20:42 There's a reason they don't bring it out very often, because its license only allows a handful of views before it self-destructs and needs to be re-purchased, and a license to make copies is even more expensive. Also, as you saw, it didn't come bound, it was loose and requires you to provide your own container. (At least it didn't seem to have pictures that are just broken links or thumbnails to embedded videos… at least not that we saw.)
Atleast most texts these days mention the abbreviation before it's used, checking the appendix when reading through a text is tedious. I can't even imagine having to read through a text that doesn't explain the used abbreviations.
@@euromicelli5970 Don't forget that this is a text about fractions. Also it's fitting that the author of the text would use values that make no practical sense. I guess that practice is as old as time.
@@euromicelli5970 I knew the parts were 100/8 and 700/8, so that immediately told me we were working with fractional shares, but it took me a few minutes to work out the algebra after that.
Basically everytime the teacher wrote something with pi on the blackboard - or on the overhead projector (or whatever those things with the glase plate, the light and the plastic film you write on are called in English)...
If anyone’s curious, the amount of loaves each of the 5 partitions would get would be: 1) 1 and 2/3 2) 10 and 5/6 3) 20 4) 29 and 1/6 5) 38 and 1/3 The arithmetic interval is 55/6, which in practical terms is 9 and 1/6 loaves. The smallest partition is 5/3 or 1 and 2/3 loaves. This can be found by solving this system of equations, with x representing the smallest partition and y representing the arithmetic interval: 1) x + (x+y) + (x+2y) + (x+3y) + (x+4y) = 100 2) x + (x+y) = 1/7((x+2y) + (x+3y) + (x+4y))
I used 20 for the median value, which I knew from the statement of the problem, and then the 5 shares were 20-2x, 20-x, 20, 20+x, and 20+2x. This used only one variable instead of two. The equations, then, were: (60+3x)/7 = 40-3x 60+3x = 280-21x 24x = 220 x = 55/6, which is the difference in the sequence.
Nice, but still a bit weird. It works because several independent values are remarkably close to each other, and possibly because 3^-4 is a consecutive list of the natural numbers. Specifically, the square roots of both non-perfect squares can be approximated relatively precisely as RATIOS of the square roots of two perfect squares whose own ratio equals the imperfect square. That is, 2^½~7/5~10/7, because 49/25 and 100/49 are very close to 2 and, better yet, differ by nearly the same amount but in opposite directions, so their average, 99/70 is far better still. Likewise, the squares of both (9/4) and (20/9) are about the same small amount above and below 5, so their average, 161/72, is also very close. The product of the two very close to the square root of 10, which is about as close to squaring the circle as we're ever going to get, because... : Another old crude value of pi, 22/7, is also the rational number pi is not, and that ratios square is 484/49. Not quite 10, but close enough that we can create an equation like π²=(10x²-x+1)/x². Unfortunately, feeding that into Wolfram Alpha produces a pair of unholy incantations whose values. if we take π to be EXACTLY √10, are 0/2 (potentially a problem) and 2/0 (definitely a problem.) Note that this somehow STILL wouldn't square the circle, since it would give a circle of radius π an area of π³ and a square with sides of length π an area of π², so we're just relating two irrational numbers to each other in hopes their irrationality cancels out somehow.
I'm so glad that some ancient dude was sitting out there wondering who those insane people were with something like "21 watermelons, 11 pyramid blocks and 9 whole and 1/6 of bread loaf"
Not sure if it's me getting older, but I'm a bit overwhelmed by this. We talk of touched by the hand of the artist, but to have a 3000 year old text of anything, let alone a maths text is amazing. Thank you for finding and sharing this.
800 years before this text was authored, the architects of Akhet-Khufu (the Great Pyramid) chose a slope angle that gave it a perimeter/height ratio of 44/7.
The "Friendship ended with Mudasir" meme is almost 10 years old! Sometimes I just wonder if at some point people in the nursing home would think I am insane, while I would be just referencing obscure memes from the 2010s.
It's a little of both. Translation isn't a one to one science, and the further apart the cultures, the more art there is in intuiting meaning. Even though this is a mathematical text, that will still have bearing. You want to see wild? Translation of poetry so that it retains the feel and emotion of the original while attempting to make it flow like poetry in the translation is WILD. And difficult.
The title saying "the last" maths author implies there will be some apocalyptic cataclysm when Matt's book is released. Maybe "the latest" would have been less grim.
1:50 "I think I can probably decipher the maths but I have no idea what the hieroglyphics are saying." This from the man who said that he can't read French, but maths is maths . . .
I can't help but think what an equally distant mathematician from the future would say about our textbooks. "As you can see here, solving for the number of watermelons was an important practical problem in their day to day life, as one person could easily eat 100 of them."
I wonder how he would have felt, had he known that 3500 years later, 2 people in a land he barely even knew existed if at all, were pouring over his math tests.
I wonder how he would have felt, had he known that 3500 years later, over a million people all over the world would see his document, transmitted basically instantly and magically to appear on a small device in their hand?
Get a hold of Max Miller from Tasting History, find a papyrus, and bake bread using Egyptian proportions while talking about math and history. I can't think of much more I would want to see than that.
I stopped at 12:33 to try to figure it out, it's a fun little algebra puzzle. If I understood the assignment correctly, then the difference between the shares is 55/6. To check the answer, person 1 gets 10/6 loaves, person 2 gets 65/6 loaves, person 3 gets 120/6 loaves, person 4 gets 175/6 loaves, and person 5 gets 230/6 loaves. 10+65=75, 120+175+230=525, 75/525=1/7, thus persons 1 and 2 combined have 1/7 the loaves of persons 3, 4, and 5 combined. And, 75+525 = 600, thus there are 100 loaves between them. I used a system of equations to solve, 6 equations with 6 unknowns: a + b + c + d + e = 100 7(a + b) = c + d + e b = a + n c = a + 2n d = a + 3n e = a + 4n I wouldn't be surprised if there was an easier way. And, admittedly, I did have to redo it once I looked up arithmetic progression - my first attempt assumed the progression was b=a*n instead of b=a+n for some reason 😅 This is also, sadly, the only maths puzzle on this channel I've managed to actually do (seemingly) correctly. Turns out, I _can_ participate! ...so long as we're doing millennia-old algebra puzzles!
I had the same approach and also had to look up arithmetic progression :) "I wouldn't be surprised if there was an easier way" In fact there is a very nice solution here in comments by @psiphiorg. Normally you think about the shares as a (b)ase value + an (a)ddition like in your post x1 = b x2 = b + 1a x3 = b + 2a x4 = b + 3a x5 = b + 4a There you have to find both base and addition values - so youbuild set of equations with two unknowns. But you can also look at it from a differend view. a sharer in the middle will get exactly average amount - lets call it (m)iddle. So: x1 = m - 2a x2 = m - a x3 = m x4 = m + a x5 = m + 2a because (a) negates itself we get that m = 100/5 = 20 so (60 + 3a) / 7 = 40 - 3a 60 + 3a = 280-21a 24a = 220 a = 55/6 Same result, but we have one unknown for free :)
Matt says they got it out of storage for him, so this is presumably just some office space that happened to have a couple of big tables in it, not where the papyrus is usually kept.
Well, remember, this is the British Museum. They wax philosophical about the importance of the artifacts and the care they give them, and that's why they can't give the obviously stolen stuff back, and then they constantly destroy or hide the stuff out of neglect.
I'm not following the Egyptian text as well as I like to. I think that's because I don't see what Ilona is pointing to. It would have been helpful to have those bits show separately on the screen.
I'd guess that the British Library were not too keen to have him shove the necessary lighting directly at the ancient scroll to be able to film it in close-up like that.
@@MrDannyDetail There are scans of the document which could be cropped from and overlayed onto the video in post. It would be a bit of extra work for the editor but probably worth it. It's possible that this was intended, but would be too much trouble to get licensing to use the images, which seems silly since he was allowed to film the document directly but that's often how things are if you want to make truly legitimate UA-cam content.
@@rustymustard7798 Srsly. The camera operator seemed to have only a vague idea of what specific part of the text was being talked about at any point. That, or they just decided this video would be about the atmospherics of "check it out, it's a cool old papyrus" rather than what Matt and Ilona were _actually_ trying to talk about.
