecting it over and over. But in math, you can make the cuts arbitrarily close to one another and the regions arbitrarily small but greater than 0 in area.
@@aimeeriverswrong. Pi times z squared (which is the same as z times z) times a is correct. So it of course does not matter how long z is. It can be 1cm or 349388383 lightyears. 😮
@@kennygeheim4230 oh i thought the equation was the diameter times pi (the diameter being 2 times the radius) ... but it is a very long time since i went to school 😂
@@aimeerivers Diameter times pi gives you the circumference. The surface (and then volume) requires the radius squared. Hopefully that clears up the confusion :)
Based on this I now understand that the total size of a cut pizza is 11/12 pizzas. I've weighed the crumbs left on the pan after cutting and believe this to be accurate.
@plackt this is what is beautiful about mathematics, such an unintuitive answer that it seems impossible but then apply it to a real world situation, and the surprising result confirms that what we once thought of as nonsense is indeed innovation.
I have not attempted to learn mathematics or work with mathematics since I last studied it a couple years ago and this is my first time in a long time I have had to think mathematically. It really feels uncanny but awesome how all of this kind of stuff just *makes sense*. Thank you for presenting these kind of simplified versions for people like me to get back into mathematics!
I found an easy way to cut the pizza into the maximal number of pieces from n cuts (I don't know if this is how they do it in the paper or if this is even correct): if n is odd, construct a regular n-gon, and extend all the sides as infinite lines. Then, draw your circle big enough around the n-gon so that all the line intersections are contained. Each line will intersect every other line in the circle, as needed. If n is even, do the same with n+1 and remove a line. Kinda cool!
I bet the dialogue of that teacher with the young Gauss went like this: "Can you calculate the sum of all integers from 1 to 100" - "Ahh, well, fifty-fifty"
After watching this I realized making the least number of pieces is incredibly simple, since you just keep cutting parallel lines to add one strip at a time.
This is a greedy algorithm, you try to maximize the number of new regions in each step, but the greedy approach does not guarantee optimal solution in general, so its use should be justified in a case by case basis, right?
You are right, the greedy algorithm is not always correct, but I think it is proven in the video, that the greedy approach works here, perhaps not explicitly enough 0) f(n) is defined "The maximum number of slices using n cuts" 1) You have to do exactly n-1 cuts before making the nth cut 2) (1) => on the nth cut you can intersect no more than n-1 lines, making no more than n new slices 3) (1) => before nth cut there can be no more than f(n-1) slices 4) (2), (3) => f(n) = f(n-1) + n or less 5) f(n-1) + n slices is achievable by doing the greedy algorithm => f(n) = f(n-1) + n or more (or it would not be maximum) 6) (4), (5) => f(n) = f(n-1) + n (6) states, that the formula at 7:20 is correct and the rest is certainly in the video
2:10 There should be a follow up video on if it’s possible to “Lazy Cut” a pizza into equal area slices. Intuitively, this seems possible with 7 slices and 3 cuts, but with 4 or more cuts, it becomes a lot less clear whether or not it can be fairly divided.
I propose a method to get the most pieces. (5 mins into video) 1) Initial cut 2) Almost parallel but intersects near edge of pie 3) Same thing but intersects the previous one 4) Same thing but intersects the previous two 5) Repeat til you get a dreamcatcher. Your intersection with the first cut changing location slightly each new cut that if linked form a bent chord.
My intuition is that the intermediate value theorem would allow you to make equal areas for any given solution by rotating and translating the cuts, without changing which overlap which.
I highly doubt that it could possibly be fair even for the 3-cut case. An interesting query is how fair can it get for a number of cuts, in terms of narrowing the range of slice areas? Also, how many fair slices can be achieved? (For that, we have the lower bound of 2n for the normal radial cutting method and the upper bound provided here. I don't expect the actual upper bound to be better than linear).
The slices look a lot like life, though. The "normal"/"fair" way is the PC version. This version is how life really is. Mine is the the tinny tiny one in the left corner.
if you don't want the middle triangle from the third cut to be too small, don't make the first two cuts meet near the center! that minimizes the possible size of that triangle!
I would quite enjoy that "pizza" as a snack, despite what haters say. Tortilla bread is delicious. Amazing video by the way. I had taken a hiatus from math, but this reminded me why I love the subject.
2:05 "but obviously this one's quite small and no one wants it" I'd say the size isn't even the biggest issue; after all you could rearrange the cuts to make that middle piece bigger if you want. But it'd still be a bad slice of pizza since it doesn't have any outer crust, and thus cannot be easily picked up to eat without getting toppings all over your hands
real world pizza actually has thickness so you can for example make a cut parallel to the plane of the table, but then the solution to the number of pieces you get with n cuts is (n+1)(n^2-n+6)/6
This is the only numberphile video that I guessed anything. I had a feeling it was how many intersections there were. Im no math expert so that was exciting for me
I had to solve this looong ago when I was a teenager. I did as in the video and I was rightfully happy with it. Another one did better, though somewhat out of the box. Cut the pizza in half, place one half on top of the other, repeat. We'll get 2^n equally sized pieces of pizza.