For those trying to work out the problem, if you're stuck: instead of thinking about 5 different numbers / variables (e.g. x, y, z, a, b), think about how they're related. If x is the largest, y = x minus some value, z = x minus twice that value, etc.. So instead of five variables, you have only two (say x and c). There's enough info to create two separate equations using the two variables. System of equations!
@@linknlogs2273 I wonder if they used the middle item of an odd number of sequential terms in an arithmetic sequence being equal to the average of those terms? So the share in the middle has (100/5) loaves, and once you find the total of the three with the biggest shares (seven-eighths of the total 100) the share in the middle of those (the second-largest overall) is a third that total of the three biggest? This is fun...sometimes "let's come up with different approaches" is as interesting (and educational!) as finding a single efficient way.
Ilona is Belgian, I'm sure from the accent! What a wonderful video to find someone from my own country in, and with such fascinating expertise as well! These kinds of artefacts almost make me teary-eyed, because you feel a kinship with these people who lived so long ago. Even then they were doing these kinds of problems and thinking about them in sometimes very similar and sometimes very unique ways, some undoubtedly even lost to time entirely. What a privilege, Matt. I'll bet you loved it!
@@98Mikemaster A majority of the Belgian population speak Flemish, which is more or less the same as Dutch, as their native language. Belgium is famously bilingual, a bit similar to Canada.
@@rudyvigil6928 To get 55/6: Effectively you are solving 2 simultaneus equations: a+(a+r)+(a+2r)+(a+3r)+(a+4r)=100 and a+(a+r)=(1/7)[(a+2r)+(a+3r)+(a+4r)], where a is the smallest share and r is the difference between each consecutive share. Simplifying the first equation, one gets: a+2r=20. Simplifyng the second equation gives: a=(2/11)r Then substitute the second equation into the first one to get r=55/6. The smallest share, a, is 5/3 Once one has the equation it is not too hard, it took me a little while as i decided to do it without a calculator for added authenticity (I ended up having to relearn long devision for the first time in a long time)
@@rudyvigil6928 I can. Say "X" is the smallest share and "n" is the difference between shares. So 100 loaves is then equal to 5X + 10n. We also know that 1/7 of 3X + 9n is equal to 2X + n. There might be a more clever way to solve it but you can solve "n" from this system of equations and that's the answer to the problem.
*Summary* *About the Papyrus:* * *0:00* This 3,500-year-old papyrus is the earliest known mathematical text attributed to an author: Ahmes. * *0:46* It's essentially a textbook with over 80 worked problems and hints for solutions. * *1:03* Many problems deal with practical applications, like dividing bread, calculating field areas, grain volume, and even pyramid slopes. *Egyptian Mathematics Highlights:* * *5:18* Used unit fractions extensively (though 2/3 and 3/4 appear too). * *10:26* Employed a doubling method for multiplication. * *11:19* Demonstrated an understanding of arithmetic progressions. * *15:45* Calculated pi to a surprisingly accurate value (~3.16) for practical applications. *Interesting Problem:* * *11:19* The video challenges viewers to solve an ancient Egyptian problem involving dividing 100 loaves of bread among 5 men in an arithmetic progression with a specific condition. *Matt Parker's Take:* * *19:38* He marvels at the papyrus's similarity to a modern math textbook in terms of structure and content. * *19:52* He emphasizes the practical nature of the problems while noting the potential for early recreational math. * *20:37* Parker expresses his gratitude to the British Museum for access to this rarely displayed artifact. i used gemini 1.5 pro to summarize the transcript
For the loaves problem there isn't an integer solution to divide the loaves, but the solution 1 2/3 to the first man, 10 5/6 to the second, 20 to the 3rd, 29 1/6 to the 4th, and 38 1/3 to the for 5th satisfies both conditions. The difference between each share is 9 1/6.
For the Problem at 12:33 First Equation: 100 = x + (x+y) + (x+2y) + (x+3y) + (x+4y) => 100 = 5x + 10y Second Equation: [(x+4y) + (x+3y) + (x+2y)] * 1/7 = (x+y) + x => 3/7x + 9/7y = 2x + y Solve second equation for x, substitute into first equation x = 5/3 y = 55/6 Result, the 5 men get the following amount of bread: 5/3 = 1 + 2/3 65/6 = 10 + 5/6 20 175/6 = 29 + 1/6 115/3 = 38 + 1/3
I fall asleep every night to "Kushim and the earliest known maths mistake", and I smile every time Matt says, "earliest KNOWAN maths mistake." I've started watching this one with the subtitles on and it actually got transcribed as "all of know and authorship..." So Matt hasn't got out of that habit then!
for the word problem: you have 5 men (a, b, c, d, e) that get a portion of the 100 loaves. First we are going to split the group into 2 parts where a+b+c is the "largest 3 shares" as A and d+e is the "smallest 2" as B. This gives us A/7 = B (where 1/7th of the largest 3 are equal to the smallest 2) and we also have A+B = 100 (all groups added together equals the loaves. This gives us 2 variables and 2 equations which results in A = 175/2 and B = 25/2. next, we have a progression sequence, assuming additive would be a new variable F, where a=b+F, b=c+F, and so on....if we subtract out a common value G from all sums, and coalesce the F, we have something like a=G+4F, b=G+3F, c=G+2F, d=G+F, and e=G. we can pick one of the sides of shares that we know of (either largest or smallest) against the full total, which we will pick the smallest since it's easier....so we get B=2G+F and 100=5G+10F. We already know B is 25/2 so we get F=25/2-2G and F=10-1/2G. Equate them to find G=5/3 and F=55/6. Since e=G, that starts the progression at 5/3 (or 10/6), then the others follow every 55/6, giving: 10/6, 65/6, 120/6, 175/6, and 230/6....which is reduced to 1.2/3, 10.5/6, 20, 29.1/6, and 38.1/3. And because this is calculated resources, it means that 1 loaf would be split into 6 and 1 loaf into 3, the rest are given out in whole amounts for payment.
4*(1-1/13)^3 and 4*(1-1/17)^4 are the best possible solutions with 3rd and 4th powers, and both are technically better approximations, but they are hard to do in practice (working with 17 to the 4th power isn't great). And the quadratic case was good enough, so why bother?
Maybe it was due to the fact that they used primarily unit fractions, as Matt said? And since 8/9 is a fancy way of saying 1 - 1/9, it was more natural for them to arrive at this
@@Faroshkasvery good point. Also, how well could they check their approximations of pi? As far as they knew that might have been as close as they could ever get
I think the partial answer to the first one is the 2 with the smallest share added together would have 12.5 loaves, with the remaining 87.5 going to the other 3. (12.5 * 7 = 87.5) I haven't figured out what the individual shares would be, but that's what I've come up with so far. It's a confusingly worded puzzle to be sure, but having it written out in English does make it easier.
I love that they used a beetle (presumably because it transforms from grub into adult) to mean 'turns into'. MUCH better than our boring equals sign these days!
Is totally possible, classic restaurants share the tips between the workers in a similar fashion, depending on the role. Maybe they calculated the wages like this?
@@framegrace1 That's not unlikely that they distributed wages in this way in some instances. The specific numbers in this example seem to be not very realistic though. Giving the 2nd lowest paid worker more than 6 times as much as the lowest paid one is a bit ridiculous unless it's 1 slave and 4 non-slaves. Maybe if it's a bonus on top of another payment, but even then the 1 2/3 loaf feels more like a middle finger to the bottom guy than giving him no bonus at all
@@kimlground206 They are getting 1 4/6 10 5/6 20 29 1/6 38 2/6 The last one divided by 1 2/3 = 23, so he is getting 23 times as much as the bottom guy to be exact (2nd guy is also not 6 but 6.5 times as much as the bottom guy)
If its in AP, then, let a be the first part and common difference be x, so we have, a,a+x,a+2x,a+3x,a+4x will add upto 100. i.e. 5a+10x=100 And 3a+9x=7(2a+x), so a=(2/11)x. Substituting that we get, x=55/6 and a=10/6.
The knowledge needed for calculating the distribution of bread and the volume of granaries was something that sedimented Egypt as a superpower. It was behind one of the largest voluntary governmental property acquisition in history. And it was the primary contributing factor to Egyptians surviving one of the greatest famines in the ancient history. You can read about the incredible story to the guy who was behind this logistical marvel in an old book that many people discard.