If you use the whole plane instead of a pizza, then it’s easier to draw the lines so they all intersect each other with no triple points, and then you can draw a circle around all the intersections, and then scale down to the actual size of the pizza.
Also a classic for the whole "any finite sequence is the first n terms of infinitely many other sequences". Complete the sequence: "1,2,4..." you ask people and they invariably say 8, but if were using this, its 7!
My profesor after watching this video. A pizza has a radius of x cm. Derive a formula for the maximum number of straight linear cuts that can be made in the pizza, under the assumption that each piece of the pizza must have an area greater than or equal to 1cm^2. Prove the formula.
5:50 "It's definitely possible [to draw a new line that intersects all previous lines]." Proof: It's (obviously) possible to draw n lines such that they are pairwise non-parallel and that no three lines intersect each other in a common point. In general, not all intersection points will lie inside the pizza, but there's always a larger circle which contains them. Now shrink the entire configuration of lines and that circle in such a manner that the circle coincides with the given pizza. Then the "new" lines have the required properties and cut the pizza into f(n) pieces.
“I like to derive my own formulas” … this is why my grade went from 64% to 98% between my first and second semesters of calculus. We had a giant list of trig integrals to memorize for the first end of term 1, and I can’t memorize at all well. Term two, we learned integration by parts and I could derive them instead! I also did this for many years with the quadratic formula
I was confused for a moment since I vaguely remembered a similar topic in a 3b1b video that results in a different sequence but I realize my mistake now. This sequence is constructed by placing lines through a circle, while the 3b1b sequence was constructed by placing points on the circumference of a circle and drawing lines between those points
Natural follow-up question: what pattern of cuts gives you the biggest small pieces/most even pieces. Like for one and two cuts, the centre is obviously the most even place, but for three cuts, it works better if the first two are not central.
I am a simple man with complicated pleasures. I enjoy math, science and cooking. Watching someone spread raw tomato paste on dough with a trowel then applying pre shredded mozzarella on top is devestating. 3:40 well with the hyperbolic way that you treated that beautiful piece of dough I think you can find a way.
Pausing at 6:47 to think about this going the other direction. If I get N new regions when I have made N cuts this suggests when I made 0 cuts I got 0 new regions. This means -1 cuts should ALSO have 1 region. But that would imply that when I made -1 cuts that I lost a region at that time. So if I had -2 cuts I should have 2 regions. And -3 cuts would have been 4 regions and so on.
If this pizza was truly a 3d pizza, you can get more pieces starting with f(3) f(1) and f(2) are still normal cuts across the surface But where Tom showed f(3) = 7... Instead of cutting along the surface, you can cut along the same plane as the pizza. This doubles the number of slices since it manages to cut every slice in half, yielding f(3) = 8
Mammamia ! 😱 If I were an Italian mathematician (which I'm not), I'd challenge you to find: "The Lazy Way to Cut Christmas Pudding". In 3D, with planes instead of lines. ... and then label the slices of pudding with pieces of tomato or salami. 😝
That's the saddest "pizza" I've ever seen, but Professor guy gets a pass from Italy anyway - because of the interesting content as usual and the strikingly perfect free hand circle 😅
Now the next step... Figuring out algorithm to maximize the cut area (of the final pieces) as much as possible, and then determining the distribution of area across all the cuts and see if it fits a pattern.
You can't make pizza with tortilla but, if there's no choice, mix the tomato purée with smooth peach chutney, garlic and oregano then throw on olives, capers, anchovies and grated emmental cheese. lastly, sprinkle on olive oil and salt before sticking under the grill.
I'd recommend the "lazy AND lase-y" way to cut a pizza. Using an industrial laser would be much less messy... unless you accidentally aimed it across the room of course.
Fun an all, but being lazy means NO MATHS! Slapping down 6 cuts and not caring, and it is actually easier to grid it with 3 lines across then 3 vertical (giving the pizza a slight flick so it wonkily ends up between 45 and 135deg-ish from previous orientation. Result with 6 lines is just 16 pieces, 7 lines gives 20 and it takes but 2 seconds and best of all, you won't get fired. 🍕
There also at least two ways of "beating" these numbers. The messy way would be to fold the pizza multiple times first: in theory you can then get any number of pieces from a single cut, albeit mostly slivers of crust. A second way is to move the cut pieces between cuts. Then each cut can double the number of pieces. Of course the method given is far more interesting... :-)
I had the same thought! Took me a minute to work it out, but I figured out what happened. `f(1) = 2` because `f(1) = f(0) + 1`, and `f(0) = 1`, so `f(1) = 1 + 1`. Tom misspoke at 7:37 saying that `f(1) = 1`, but everything that was written down was correct. The formula `f(1) + 2 + ... + n` is correct, and expanding `f(1)` and `f(0)` at the same time does lead to `1 + 1 + 2 + ... + n`. TL;DR: the math written down is correct, Tom just misspoke.