Matt, you looked like you were having the time of your life with the math papyrus. And with good reason... this was just incredible to see! And to realize how advanced the Egyptians were in 1500 BC! I mean, calculating pi to be 3.16 is amazing. If I'm not mistaken, I think there might have been a year or two where your pi day calculations were a tad less accurate... Should we refer to that as Parker Pi, anyone with me here? 😁 PS I love your Pi Day videos so much. The one with actual pies was incredible. -- Long time subscriber...
1000 years before that, we have examples of Sumarian math homework. Even some with corrections made by the teacher. Now if you have trouble with your math homework, imagine if you had to finish it before the clay dries and it becomes literally set in stone.
I love the joke, but I do think it's worth taking a moment to talk about what the Sumerians actually had; the students would likely have had access to wax tablets for note-taking and drafting as well; clay tablets were typically only used for work you specifically wanted preserved, whereas wax could be marked up and made flat (blank) again and again without needing to buy new materials. If you had a truly difficult problem, you could work it out on wax with the ability to erase mistakes as you went, and finally, once you had determined what you wanted to put down on clay (if you even needed to), you could just copy/transfer it all from your rough wax draft. This process of copy/transfer could also be done before carving something in stone! You don't want to get half way through a verse of your poem for the King's tomb and find out that you ran out of space on the wall! Lastly, when using clay tablets, you can always put a wet cloth over the clay and it will remain soft.
@@M4TCH3SM4L0N3in other words they had scratch paper, just like we do now? That's awesome! Actually it's more like having a whiteboard to do your scratch work on but same idea
Answer to the question : 55/6th of loafs. You give 10/6th of loafs to the first. and then 65/6th, 120/6th, 175/6th and 230/6th respectively. Method: You start with (3C+9K)/7=2C+K Where C is the first term and K the progression. The first whole integer solution is C=2 , K=11. So you get a temporary serie: 2,13,24,35,46 Which satisfy the 1/7th rule, but is 120 loafs. So you multiply everything by 100/120 loafs or 5/6th. Satisfying part is: You will only have to cut 2 loafs. :D
I coded a couple of Matt Parker's maths examples estimating the perimeter of an ellipse. The algorithms worked as shown, so it's fun now having a Common Lisp library that does something no other coding library does!
Perhaps the beetle glyph which is translated as "becomes" could be interpreted as our modern "equals or =". I was also impressed to see diagrams on the papyrus, showing something like the shape of a pyramid, next to the calculations about pyramids. Early graphing?
5:54 “You have SEKKEM, which means something like… to calculate” In modern Hebrew (my native tongue), SKHUM means summation. This text is 3600 years old. I think it’s pretty dope.
I found that ∆B which is what I named the change in bread amounts to be (55/6), I calculated it all by hand through the day because I had exams today. Here is some calculations: Eqn1: B1+B2+B3+B4+B5 = 100 Eqn2: (B1+B2+B3)/7 = B4+B5 Eqn3: Bn-∆B = Bn-1 Substitute Eqn3 into Eqn1 to get Eqn4. Eqn4: B1 = 20+2∆B Substitute Eqn3 into Eqn2 to get Eqn5. Eqn5: 46∆B = 11B1 Then substitute Eqn4 into Eqn5 to get ∆B ∆B = (55/6)
dif of the shares is 55/6, with the first guy taking 10/6. it would be whole numbers if we were splitting total of 120 loaves with 11 dif of shares and the first guy taking 2 loaves.
Fun problem at 12:00! For me, the thing that made it click is that if the first two shares are 1/7 of the next three, then they are 1/8 of the total of 100, or 12.5. That, plus the fact that the middle share needs to be 20 to make it add up.
For those struggling with the answer: Translating to algebraic notation: x = n of loaves y = n of shares The first condition: x + (x+y) + (x+2y) + ... + (x+4y) = 100 --> 5x + 10y = 100 The second condition: ((x+4y)+(x+3y)+(x+2y))/7 = x+(x+y) --> 3x + 9y = 7*(2x+y) --> 2y = 11x Solving from here (substitution for ex.) we get x = 5/3 = 1.67 (ish) and y = 55/6 = 9.16 (ish)
That was kinda fun. Five number in arithmetical progression means we get these 5 shares: a, a + b, a + 2b, a + 3b, a + 4b Then, following the imposed rule: 1/7 * (a + 2b + a + 3b + a + 4b) = 2 * (a + a + b) 3a + 9b = 14a + 7b 11a = 2b Because the whole makes 100, you also have: 5a + 10b = 100 a + 2b = 20 2b = 20 - a So then 11a = 20 - a 12a = 20 a = 5/3 So then 2b = 20 - a b = 10 - a/2 = 55/6
I don't exactly know how it was in ancient Egypt, however, ancient greek mathematics didn't use angles & length when doing trigonometrics but surfaces & spreads instead, called rational trigonometrics. Interesting to see how the Egyptians did it.
So I've got: 100 = a +b +c +d +e a < b < c < d < e a +b = (c +d +e) / 7 That's what I started out with. Let: a + b = f c + d + e = g 100 = f + g f = g / 7 -> 7f = g 100 = f + (7f) 100 = 8f -> 100 / 8 = f f = 12.5 100 - 12.5 = g g = 87.5 a + b = 12.5 c + d + e = 87.5 From here, I took the Average for each. Avg(a,b) = 6.25 a < 6.25 < b Avg(c,d,e) = 29.1666... c < 29.1666... < e d is approx. 29.1666... (could be more, could be less) But I'm honestly stumped on how to make this an arithmetical progression
You just forgot to use the fact that they are in arithmetic progression, which means the difference between consecutive values of the sequence is a constant. Let's call it r. b=a+r c=b+r d=c+r e=d+r With that extra info added to what you already computed you should be able to find the exact values of c and d, and then the others.
You can reduce the problem to a very simple system of equations First, for the smallest two to be equal to 1/7 of the largest three, we have to split the whole into 1/8 and 7/8, meaning that the smallest two add up to 12.5 and the rest add up to 87.5 Next we need to find two numbers that add up to 12.5: x + y = 12.5 Suppose x is the largest of the two Now we take advantage of the fact theyre in an arithmetic progression so we can define the other numbers based on these first two. Namely: Third would be x + (x - y) Fourth would be x + 2(x - y) Last would be x + 3(x - y) All of those should add up to 87.5, so we'll end up with this: 3x + 6(x - y) = 87.5, or 9x - 6y = 87.5 And there you have it! A nice simple system of equations that you can solve using your favorite method. To reiterate: x + y = 12.5 9x - 6y = 87.5 Solving this gives you that x = 65/6 or 10 and 5/6 loaves, and y = 5/3 or just 1 and 2/3 loaves (SAD). So the answer to the problem would be 55/6 or a difference of 9 and 1/6 loaves between all people. Yay and yippee
Fantastic to get a glimpse of what was known about 3600 years ago. Imagine the design and logistics calculations needed for large construction projects like canals and the pyramids along with surveying and tracking/accounting for grain storage. Not until the creation of the printing press in 1440 (earlier still in Asia) could enough copies of books be made for wider distribution to ensure survival.
Okay, I figured out how to divide the loaves 5/3, 65/6, 20, 175/6, 115/3 Approximately 1.67, 10.83, 20, 29.17, 38.33 I had to solve this system of equations 5x + 10y = 100 2x + y = (3x + 9y) / 7 x = 5/3 y = 55/6 each portion is then x + (n-1)y
Solving the bread problem with geometry. Start with a rectangle of 5 rows (A-E) of 20 loaves. You move some loaves from row B to row D, and twice as many from row A to row E. (You now have a trapezoid.) How many loaves are moved? If you had divided the loaves into 8 piles, Rows A+B got one pile, and rows C+D+E got 7 piles, for a 1:7 split. So rows A+B are left with 1/8 of 100 loaves = 12 1/2 loaves. They started with 40 loaves, so they lost 27 1/2 loaves. Row B lost a third, or 9 1/6 loaves. This is difference between row B and row C, and between all adjacent rows.