So you're not allowed to cook in the library, but you can otherwise make a pizza? Also, for completeness, you need to show what happens with an odd number of numbers.
You can tell he doesn't do this very often from the way he squeezes the tube of tomato paste... 😭😭 You ALWAYS squeeze tubes from the closed end to the front, and very gently. Because otherwise you'll create wrinkles in the tube that cannot be straightened out again and that'll make it impossible to empty the tube properly. Hint: this advice works for mustard and toothpaste as well. 😂
It's quite easy to construct a set of line cuts. One cute possibility is... parametrically: line(n, t) := n + t*e^(i*n*🦩) 0° < 🦩 < 90°/🌼, where 🌼 is the number of cuts you want to make. Then just scale down (or up) the whole shenanigan to fit your pizza.
There’s a really nice geometric derivation for that relationship. Take n 1x1 squares in a line. Stack on top of those n - 1 more 1x1 squares, then n - 2, etc. Align them all at the left, so you have what looks like stairs. What is the area of the stairs? If you draw a diagonal down the stairs, you can pretty easily see we have a triangle with base and height of n, so n^2/2 (you can also think of this as half an nxn square), but we are missing n half squares that were cut off by the diagonal. So, f(n) = n^2/2 + n/2 = (n^2 + n)/2 = n(n + 1)/2.
I imagine Gauss' teacher not realising how he did it, so just kept giving him bigger and bigger values of n and Gauss kept coming back seconds later with the answer.
I think this is a beautiful counterpoint to the claim that the sum of all numbers equal -1/12. Because it should be theoretically possible to cut a pizza infinitely many times into less than one piece.
You should do a follow up video where you not only maximize the number of pieces, but you also figure out the cuts that would make the most equal sized pieces. In other words what cuts would result in the smallest difference between the biggest slice and the smallest slice?
I'm reminded of the peg-and-string thing where you start off with a pair of perpendicular axes (more than one axis, not more than one axe) and draw the lines (X,1), (X-1, 2), (X-2,3) and so on to (1, X). With a large enough X, you get the image of a quarter circle made out of many straight lines... however the intersections of each new line conform to this lazy algorithm too.
He keeps saying the first cut doesn't matter. But it does. You want to maximize the size of the pieces you will divide up, I think that means you want first slice to be a diameter cut.
You may just have explained why my Brazilian friends seem to all cut up their pizza in such an irregular way. I'm sure the mafia, camorra, mafia and ndrangheta are just joining forces to avenge the desecration of a national food relic.
in their defense, American style pizza is a WILDLY broad category of food (just think of NY vs Chicago style differences!), so Tom's oversized Lunchables is indeed recognizable as a pizza on our side of the lake. Definitely not a *good* pizza, but recognizably pizza. The tortilla reminded me, though, pizza toppings sans sauce makes a GREAT quesadilla with marinara dipping sauce. Maybe try cooking it first, though, Tom ;p
![[DRAGON BALL LEGENDS REVEALS ＆ STUFF] LEGENDS FESTIVAL 2024 Special Edition Part 1](http://i.ytimg.com/vi/mV5bGIswTtQ/mqdefault.jpg)
Labelling pieces using bits of pineapple is such an obvious comment bait that I'm fully on board with it.
I just realized those were pineapple slices
It never occurred to me that those were pineapple slices, although now that you say it, they clearly are. I just saw them as yellow backgrounds so the numbers would be more visible.
I thought its cheese
I thought they were mini pizzas
Evil was hiding in plain sight. 😮
Truly a mathematician's sort of pizza; reducing it to a bare minimum set of ingredients that have been known to make up a pizza (even if they're in an unconventional form), calling it a previously solved problem, and not even bothering to bake it :p
Every problem is solvable if you bend the definitions enough
@@usernametaken017 How does a mathematician catch a lion?
He steps in the cage and defines the inside to be free.
Original pizza didn't have tomato. Tomato was not known in Italy (or anywhere outside America) until the early 1500's, after the Spanish discovery / conquest. Pizza has existed in Italy since before there was a Italy, or a Roman empire.
All you need is crust and cheese technically
Assume a spherical pizza...
12:35 Missed opportunity to have a "tortilla change" transition in the style of the old "paper change" transition.
daaamn u right😂😂😂
yes
1:00 that IS a good circle!
Pure Zen!
The larger a circle gets, the more difficult it is to draw it nicely. No idea how he managed to do that, unless he had a lot of practice.
Now that we know that there is a cut pattern that makes 1+n(n+1)/2 pieces out of n cuts, we could ask about fairness in several ways:
- What is the pattern that makes the smallest piece the biggest?
- What is the pattern that makes the biggest piece the smallest?
- What is the pattern that minimizes the variance of the area of the pieces?
- Are some of those questions equivalent?
The thing to remember is that you don't want to come even close to parallel lines, because each line must intersect each other line. So change the angle the smallest possible amount on each cut, staying just shy of the limit an infinitely small (but not zero) amount. Once you pass 180 degrees you get a parallel line that cannot possibly cut all lines, so your range of motion is half a circle. So between (exclude the limits) 0 and 180 degrees you'll be making cuts with the two outermost cuts as close to perpendicular as you can get. The closer the first cut for both of these is to the far edge of the circle, the less long their adjoining pieces are. And since we're talking about cuts with an angle, we're talking triangular pieces for these, two so the longer they get, the bigger they get.
If you plot your intersections you should get a half circle with anywhere between zero or infinite surface area. of intersections. So the question becomes: When I draw half a circle inside a circle, how close away can the start and beginning of that half circle be from the edge? The closer they are, the smaller their pieces. Then again, that's only true if the half circle of intersections is for a smaller circle than the pizza. If the intersection half-circle is infinitely bigger, the intersection half-circle is effectively a straight line, so the closer the line is to the middle, the fairer the cuts. So depending on the difference between the pizza's radius and the intersection half-circle's radius, the best option is either as close to the edge or as far from the edge. I wonder what the middle point would be where the pizza and intersection radius are the same. This is probably basic, but math was a long time ago for me.
"Smallest piece the biggest" doesn't make sense. I don't know what you meant by that
@@bartolhrg7609After you make all n cuts you see what's the size of the smallest piece. Now you consider if you can cut differently such that the smallest piece is larger.
@@bartolhrg7609 maximising the size of the smallest piece
@@bartolhrg7609 He means that the smallest piece (out of the set of all pieces) has the largest area it possibly can
Librarian: "You can't cook a pizza in the library!"
Brady and Tom: "Fine, but you won't like what we'll do instead..."
Did they really have to do this in the library?
Never beating the "The English Can't Cook" allegations with this one
This was not allowed to cook ...
I think this was more "won't cook" than "can't cook".
To be fair, Brady's Australian. We can't cook either, which is why we toss shrimp on the barbie or whatever.
@@N.I.R.A.T.I.A.S. shrimp on the what now??
Congrats on 500 likes
That’s the best drawn circle I have ever seen on numberphile
ikr?
I know its not the point of the video at all, but that library is just absolutely beautiful, you have my envy over being able to see it in person.
Love how Brady jumped right in for -1/12, implying that if you take this algorithm to infinity you wind up with a pizza cut into 11/12 regions. Get the Sixty Symbols squad on this, I smell a Nobel prize for topological defects in higher dimensional pizzas!
Well that proofs then, if you cut through the middle, that infinity is the same as 6 cuts, since that leaves you 12 regions. With the lazy cutting technique infinity using the -1/12 answer is impossible since you either have 11 regions or 16, but no 12. 8-)
@@Nemelis0 it's not that you get 11 of 12 regions, it's that you get 11/12ths of "the concept of a region". Think of it in terms of countries, if I cut a country in half I wind up with two countries, I can move the line wherever but I still get out 2 whole units. If I cut an infinite number of times though, I get 11/12ths of a country, 11/12ths of whole indivisible unit.
I'm afraid you need a blade or pizza roller that can cut over 1.00 efficiency. In other words, it needs to cut into non-parallel non-curved spacial direction while being a curved-space object.
@@Yezpahr @Yezpahr I mean, moving this from the imperfect doughy world of pizza and into the mathematically pure realm of infinitely thin cuts and infinitely flat planes the algorithm still holds. The algorithm is just a greedy method of partitioning a flat region (doesn't have to be a circle) into as many sub-regions as possible by bisecting it over and over. But in math, you can make the cuts arbitrarily close to one another and the regions arbitrarily small but greater than 0 in area.
Or they would end up with enough pieces to make two pizzas as big as he original hehe (Banach-Tarski paradox)
Making the world's worst pizza in a library is exactly the kind of insanity I originally subscribed to this channel for
I once had a pizza with bananas and Nutella. I think that one was worse. It tasted as bad as it sounds.
@@skyscraperfan I mean, it should taste fine as long as all ingredients were dessert-y things
@@widmo206 it had nutella on it, therefore it did not taste fine
@@alquinn8576 So you just don't like nutella?
@@widmo206 it should be called sugarella since it is 57% sugar by mass. it is objectively garbage with respect to human metabolism.
7:52 Brady's -1/12th joke love it
Joking about a false statement is a mathematician's favorite passtime
@@mystifoxtech sum_k=1..inf(k) = -1/12 isn't a false statement, it's just the golden nugget of a true statement!