So 5 numbers in arithmetic progression will be x-2d, x-d, x, x+d and x+2d 1. x-2d+x-d+x+x+d+x+2d = 100 => 5x = 100 => x = 20 2. (x+x+d+x+2d)/7 = x-2d+x-d => (3x+3d)/7 = 2x-3d => 24d = 11x from 1 and 2 24d = 11 * 20 => d = 220/24 So, d = 55/6 Thus, the portions are: a. x - 2d = x - 2*55/6 = 20 - 55/3 = 5/3 b. x - d = x - 55/6 = 20 - 55/6 = 65/6 c. =========================> 20 d. x + d = x + 55/6 = 20 + 55/6 = 175/6 e. x + 2d = x + 2*55/6 = 20 + 55/3 = 1155/3
Maybe we have defined Pi incorrectly all these years. Instead of a constant that multiplies the square of the radius to find the area, maybe Pi should have been defined as a constant that multiplies the radius first, before squaring the result. That constant would of course be the root of Pi, which happens to equal Gamma (½). Interesting that the area of a circle is half the diameter multiplied by Gamma (½) all squared. That Gamma function just keeps popping up everywhere.
12:26 I solved it using algebra. I was curious as to how they solved it at the time without algebra. But now that I look in wikipedia, it looks like they did have rudimentary algebra at the time! I have to say what threw me off at first was that I don't think I had ever encountered a progression problem with fractions. These type of problems were all using natural numbers when I was a student as far as I remember.
The real solution to the bread problem is "make 20 more loaves of bread", that way you can have a whole number solution: 2, 13, 24, 35, 46, with a difference of 11.
Me too. But I looked up "secant" and the term (in its modern usage) was "First used by Danish mathematician Thomas Fincke in "Geometria Rotundi" (1583)". He based it on "Latin secantem (nominative secans) "a cutting," present participle of secare "to cut" (from PIE root *sek- "to cut")." (Quotes from etymonline, the on-line etymological dictionary). So I think the phonetic similarity is due to chance.
The original brown paper from numberphiles
😂
I was wondering if Brady was behind the camera
The ancient Turkish mathematician, Nam Bephiles
the brown papyrus
Change papyrus
Has anyone tried turning it over? There should be solutions for the odd numbered problems on the back.
Unfortunately, the back side only features another set of questions which was tragically missed by the author, seriously tanking his grade.
No, the holes are where the ancient Greeks wore out the "reveal" buttons
The even numbers can be bought separately as a "teachers' edition" for quadruple the price
@@sandguyman Thanks for not making the author an undefined term.
20:42 There's a reason they don't bring it out very often, because its license only allows a handful of views before it self-destructs and needs to be re-purchased, and a license to make copies is even more expensive. Also, as you saw, it didn't come bound, it was loose and requires you to provide your own container. (At least it didn't seem to have pictures that are just broken links or thumbnails to embedded videos… at least not that we saw.)
"Because it is a mathematical text they abbreviate a lot" I see very little has changed
"Assume a spherical asp in a Nile that flows without grasping...."
Atleast most texts these days mention the abbreviation before it's used, checking the appendix when reading through a text is tedious. I can't even imagine having to read through a text that doesn't explain the used abbreviations.
"the reader can easily demonstrate for themselves that...."
@@HamishBarker "The rest is left as an exercise for the student".
Q.E.D.😂
Somewhere on this papyrus there's a note saying "the proof is trivial and left as an exercise to the reader"
Also, "I have an ingenious proof for this which is too large to fit in the margin of this papyrus"
I have discovered a truly marvellous proof, which this 18ft scroll is too small to contain.
abbreviated as a dot.
The phase "which reveals all secrets" seams to me to just be "solutions manual". But it does go so much harder
Totally seems the homework of some kid doesn't it?
Difference in shares is 55/6 loaves
I really, _really_ expected to get a whole number, so I had to do it twice. I see now we all agree. False expectations I guess.
@@euromicelli5970 Don't forget that this is a text about fractions. Also it's fitting that the author of the text would use values that make no practical sense. I guess that practice is as old as time.
@@euromicelli5970 I knew the parts were 100/8 and 700/8, so that immediately told me we were working with fractional shares, but it took me a few minutes to work out the algebra after that.
Yeah, that was a bit annoying. 9⅙, with the poor first sod only getting 1⅔.
Smallest share is 5/3. Each larger share increases by 55/6ths.
“Calculating Pi by Translating Hieroglyphics”
Pieroglyphics.
Basically everytime the teacher wrote something with pi on the blackboard - or on the overhead projector (or whatever those things with the glase plate, the light and the plastic film you write on are called in English)...
Amazing pi-day idea!
And it's not even March. How lucky are we? :)
@@WaechterDerNacht An epidiascope, though it's rarely called that.
If anyone’s curious, the amount of loaves each of the 5 partitions would get would be:
1) 1 and 2/3
2) 10 and 5/6
3) 20
4) 29 and 1/6
5) 38 and 1/3
The arithmetic interval is 55/6, which in practical terms is 9 and 1/6 loaves. The smallest partition is 5/3 or 1 and 2/3 loaves.
This can be found by solving this system of equations, with x representing the smallest partition and y representing the arithmetic interval:
1) x + (x+y) + (x+2y) + (x+3y) + (x+4y) = 100
2) x + (x+y) = 1/7((x+2y) + (x+3y) + (x+4y))
Now write it up using hieroglyphs
I used 20 for the median value, which I knew from the statement of the problem, and then the 5 shares were 20-2x, 20-x, 20, 20+x, and 20+2x. This used only one variable instead of two. The equations, then, were:
(60+3x)/7 = 40-3x
60+3x = 280-21x
24x = 220
x = 55/6, which is the difference in the sequence.
@@psiphiorg Nice, that’s a much more clever method, I like it.
Explain then how tf I got 0, 10, 20, 30 and 40...
@@andrry_armorThe last part of the problem states that the two smallest #s must be 1/7 of the three larger #s.
(0+10) = 70
20+30+40 = 90
90 =! 70
It's nice that 4:3 is an old aspect ratio, 16:9 is its square and is a newer aspect ratio, and squaring it again gives an old approximation of pi
π:1 is the ultimate aspect ratio /j
Cool observation! Thanks for sharing.
Nice, but still a bit weird. It works because several independent values are remarkably close to each other, and possibly because 3^-4 is a consecutive list of the natural numbers. Specifically, the square roots of both non-perfect squares can be approximated relatively precisely as RATIOS of the square roots of two perfect squares whose own ratio equals the imperfect square. That is, 2^½~7/5~10/7, because 49/25 and 100/49 are very close to 2 and, better yet, differ by nearly the same amount but in opposite directions, so their average, 99/70 is far better still. Likewise, the squares of both (9/4) and (20/9) are about the same small amount above and below 5, so their average, 161/72, is also very close. The product of the two very close to the square root of 10, which is about as close to squaring the circle as we're ever going to get, because... :
Another old crude value of pi, 22/7, is also the rational number pi is not, and that ratios square is 484/49. Not quite 10, but close enough that we can create an equation like π²=(10x²-x+1)/x². Unfortunately, feeding that into Wolfram Alpha produces a pair of unholy incantations whose values. if we take π to be EXACTLY √10, are 0/2 (potentially a problem) and 2/0 (definitely a problem.) Note that this somehow STILL wouldn't square the circle, since it would give a circle of radius π an area of π³ and a square with sides of length π an area of π², so we're just relating two irrational numbers to each other in hopes their irrationality cancels out somehow.
@@aylen7062which is why we’ll all be watching circular TVs by the year 3142
Actually it's close to π/4, not π @yoavshati.
I love that the whole description with dividing loaves in specific ways sounds exactly like Highschool math word problems
I'm so glad that some ancient dude was sitting out there wondering who those insane people were with something like "21 watermelons, 11 pyramid blocks and 9 whole and 1/6 of bread loaf"
Not sure if it's me getting older, but I'm a bit overwhelmed by this. We talk of touched by the hand of the artist, but to have a 3000 year old text of anything, let alone a maths text is amazing. Thank you for finding and sharing this.
As an Egyptologist, this video is really exiting! this is a very famous papyrus, what a treat you could see it
Friendship ended with 22/7, now 256/81 is my best friend.
The fact that it's (4/3)^4 is oddly satisfying.