@@rainerzufall42 It's only true for very specific definitions of the words "sum", "equals", and "number"
@@mystifoxtech You didn't get my point, or you don't know the history of this channel well enough!
It wasn’t that funny, really.
Did you know that the volume of a pizza radius "z", thickness "a" equals pi.z.z.a
technically only true when z = 2
@@aimeeriverswrong. Pi times z squared (which is the same as z times z) times a is correct. So it of course does not matter how long z is. It can be 1cm or 349388383 lightyears. 😮
@@kennygeheim4230 oh i thought the equation was the diameter times pi (the diameter being 2 times the radius) ... but it is a very long time since i went to school 😂
@@aimeerivers Diameter times pi gives you the circumference. The surface (and then volume) requires the radius squared. Hopefully that clears up the confusion :)
@@Mechanikatt thank you, that does help!! and now i can always remember the equation thanks to the pi.z.z.a formula!!
Numberphile 2014:
The scientific way to cut a cake.
Numberphile 2024:
The lazy way to cut a pizza.
i thought this sounded familiar...
who
Numberphile 2034:
The lazy way to cut a cake
Numberphile 2044:
The scientific way to cut a pizza.
Dr Hannah Fry, our queen
The pineapple numbers! 😂 I love this attempt at keeping the engagement in the comments high. 😁
More like getting peoples hackles up.
Based on this I now understand that the total size of a cut pizza is 11/12 pizzas. I've weighed the crumbs left on the pan after cutting and believe this to be accurate.
I refuse to believe that cutting a single pizza turns it into almost 12 pizzas. =P
@@plackt it can, but only if the axiom of choice is true. Also you end up with some pretty weird cuts.
@plackt this is what is beautiful about mathematics, such an unintuitive answer that it seems impossible but then apply it to a real world situation, and the surprising result confirms that what we once thought of as nonsense is indeed innovation.
Cutting a pizza like a drunk person, with a metal ruler in a library, while being recorded with a big camera. Absolutely amazing setup.
I have not attempted to learn mathematics or work with mathematics since I last studied it a couple years ago and this is my first time in a long time I have had to think mathematically.
It really feels uncanny but awesome how all of this kind of stuff just *makes sense*. Thank you for presenting these kind of simplified versions for people like me to get back into mathematics!
What happens if you have a non-euclidian pizza?
oof, then I hope you didn't invite Euclid to the pizza party.
He's going to be mighty annoyed that he didn't get any.
thanks. now some of my 22 guests are angry at me after I handed them their "piece"
more like 19 are unhappy but 3 people are quite happy with their amount
@@randomjapsiYes, 19 is some of 22.
Love the detail of the little pizza-loving rat in the animations 😊
I love the enthusiasm of this guy.
I found an easy way to cut the pizza into the maximal number of pieces from n cuts (I don't know if this is how they do it in the paper or if this is even correct): if n is odd, construct a regular n-gon, and extend all the sides as infinite lines. Then, draw your circle big enough around the n-gon so that all the line intersections are contained. Each line will intersect every other line in the circle, as needed. If n is even, do the same with n+1 and remove a line. Kinda cool!
The animations are getting more sophisticated by the day. Love
I was screaming in my Head " MINUS a TWELFTH" each second, and then finally brady said it :D
I love it. I love you brady. I love Math.
I bet the dialogue of that teacher with the young Gauss went like this: "Can you calculate the sum of all integers from 1 to 100" - "Ahh, well, fifty-fifty"
After watching this I realized making the least number of pieces is incredibly simple, since you just keep cutting parallel lines to add one strip at a time.
They don't even have to be parallel so long as their crossing points are on or outside the rim of the pizza
Who says you need the cut lines to intersect the pizza? Or be distinct from each other?
just cut outside of the pizza
I think it could strongly be argued that a line that does not in any way divide the pizza does not fit a reasonable definition of the word "cut"...
Simple. Eat a slice every time you cut so answer is always 1
This is a greedy algorithm, you try to maximize the number of new regions in each step, but the greedy approach does not guarantee optimal solution in general, so its use should be justified in a case by case basis, right?
You are right, the greedy algorithm is not always correct, but I think it is proven in the video, that the greedy approach works here, perhaps not explicitly enough
0) f(n) is defined "The maximum number of slices using n cuts"
1) You have to do exactly n-1 cuts before making the nth cut
2) (1) => on the nth cut you can intersect no more than n-1 lines, making no more than n new slices
3) (1) => before nth cut there can be no more than f(n-1) slices
4) (2), (3) => f(n) = f(n-1) + n or less
5) f(n-1) + n slices is achievable by doing the greedy algorithm => f(n) = f(n-1) + n or more (or it would not be maximum)
6) (4), (5) => f(n) = f(n-1) + n
(6) states, that the formula at 7:20 is correct and the rest is certainly in the video
@@egorkarpuhin7053 Thank you! It's very convincing.