@@hughcaldwell1034 a power of two over a power of nine is also very satisfying
800 years before this text was authored, the architects of Akhet-Khufu (the Great Pyramid) chose a slope angle that gave it a perimeter/height ratio of 44/7.
The "Friendship ended with Mudasir" meme is almost 10 years old! Sometimes I just wonder if at some point people in the nursing home would think I am insane, while I would be just referencing obscure memes from the 2010s.
I was always friends to 355/113 because it is so easy to remember. Much easier than 3.141592653589793...
Live translation is wild. I always thought people decoded what each part meant and slowly worked through it like a puzzle.
That’s usually what happens, but if you’ve worked with a language long enough, that working out comes more easily.
It's a little of both. Translation isn't a one to one science, and the further apart the cultures, the more art there is in intuiting meaning. Even though this is a mathematical text, that will still have bearing. You want to see wild? Translation of poetry so that it retains the feel and emotion of the original while attempting to make it flow like poetry in the translation is WILD. And difficult.
@@victoriaeads6126 , like translating a song.
@SgtSupaman yes, very similar.
pa-π-rus, you say?
The title saying "the last" maths author implies there will be some apocalyptic cataclysm when Matt's book is released. Maybe "the latest" would have been less grim.
Matt has plans, devious devious plans ;)
love digging up the history of youtube videos in the comments (title has since changed)
Last can mean most recent. Like "last night I had pizza". Doesn't mean it's my last night ever.
@@theadamabrams "last" can mean most recent, but "the last" generally doesn't.
@@gmalivuk 'the last' time I checked, that's not necessarily true. But I'll check again to make sure.
1:50 "I think I can probably decipher the maths but I have no idea what the hieroglyphics are saying."
This from the man who said that he can't read French, but maths is maths . . .
He's consistent.
Roman alphabet is a necessary but not sufficient condition for understanding.
French also uses arabic numerals like the rest of us, hieroglyphs apparently don’t.
Hieroglyphs! Not hieroglyphics.
Hieroglyphics is like saying alphabetics when you mean to say alphabets.
I can't help but think what an equally distant mathematician from the future would say about our textbooks.
"As you can see here, solving for the number of watermelons was an important practical problem in their day to day life, as one person could easily eat 100 of them."
"While we have not discovered the ruins, there was clearly a critical high-speed rail line from Chicago to Miami."
they got order of operations wrong with pemdas ... didnt they know belmdas was right
How do you say "Could I get more paper, Brady?" in ancient Egyptian?
There are multiple layers of nerdiness in this video, and I'm ALL IN! 😂😂❤❤❤
I wonder how he would have felt, had he known that 3500 years later, 2 people in a land he barely even knew existed if at all, were pouring over his math tests.
Anxious, probably
"Bout damn time students started paying attention. These interdynastic period kids... no discipline, no discipline at all."
I wonder how he would have felt, had he known that 3500 years later, over a million people all over the world would see his document, transmitted basically instantly and magically to appear on a small device in their hand?
"That island in the far western sea where 1/4 of our tin supply comes from."
Get a hold of Max Miller from Tasting History, find a papyrus, and bake bread using Egyptian proportions while talking about math and history. I can't think of much more I would want to see than that.
People need to link them both up ASAP
+
Very good idea
I also agree that this is a great idea!!
Me too!!!!
I stopped at 12:33 to try to figure it out, it's a fun little algebra puzzle.
If I understood the assignment correctly, then the difference between the shares is 55/6.
To check the answer, person 1 gets 10/6 loaves, person 2 gets 65/6 loaves, person 3 gets 120/6 loaves, person 4 gets 175/6 loaves, and person 5 gets 230/6 loaves. 10+65=75, 120+175+230=525, 75/525=1/7, thus persons 1 and 2 combined have 1/7 the loaves of persons 3, 4, and 5 combined. And, 75+525 = 600, thus there are 100 loaves between them.
I used a system of equations to solve, 6 equations with 6 unknowns:
a + b + c + d + e = 100
7(a + b) = c + d + e
b = a + n
c = a + 2n
d = a + 3n
e = a + 4n
I wouldn't be surprised if there was an easier way. And, admittedly, I did have to redo it once I looked up arithmetic progression - my first attempt assumed the progression was b=a*n instead of b=a+n for some reason 😅
This is also, sadly, the only maths puzzle on this channel I've managed to actually do (seemingly) correctly. Turns out, I _can_ participate! ...so long as we're doing millennia-old algebra puzzles!
I had the same approach and also had to look up arithmetic progression :)
"I wouldn't be surprised if there was an easier way"
In fact there is a very nice solution here in comments by @psiphiorg.
Normally you think about the shares as a (b)ase value + an (a)ddition like in your post
x1 = b
x2 = b + 1a
x3 = b + 2a
x4 = b + 3a
x5 = b + 4a
There you have to find both base and addition values - so youbuild set of equations with two unknowns.
But you can also look at it from a differend view.
a sharer in the middle will get exactly average amount - lets call it (m)iddle.
So:
x1 = m - 2a
x2 = m - a
x3 = m
x4 = m + a
x5 = m + 2a
because (a) negates itself we get that m = 100/5 = 20
so
(60 + 3a) / 7 = 40 - 3a
60 + 3a = 280-21a
24a = 220
a = 55/6
Same result, but we have one unknown for free :)
It's a little wild to see this priceless ancient artifact in this cluttered room with a dropped ceiling and bunch of moving boxes
Matt says they got it out of storage for him, so this is presumably just some office space that happened to have a couple of big tables in it, not where the papyrus is usually kept.
Well, remember, this is the British Museum. They wax philosophical about the importance of the artifacts and the care they give them, and that's why they can't give the obviously stolen stuff back, and then they constantly destroy or hide the stuff out of neglect.
Matt, you missed a great opportunity to bring the oldest Brown Paper to Numberphile !
I would hate to see them use a sharpie on the papyrus :D
I'm not following the Egyptian text as well as I like to. I think that's because I don't see what Ilona is pointing to. It would have been helpful to have those bits show separately on the screen.
I'd guess that the British Library were not too keen to have him shove the necessary lighting directly at the ancient scroll to be able to film it in close-up like that.
They are however happy to cover up any wrong doings going on and blame the person who told them about the recent thefts instead.
@@MrDannyDetail There are scans of the document which could be cropped from and overlayed onto the video in post. It would be a bit of extra work for the editor but probably worth it. It's possible that this was intended, but would be too much trouble to get licensing to use the images, which seems silly since he was allowed to film the document directly but that's often how things are if you want to make truly legitimate UA-cam content.
Cameraperson had way too much coffee lol. Just stop shaking and maybe we can see it lol.
@@rustymustard7798 Srsly. The camera operator seemed to have only a vague idea of what specific part of the text was being talked about at any point. That, or they just decided this video would be about the atmospherics of "check it out, it's a cool old papyrus" rather than what Matt and Ilona were _actually_ trying to talk about.
For those trying to work out the problem, if you're stuck: instead of thinking about 5 different numbers / variables (e.g. x, y, z, a, b), think about how they're related. If x is the largest, y = x minus some value, z = x minus twice that value, etc.. So instead of five variables, you have only two (say x and c). There's enough info to create two separate equations using the two variables. System of equations!
Now the questions is, how did they do it without algebra?
@@linknlogs2273 I wonder if they used the middle item of an odd number of sequential terms in an arithmetic sequence being equal to the average of those terms? So the share in the middle has (100/5) loaves, and once you find the total of the three with the biggest shares (seven-eighths of the total 100) the share in the middle of those (the second-largest overall) is a third that total of the three biggest?
This is fun...sometimes "let's come up with different approaches" is as interesting (and educational!) as finding a single efficient way.
@@linknlogs2273 geometry probably
Ilona is Belgian, I'm sure from the accent! What a wonderful video to find someone from my own country in, and with such fascinating expertise as well! These kinds of artefacts almost make me teary-eyed, because you feel a kinship with these people who lived so long ago. Even then they were doing these kinds of problems and thinking about them in sometimes very similar and sometimes very unique ways, some undoubtedly even lost to time entirely. What a privilege, Matt. I'll bet you loved it!
I thought Dutch
@@98Mikemaster A majority of the Belgian population speak Flemish, which is more or less the same as Dutch, as their native language. Belgium is famously bilingual, a bit similar to Canada.
Born in Hasselt it is. The Belgian Hasselt.
I came to the comments for this…
It's just amazing that there are people like Ilona who can read 3500 texts. Very cool
Is the "Hugh thanks" in the description a pun or a typo?
No, he's one of our conservators. Really nice bloke
@@britishmuseumOh my.
@@britishmuseum I think I went to school with his brother, Many.
it's a Parker Pun
@@britishmuseum Looks like I got ratioed by the British Museum.
The difference of the shares is 55/6 loaves.
That's what I got too
Could you provide a basic work of how you determined that if it's not too much trouble?
@@rudyvigil6928a few comments near the top (for me) have shown their work or a good starting point at least. Maybe see if you can find some of those?
@@rudyvigil6928 To get 55/6: Effectively you are solving 2 simultaneus equations:
a+(a+r)+(a+2r)+(a+3r)+(a+4r)=100 and a+(a+r)=(1/7)[(a+2r)+(a+3r)+(a+4r)], where a is the smallest share and r is the difference between each consecutive share.
Simplifying the first equation, one gets: a+2r=20.
Simplifyng the second equation gives: a=(2/11)r
Then substitute the second equation into the first one to get r=55/6. The smallest share, a, is 5/3
Once one has the equation it is not too hard, it took me a little while as i decided to do it without a calculator for added authenticity (I ended up having to relearn long devision for the first time in a long time)
@@rudyvigil6928 I can. Say "X" is the smallest share and "n" is the difference between shares. So 100 loaves is then equal to 5X + 10n. We also know that 1/7 of 3X + 9n is equal to 2X + n. There might be a more clever way to solve it but you can solve "n" from this system of equations and that's the answer to the problem.
2 of my absolute favourite experts / presenters! Great vid Ilona and Matt, many thanks!
Yes, easily. [5 minute later] I give up
Amateur. I go at least 30 minutes before conceding I was over-confident and giving up. 💪💪
What an awesome glimpse into the past, a journey back in time, and not just a little, but 3,500 years 👍
*Summary*
*About the Papyrus:*
* *0:00* This 3,500-year-old papyrus is the earliest known mathematical text attributed to an author: Ahmes.
* *0:46* It's essentially a textbook with over 80 worked problems and hints for solutions.
* *1:03* Many problems deal with practical applications, like dividing bread, calculating field areas, grain volume, and even pyramid slopes.
*Egyptian Mathematics Highlights:*
* *5:18* Used unit fractions extensively (though 2/3 and 3/4 appear too).
* *10:26* Employed a doubling method for multiplication.
* *11:19* Demonstrated an understanding of arithmetic progressions.
* *15:45* Calculated pi to a surprisingly accurate value (~3.16) for practical applications.
*Interesting Problem:*
* *11:19* The video challenges viewers to solve an ancient Egyptian problem involving dividing 100 loaves of bread among 5 men in an arithmetic progression with a specific condition.
*Matt Parker's Take:*
* *19:38* He marvels at the papyrus's similarity to a modern math textbook in terms of structure and content.
* *19:52* He emphasizes the practical nature of the problems while noting the potential for early recreational math.
* *20:37* Parker expresses his gratitude to the British Museum for access to this rarely displayed artifact.
i used gemini 1.5 pro to summarize the transcript
Oldest known maths papyrus signed by the author, and it turns out it’s not even a first edition
For the loaves problem there isn't an integer solution to divide the loaves, but the solution 1 2/3 to the first man, 10 5/6 to the second, 20 to the 3rd, 29 1/6 to the 4th, and 38 1/3 to the for 5th satisfies both conditions. The difference between each share is 9 1/6.
For the Problem at 12:33
First Equation: 100 = x + (x+y) + (x+2y) + (x+3y) + (x+4y) => 100 = 5x + 10y
Second Equation: [(x+4y) + (x+3y) + (x+2y)] * 1/7 = (x+y) + x => 3/7x + 9/7y = 2x + y
Solve second equation for x, substitute into first equation
x = 5/3
y = 55/6
Result, the 5 men get the following amount of bread:
5/3 = 1 + 2/3
65/6 = 10 + 5/6
20
175/6 = 29 + 1/6
115/3 = 38 + 1/3
I fall asleep every night to "Kushim and the earliest known maths mistake", and I smile every time Matt says, "earliest KNOWAN maths mistake." I've started watching this one with the subtitles on and it actually got transcribed as "all of know and authorship..." So Matt hasn't got out of that habit then!
Guess it’s a WA thing
Is he a Perth boy?
@@gurrrn1102
for the word problem: you have 5 men (a, b, c, d, e) that get a portion of the 100 loaves. First we are going to split the group into 2 parts where a+b+c is the "largest 3 shares" as A and d+e is the "smallest 2" as B. This gives us A/7 = B (where 1/7th of the largest 3 are equal to the smallest 2) and we also have A+B = 100 (all groups added together equals the loaves. This gives us 2 variables and 2 equations which results in A = 175/2 and B = 25/2.
next, we have a progression sequence, assuming additive would be a new variable F, where a=b+F, b=c+F, and so on....if we subtract out a common value G from all sums, and coalesce the F, we have something like a=G+4F, b=G+3F, c=G+2F, d=G+F, and e=G. we can pick one of the sides of shares that we know of (either largest or smallest) against the full total, which we will pick the smallest since it's easier....so we get B=2G+F and 100=5G+10F. We already know B is 25/2 so we get F=25/2-2G and F=10-1/2G. Equate them to find G=5/3 and F=55/6.
Since e=G, that starts the progression at 5/3 (or 10/6), then the others follow every 55/6, giving: 10/6, 65/6, 120/6, 175/6, and 230/6....which is reduced to 1.2/3, 10.5/6, 20, 29.1/6, and 38.1/3. And because this is calculated resources, it means that 1 loaf would be split into 6 and 1 loaf into 3, the rest are given out in whole amounts for payment.
I've found the same result (which is pretty unsatisfying, if you ask me)
*_BREAD-SHARING PROBLEM _**_12:24_**_ SOLVED WITH SECONDARY SCHOOL MATHS:_*
*Problem statement, as literally stated:*
share: “s”
difference: “d”
100 = s + (s + 1d) + (s + 2d) + (s + 3d) + (s + 4d)
1/7 * ( (s + 2d) + (s + 3d) + (s + 4d) ) = s + (s + 1d)
d = ?
*Problem statement, simplified:*
100 = 5s + 10d
(3s + 9d)/7 = 2s + d
d = ?
*Equation 1, isolate the variables:*
100 = 5s + 10d
20 = s + 2d
s = 20 - 2d ✔
2d = 20 - s
d = 10 - 1/2s ✔
*Equation 2, isolate the variables:*
(3s + 9d)/7 = 2s + d
3/7s + 9/7d = 2s + d
9/7d - d = 2s - 3/7s
9/7d - 7/7d = 14/7s - 3/7s
2/7d = 11/7s
2d = 11s
d = 11/2s ✔
s = 2/11d ✔
*Find the value of d:*
s = 2/11d _and_ s = 20 - 2d
2/11d = 20 - 2d
2/11d + 2d = 20
2/11d + 22/11d = 20
24/11d = 20
24d = 220
d = 220/24
d = 55/6 = 9 + 1/6 ✔
_The difference of the shares is 9 + 1/6 loaves, that is, each man gets 9 + 1/6 loaves more than the previous man._
*Find the value of s:*
d = 11/2s _and_ d = 10 - 1/2s
11/2s = 10 - 1/2s
11/2s + 1/2s = 10
6s = 10
s = 10/6 = 5/3 = 1 + 2/3 ✔
_The first share is 1 + 2/3 loaves._
*The shares of the five men:*
s = 1 + 2/3 _loaves are given to the first man._
s + 1d = (1 + 2/3) + 1*(9 + 1/6) = 1 + 4/6 + 9 + 1/6 = 10 + 5/6 _loaves are given to the second man._
s + 2d = (1 + 2/3) + 2*(9 + 1/6) = 1 + 4/6 + 18 + 2/6 = 19 + 6/6 = 20 _loaves are given to the third man._
s + 3d = (1 + 2/3) + 3*(9 + 1/6) = 1 + 4/6 + 27 + 3/6 = 28 + 7/6 = 29 + 1/6 _loaves are given to the fourth man._
s + 4d = (1 + 2/3) + 4*(9 + 1/6) = 1 + 4/6 + 36 + 4/6 = 37 + 8/6 = 38 + 2/6 _loaves are given to the fifth man._
*Check the result:*
1 + 4/6 + 10 + 5/6 + 20 + 29 + 1/6 + 38 + 2/6 =
1 + 10 + 20 + 29 + 38 + (1/6 + 5/6) + (2/6 + 4/6) =
98 + 1 + 1 = 100 ✔
I'm surprised given their proficiency in fractions they didn't use 22/7 for pi
8 and 9 are numbers a lot nicer to work with than 7 back in those days, I imagine
4*(1-1/13)^3 and 4*(1-1/17)^4 are the best possible solutions with 3rd and 4th powers, and both are technically better approximations, but they are hard to do in practice (working with 17 to the 4th power isn't great). And the quadratic case was good enough, so why bother?