2:10
There should be a follow up video on if it’s possible to “Lazy Cut” a pizza into equal area slices.
Intuitively, this seems possible with 7 slices and 3 cuts, but with 4 or more cuts, it becomes a lot less clear whether or not it can be fairly divided.
"Pizza is not one of my favorites." Literally cannot be trusted now.
Distrust any "man" who dislikes pizza
I thought the same thing. I was immediately like, "Y'know, Tom, I respected you, but now I'm not sure I can anymore..." 😂
Yeah he lost me the thing he said
0:03 "It's not one of my favorites" and you've lost me
Problem showed in one of the greatest math books "Concrete Mathematics" by R. Graham, D. Knuth and O. Patashnik. Great video!
I propose a method to get the most pieces. (5 mins into video)
1) Initial cut
2) Almost parallel but intersects near edge of pie
3) Same thing but intersects the previous one
4) Same thing but intersects the previous two
5) Repeat til you get a dreamcatcher. Your intersection with the first cut changing location slightly each new cut that if linked form a bent chord.
Next subject : The Lazy Way to Cut Pizza "fairly".
My intuition is that the intermediate value theorem would allow you to make equal areas for any given solution by rotating and translating the cuts, without changing which overlap which.
I assume that calculating the sequence of angles g(n) that ensures the f(n) portions have the same area is out of the scope of this channel.
I highly doubt that it could possibly be fair even for the 3-cut case. An interesting query is how fair can it get for a number of cuts, in terms of narrowing the range of slice areas? Also, how many fair slices can be achieved? (For that, we have the lower bound of 2n for the normal radial cutting method and the upper bound provided here. I don't expect the actual upper bound to be better than linear).
Yeah, what's the maximum number per cut when you have to make every slice equal area?
The slices look a lot like life, though.
The "normal"/"fair" way is the PC version.
This version is how life really is. Mine is the the tinny tiny one in the left corner.
That library's going to smell like tomato paste for hours. :)
if you don't want the middle triangle from the third cut to be too small, don't make the first two cuts meet near the center! that minimizes the possible size of that triangle!
I would quite enjoy that "pizza" as a snack, despite what haters say. Tortilla bread is delicious. Amazing video by the way. I had taken a hiatus from math, but this reminded me why I love the subject.
2:05 "but obviously this one's quite small and no one wants it" I'd say the size isn't even the biggest issue; after all you could rearrange the cuts to make that middle piece bigger if you want. But it'd still be a bad slice of pizza since it doesn't have any outer crust, and thus cannot be easily picked up to eat without getting toppings all over your hands
I prefer pizza without crust. I don't mind getting toppings on my hands or eating pizza with a fork.
It's not about the size of the piece, it's what you make of it that counts. 🤣
That's the problem with rectangular pizzas (cut into rectangles) in general: the bigger they get, the more crustless pieces you have.
@@mal2ksc Based on how many people don't eat pizza crust I'd say it's a mixed loss and win.
@@mal2ksc Having crustless pieces is more of an advantage than a problem.
This max cuts sequence is excelent for distribute potential over dimentions..
real world pizza actually has thickness so you can for example make a cut parallel to the plane of the table, but then the solution to the number of pieces you get with n cuts is (n+1)(n^2-n+6)/6
Yep and then 4th cut in another plane orthogonal to all three. But probably 4-dimensional cuts are not allowed in the library either.
@@jurajvariny6034 Here's the number if you can make 4d cuts: (24+14n+11n^2-2n^3+n^4)/24
This is the only numberphile video that I guessed anything. I had a feeling it was how many intersections there were. Im no math expert so that was exciting for me
0:14 the Erdos graffiti on the edge of the building!
?
@@MagmaBow Erdös is a mathematician whose work has been covered by Numberphile a few times
@@word6344 ah alright
I had to solve this looong ago when I was a teenager. I did as in the video and I was rightfully happy with it. Another one did better, though somewhat out of the box. Cut the pizza in half, place one half on top of the other, repeat. We'll get 2^n equally sized pieces of pizza.
We really need the discussion for the higher dimensions versions of this problem. What about 2d slices of a 3d sphere?
Doesn't even need to be sphere. A sufficiently skilled caterer can cut a pizza in half horizontally. Way more slices that way.
Crawford circles are much more circular than Parker circles
And don't get me started on parker squares!😂
The important thing is to give it a go!
Why theory is immeasurably more satisfying than practice.
The laziest way to cut a pizza is don't, eat the whole thing, in one.
Apologising to the italians and then setting the numbers as pineapples. 10/10
*actual footage of teachers before the class pizza party*
If you use the whole plane instead of a pizza, then it’s easier to draw the lines so they all intersect each other with no triple points, and then you can draw a circle around all the intersections, and then scale down to the actual size of the pizza.
Also a classic for the whole "any finite sequence is the first n terms of infinitely many other sequences". Complete the sequence: "1,2,4..." you ask people and they invariably say 8, but if were using this, its 7!
8:01 Good one, Brady lmao
Mathematician: Well yes, but actually no
Around 7:49 there is the mistake that f(0) is counted twice. f(0) is indeed 1 but it's the only way to have just 1 slice
My profesor after watching this video.
A pizza has a radius of x cm. Derive a formula for the maximum number of straight linear cuts that can be made in the pizza, under the assumption that each piece of the pizza must have an area greater than or equal to 1cm^2. Prove the formula.
5:50 "It's definitely possible [to draw a new line that intersects all previous lines]."
Proof: It's (obviously) possible to draw n lines such that they are pairwise non-parallel and that no three lines intersect each other in a common point. In general, not all intersection points will lie inside the pizza, but there's always a larger circle which contains them. Now shrink the entire configuration of lines and that circle in such a manner that the circle coincides with the given pizza. Then the "new" lines have the required properties and cut the pizza into f(n) pieces.
10:10 "Cooking a pizza not permitted in college library" Fair enog...wait, why is it okay to bring food into the library?
“I like to derive my own formulas” … this is why my grade went from 64% to 98% between my first and second semesters of calculus.
We had a giant list of trig integrals to memorize for the first end of term 1, and I can’t memorize at all well.
Term two, we learned integration by parts and I could derive them instead!
I also did this for many years with the quadratic formula
I was confused for a moment since I vaguely remembered a similar topic in a 3b1b video that results in a different sequence but I realize my mistake now. This sequence is constructed by placing lines through a circle, while the 3b1b sequence was constructed by placing points on the circumference of a circle and drawing lines between those points
Yes, that also confused me. I remember the sequence 1,2,4,8,16,31.
Natural follow-up question: what pattern of cuts gives you the biggest small pieces/most even pieces. Like for one and two cuts, the centre is obviously the most even place, but for three cuts, it works better if the first two are not central.
So are you going to put that under "special instructions" when ordering the pizza online?
I am a simple man with complicated pleasures. I enjoy math, science and cooking. Watching someone spread raw tomato paste on dough with a trowel then applying pre shredded mozzarella on top is devestating. 3:40 well with the hyperbolic way that you treated that beautiful piece of dough I think you can find a way.
Scissoring pizza into power of three, is a sign of actual competence imo 😌
Pausing at 6:47 to think about this going the other direction. If I get N new regions when I have made N cuts this suggests when I made 0 cuts I got 0 new regions. This means -1 cuts should ALSO have 1 region. But that would imply that when I made -1 cuts that I lost a region at that time. So if I had -2 cuts I should have 2 regions. And -3 cuts would have been 4 regions and so on.
I work at BlackJackPizza. I'm always pondering different ways of cutting a pizza 🍕 so this video is one I cannot skip 😂
Please do a video on p-adic numbers ❤
The transition from pizza cutting to sums was so smooth
Tom makes another appearance! yay!
You are not a true mathematician unless you spend hours playing with your food.
If this pizza was truly a 3d pizza, you can get more pieces starting with f(3)
f(1) and f(2) are still normal cuts across the surface
But where Tom showed f(3) = 7...
Instead of cutting along the surface, you can cut along the same plane as the pizza. This doubles the number of slices since it manages to cut every slice in half, yielding f(3) = 8
Mammamia ! 😱 If I were an Italian mathematician (which I'm not), I'd challenge you to find: "The Lazy Way to Cut Christmas Pudding".
In 3D, with planes instead of lines. ... and then label the slices of pudding with pieces of tomato or salami. 😝
That's the saddest "pizza" I've ever seen, but Professor guy gets a pass from Italy anyway - because of the interesting content as usual and the strikingly perfect free hand circle 😅
Now the next step... Figuring out algorithm to maximize the cut area (of the final pieces) as much as possible, and then determining the distribution of area across all the cuts and see if it fits a pattern.
You can't make pizza with tortilla but, if there's no choice, mix the tomato purée with smooth peach chutney, garlic and oregano then throw on olives, capers, anchovies and grated emmental cheese. lastly, sprinkle on olive oil and salt before sticking under the grill.
I'd recommend the "lazy AND lase-y" way to cut a pizza. Using an industrial laser would be much less messy... unless you accidentally aimed it across the room of course.
Fun an all, but being lazy means NO MATHS! Slapping down 6 cuts and not caring, and it is actually easier to grid it with 3 lines across then 3 vertical (giving the pizza a slight flick so it wonkily ends up between 45 and 135deg-ish from previous orientation. Result with 6 lines is just 16 pieces, 7 lines gives 20 and it takes but 2 seconds and best of all, you won't get fired. 🍕
There also at least two ways of "beating" these numbers. The messy way would be to fold the pizza multiple times first: in theory you can then get any number of pieces from a single cut, albeit mostly slivers of crust. A second way is to move the cut pieces between cuts. Then each cut can double the number of pieces. Of course the method given is far more interesting... :-)
6:32 f(1) = 2 but at 7:35 He says f(1) = 1, but formula works fine how ?