I'm surprised they didn't use 8*(1-1/3-1/25)^2, which is off by just 0.003%. Again, the answer is probably "why bother".
Maybe it was due to the fact that they used primarily unit fractions, as Matt said? And since 8/9 is a fancy way of saying 1 - 1/9, it was more natural for them to arrive at this
@@Faroshkasvery good point. Also, how well could they check their approximations of pi? As far as they knew that might have been as close as they could ever get
I think the partial answer to the first one is the 2 with the smallest share added together would have 12.5 loaves, with the remaining 87.5 going to the other 3. (12.5 * 7 = 87.5)
I haven't figured out what the individual shares would be, but that's what I've come up with so far.
It's a confusingly worded puzzle to be sure, but having it written out in English does make it easier.
I love that they used a beetle (presumably because it transforms from grub into adult) to mean 'turns into'. MUCH better than our boring equals sign these days!
I have to say the pi calculation part is pretty amazing indeed ! smallest share is 5/3 and difference of the shares is 55/6
I believe the difference is 55/6, and the smallest share is 5/3
I got the same
How? I’ve been trying to work it out for a while with no luck aha
Yeah I'm wondering how to work this one out
100=a+b+c+d+e
1/7(a+b+c)=d+e
And the arithmetic progression means
a-b=b-c=c-d=d-e=n
Or
d=n+e
c=2n+e
b=3n+e
a=4n+e
2,11,20,29,38; progression of 9
That poor one guy has to go home with less than 2 loaves of bread while the other 4 get 6 - 20 times as much
Is totally possible, classic restaurants share the tips between the workers in a similar fashion, depending on the role.
Maybe they calculated the wages like this?
@@framegrace1 That's not unlikely that they distributed wages in this way in some instances. The specific numbers in this example seem to be not very realistic though. Giving the 2nd lowest paid worker more than 6 times as much as the lowest paid one is a bit ridiculous unless it's 1 slave and 4 non-slaves. Maybe if it's a bonus on top of another payment, but even then the 1 2/3 loaf feels more like a middle finger to the bottom guy than giving him no bonus at all
Sorry, I don't see why you say "20 times as much"
@@kimlground206 They are getting
1 4/6
10 5/6
20
29 1/6
38 2/6
The last one divided by 1 2/3 = 23, so he is getting 23 times as much as the bottom guy to be exact (2nd guy is also not 6 but 6.5 times as much as the bottom guy)
If its in AP, then, let a be the first part and common difference be x, so we have,
a,a+x,a+2x,a+3x,a+4x will add upto 100.
i.e. 5a+10x=100
And 3a+9x=7(2a+x), so a=(2/11)x.
Substituting that we get, x=55/6 and a=10/6.
The progression f(n+1) := a + b×n is defined by (a = ⁵/₃, b = 9¹/₆).
The knowledge needed for calculating the distribution of bread and the volume of granaries was something that sedimented Egypt as a superpower. It was behind one of the largest voluntary governmental property acquisition in history. And it was the primary contributing factor to Egyptians surviving one of the greatest famines in the ancient history. You can read about the incredible story to the guy who was behind this logistical marvel in an old book that many people discard.
I loved this!!!! Thank you. One of your best.
Matt, you looked like you were having the time of your life with the math papyrus. And with good reason... this was just incredible to see! And to realize how advanced the Egyptians were in 1500 BC! I mean, calculating pi to be 3.16 is amazing. If I'm not mistaken, I think there might have been a year or two where your pi day calculations were a tad less accurate... Should we refer to that as Parker Pi, anyone with me here? 😁 PS I love your Pi Day videos so much. The one with actual pies was incredible. -- Long time subscriber...
Amazing that we have people that can literally read hieroglyphics. So cool!!!
Matt, great British Museum video as well. as I remarked there... that building's walls shapes are some kind of geometric progression...
loved this episode of objectivity!
2:55 - Hieroglyphs are good for writing crypt-ic things
1000 years before that, we have examples of Sumarian math homework. Even some with corrections made by the teacher.
Now if you have trouble with your math homework, imagine if you had to finish it before the clay dries and it becomes literally set in stone.
Only one way to take your time thinking. Keep the tablet submerged until you get the right answer then bake it.
I love the joke, but I do think it's worth taking a moment to talk about what the Sumerians actually had; the students would likely have had access to wax tablets for note-taking and drafting as well; clay tablets were typically only used for work you specifically wanted preserved, whereas wax could be marked up and made flat (blank) again and again without needing to buy new materials. If you had a truly difficult problem, you could work it out on wax with the ability to erase mistakes as you went, and finally, once you had determined what you wanted to put down on clay (if you even needed to), you could just copy/transfer it all from your rough wax draft. This process of copy/transfer could also be done before carving something in stone! You don't want to get half way through a verse of your poem for the King's tomb and find out that you ran out of space on the wall!
Lastly, when using clay tablets, you can always put a wet cloth over the clay and it will remain soft.
@@M4TCH3SM4L0N3in other words they had scratch paper, just like we do now? That's awesome!
Actually it's more like having a whiteboard to do your scratch work on but same idea
Answer to the question : 55/6th of loafs.
You give 10/6th of loafs to the first. and then 65/6th, 120/6th, 175/6th and 230/6th respectively.
Method:
You start with (3C+9K)/7=2C+K Where C is the first term and K the progression.
The first whole integer solution is C=2 , K=11.
So you get a temporary serie: 2,13,24,35,46 Which satisfy the 1/7th rule, but is 120 loafs.
So you multiply everything by 100/120 loafs or 5/6th.
Satisfying part is: You will only have to cut 2 loafs. :D
I coded a couple of Matt Parker's maths examples estimating the perimeter of an ellipse. The algorithms worked as shown, so it's fun now having a Common Lisp library that does something no other coding library does!
The papyrus looks a lot better than my two year old printouts. And that regard the technology hasn’t gotten much better.
Perhaps the beetle glyph which is translated as "becomes" could be interpreted as our modern "equals or =". I was also impressed to see diagrams on the papyrus, showing something like the shape of a pyramid, next to the calculations about pyramids. Early graphing?
5:54 “You have SEKKEM, which means something like… to calculate”
In modern Hebrew (my native tongue), SKHUM means summation.
This text is 3600 years old. I think it’s pretty dope.
Both Hebrew and ancient Egyptian are semitic languages, aren't they?
@@AelwynMr turns out Old Egyptian isn’t Semitic, but it’s related. Hebrew is much more similar to Akkadian
If you divide 100 loaves among 5 men as following: 2+11+20+29+38=100, the difference of the shares is 9. While 2+11=13; (20+29+38)/7 = 12,43...