I had the same thought! Took me a minute to work it out, but I figured out what happened. `f(1) = 2` because `f(1) = f(0) + 1`, and `f(0) = 1`, so `f(1) = 1 + 1`. Tom misspoke at 7:37 saying that `f(1) = 1`, but everything that was written down was correct. The formula `f(1) + 2 + ... + n` is correct, and expanding `f(1)` and `f(0)` at the same time does lead to `1 + 1 + 2 + ... + n`.
TL;DR: the math written down is correct, Tom just misspoke.
@@thislooksfun1 understood Thank You very much
I was thinking the 2 after f(1) should be value for f(1)
Awesome video, I can't wait to amaze my friends with the efficiency of this trick at our next pizza party :P
So you're not allowed to cook in the library, but you can otherwise make a pizza?
Also, for completeness, you need to show what happens with an odd number of numbers.
You can tell he doesn't do this very often from the way he squeezes the tube of tomato paste... 😭😭 You ALWAYS squeeze tubes from the closed end to the front, and very gently. Because otherwise you'll create wrinkles in the tube that cannot be straightened out again and that'll make it impossible to empty the tube properly. Hint: this advice works for mustard and toothpaste as well. 😂
6:50 I like how fast he changed the upper bound for f(n) from 2^n to n(n+1)/2+1
Why is f(1) = 1 and not 2 if we can make 1 cut the no regions should be 2
It's quite easy to construct a set of line cuts.
One cute possibility is... parametrically:
line(n, t) := n + t*e^(i*n*🦩)
0° < 🦩 < 90°/🌼, where 🌼 is the number of cuts you want to make.
Then just scale down (or up) the whole shenanigan to fit your pizza.
There’s a really nice geometric derivation for that relationship. Take n 1x1 squares in a line. Stack on top of those n - 1 more 1x1 squares, then n - 2, etc. Align them all at the left, so you have what looks like stairs. What is the area of the stairs? If you draw a diagonal down the stairs, you can pretty easily see we have a triangle with base and height of n, so n^2/2 (you can also think of this as half an nxn square), but we are missing n half squares that were cut off by the diagonal. So, f(n) = n^2/2 + n/2 = (n^2 + n)/2 = n(n + 1)/2.
Or you can stack two sets of stairs (one upside down) to make an n x n+1 rectangle...
@@rmsgrey yup, that works too
i literally just found out about this today cuz i saw it on a problem and apparently numberphile made a video about it on the same day
I imagine Gauss' teacher not realising how he did it, so just kept giving him bigger and bigger values of n and Gauss kept coming back seconds later with the answer.
When cutting horizontally with the last cut, you can double up the amount of pieces you already have :D
Mathematics wants to show up everywhere.
I think this is a beautiful counterpoint to the claim that the sum of all numbers equal -1/12. Because it should be theoretically possible to cut a pizza infinitely many times into less than one piece.
You should do a follow up video where you not only maximize the number of pieces, but you also figure out the cuts that would make the most equal sized pieces. In other words what cuts would result in the smallest difference between the biggest slice and the smallest slice?
I enjoyed it.
This video got Tom Crawford sanctioned by the Republic of Italy for crimes against gastronomy
An interesting puzzle would be to determine the method to slice where the smallest piece has the largest possible area.
I'm reminded of the peg-and-string thing where you start off with a pair of perpendicular axes (more than one axis, not more than one axe) and draw the lines (X,1), (X-1, 2), (X-2,3) and so on to (1, X). With a large enough X, you get the image of a quarter circle made out of many straight lines... however the intersections of each new line conform to this lazy algorithm too.
Is there some approach that allows you to make a minimal difference between the sizes of the pieces?
I applaud you for not getting Fibonacci involved in this spherical cow of a pizza.
He keeps saying the first cut doesn't matter. But it does. You want to maximize the size of the pieces you will divide up, I think that means you want first slice to be a diameter cut.
You may just have explained why my Brazilian friends seem to all cut up their pizza in such an irregular way.
I'm sure the mafia, camorra, mafia and ndrangheta are just joining forces to avenge the desecration of a national food relic.
the real question that should be asked here to make it more real to real life is....."What should always be done BEFORE cutting a pizza?"
Tom made a right ParkerPizza of that first attempt
in their defense, American style pizza is a WILDLY broad category of food (just think of NY vs Chicago style differences!), so Tom's oversized Lunchables is indeed recognizable as a pizza on our side of the lake. Definitely not a *good* pizza, but recognizably pizza.
The tortilla reminded me, though, pizza toppings sans sauce makes a GREAT quesadilla with marinara dipping sauce. Maybe try cooking it first, though, Tom ;p