I found part = 10/6 of a loaf and arithmetic constant is 55/6
sequence is 5/3 + 65/6 + 20 + 175/6 + 230/6 = 100 where 7 * (5/3 + 65/6) = (20 + 175/6 + 230/6) and each portion is different by 55/6 pieces of bread at 12:21
arithmetic sequence is calculated as A(n) = A(1) + (n-1)*d
A(1) + A(2) + A(3) + A(4) + A(5)=100
A(1) +(A(1)+d) + (A(1)+2d) + (A(1)+3d) + (A(1)+4d) = 100
therefore
5*A(1) + 10d = 100
7 * (A(1) + A(2)) = A(3) + A(4) + A(5)
7 * (2*A(1)+d) = 3*A(1) + 9d
14*A(1) + 7d = 3*A(1) + 9d
therefore
11*A(1) = 2d
5*A(1) + 5*2d = 100
5*A(1) + 55*A(1) = 100
therefore
A(1) = 100/60 or 5/3 or 1 and 2/3
plug this value into 11*A(1) = 2d
you get 55/3 = 2d so
55/6 = d
I found that ∆B which is what I named the change in bread amounts to be (55/6), I calculated it all by hand through the day because I had exams today. Here is some calculations:
Eqn1: B1+B2+B3+B4+B5 = 100
Eqn2: (B1+B2+B3)/7 = B4+B5
Eqn3: Bn-∆B = Bn-1
Substitute Eqn3 into Eqn1 to get Eqn4.
Eqn4: B1 = 20+2∆B
Substitute Eqn3 into Eqn2 to get Eqn5.
Eqn5: 46∆B = 11B1
Then substitute Eqn4 into Eqn5 to get ∆B
∆B = (55/6)
I am so in awe with how we almost always seem to be able to interpret these old documents into something legible/understandable. Amazing
If anyone is interest in the loaf problem, this is my working out. The difference is 55/6 and the smallest is 5/3
x + (x+y) + (x+2y) + (x+3y) + (x+4y) = 100
x + (x+y) = (x +2y + x +3y + x +4y)/7
5x + 10y = 100
2x + y = (3x + 9y)/7
10y = 100 - 5x
Y = 10-0.5x
2x + 10-0.5x = (3x +9(10-0.5x))/7
1.5x + 10 = (3x +90 - 4.5x)/7
10.5x + 70 = 3x + 90 - 4.5x
10.5x - 3x + 4.5 x = 20
12 x = 20
X = 20/12
X = 10/6
x = 5/3
5x + 10y = 100
5(5/3) + 10y = 100
10y = 100 - (25/3)
10y = 91+(2/3)
Y = (275/3)/10
Y = 55/6
X = 5/3
I could, but I dont want to.
😂
I want Matt to do it first then I'll compare my answer to his
I could, but there is not enough space in the margins
@@jonasgajdosikas1125 based
the math equivalent of "my girlfriend goes to another school" and "my uncle works for nintendo"
I see that in the last 3500 years we've been giving the same maths questions to students
dif of the shares is 55/6, with the first guy taking 10/6. it would be whole numbers if we were splitting total of 120 loaves with 11 dif of shares and the first guy taking 2 loaves.
Cool video! Missed a photo of the oldest and newest math text side by side
This is very cool
Its quite interesting seeing just how advanced they were
So the first one is like 5c+10x =100 and 9x+3c=7x+14c... Which works out to c=10/9 (intial portion) and x=85/9 ( the diff per person)
Fun problem at 12:00! For me, the thing that made it click is that if the first two shares are 1/7 of the next three, then they are 1/8 of the total of 100, or 12.5. That, plus the fact that the middle share needs to be 20 to make it add up.
For those struggling with the answer:
Translating to algebraic notation:
x = n of loaves
y = n of shares
The first condition:
x + (x+y) + (x+2y) + ... + (x+4y) = 100 --> 5x + 10y = 100
The second condition:
((x+4y)+(x+3y)+(x+2y))/7 = x+(x+y) --> 3x + 9y = 7*(2x+y) --> 2y = 11x
Solving from here (substitution for ex.) we get x = 5/3 = 1.67 (ish) and y = 55/6 = 9.16 (ish)
That writing in so beautiful! I love math :-D
That was kinda fun.
Five number in arithmetical progression means we get these 5 shares:
a, a + b, a + 2b, a + 3b, a + 4b
Then, following the imposed rule:
1/7 * (a + 2b + a + 3b + a + 4b) = 2 * (a + a + b)
3a + 9b = 14a + 7b
11a = 2b
Because the whole makes 100, you also have:
5a + 10b = 100
a + 2b = 20
2b = 20 - a
So then
11a = 20 - a
12a = 20
a = 5/3
So then
2b = 20 - a
b = 10 - a/2 = 55/6
I don't exactly know how it was in ancient Egypt, however, ancient greek mathematics didn't use angles & length when doing trigonometrics but surfaces & spreads instead, called rational trigonometrics. Interesting to see how the Egyptians did it.
Me: “Wow, what an incredible piece of Egyptian history.”
Matt: “…The British Museum…”
Me: 😕
So I've got:
100 = a +b +c +d +e
a < b < c < d < e
a +b = (c +d +e) / 7
That's what I started out with.
Let: a + b = f
c + d + e = g
100 = f + g
f = g / 7 -> 7f = g
100 = f + (7f)
100 = 8f -> 100 / 8 = f
f = 12.5
100 - 12.5 = g
g = 87.5
a + b = 12.5
c + d + e = 87.5
From here, I took the Average for each.
Avg(a,b) = 6.25
a < 6.25 < b
Avg(c,d,e) = 29.1666...
c < 29.1666... < e
d is approx. 29.1666... (could be more, could be less)
But I'm honestly stumped on how to make this an arithmetical progression
You just forgot to use the fact that they are in arithmetic progression, which means the difference between consecutive values of the sequence is a constant. Let's call it r.
b=a+r
c=b+r
d=c+r
e=d+r
With that extra info added to what you already computed you should be able to find the exact values of c and d, and then the others.
great video! Matt has an easier time accessing 3-thousand year old ancient egyptian papyrus manuscripts in the British Museum than Egypt itself!
And how do you know that -- or is that just a political wise crack ?
You can reduce the problem to a very simple system of equations
First, for the smallest two to be equal to 1/7 of the largest three, we have to split the whole into 1/8 and 7/8, meaning that the smallest two add up to 12.5 and the rest add up to 87.5
Next we need to find two numbers that add up to 12.5:
x + y = 12.5
Suppose x is the largest of the two
Now we take advantage of the fact theyre in an arithmetic progression so we can define the other numbers based on these first two. Namely:
Third would be x + (x - y)
Fourth would be x + 2(x - y)
Last would be x + 3(x - y)
All of those should add up to 87.5, so we'll end up with this:
3x + 6(x - y) = 87.5, or
9x - 6y = 87.5
And there you have it! A nice simple system of equations that you can solve using your favorite method. To reiterate:
x + y = 12.5
9x - 6y = 87.5
Solving this gives you that x = 65/6 or 10 and 5/6 loaves, and y = 5/3 or just 1 and 2/3 loaves (SAD). So the answer to the problem would be 55/6 or a difference of 9 and 1/6 loaves between all people. Yay and yippee
Thank you so much for showing this, feels like a privilege to see the maths of 3500 years ago.
Fantastic to get a glimpse of what was known about 3600 years ago. Imagine the design and logistics calculations needed for large construction projects like canals and the pyramids along with surveying and tracking/accounting for grain storage. Not until the creation of the printing press in 1440 (earlier still in Asia) could enough copies of books be made for wider distribution to ensure survival.
I never knew the spoken ancient Egyptian sounded like a bit of Dutch?
using numberphile's brown paper for a long time
Okay, I figured out how to divide the loaves
5/3, 65/6, 20, 175/6, 115/3
Approximately
1.67, 10.83, 20, 29.17, 38.33
I had to solve this system of equations
5x + 10y = 100
2x + y = (3x + 9y) / 7
x = 5/3
y = 55/6
each portion is then x + (n-1)y
Solving the bread problem with geometry. Start with a rectangle of 5 rows (A-E) of 20 loaves. You move some loaves from row B to row D, and twice as many from row A to row E. (You now have a trapezoid.) How many loaves are moved? If you had divided the loaves into 8 piles, Rows A+B got one pile, and rows C+D+E got 7 piles, for a 1:7 split. So rows A+B are left with 1/8 of 100 loaves = 12 1/2 loaves. They started with 40 loaves, so they lost 27 1/2 loaves. Row B lost a third, or 9 1/6 loaves. This is difference between row B and row C, and between all adjacent rows.
Great subject, great video! Thanks 😃
So 5 numbers in arithmetic progression will be
x-2d, x-d, x, x+d and x+2d
1. x-2d+x-d+x+x+d+x+2d = 100 => 5x = 100 => x = 20
2. (x+x+d+x+2d)/7 = x-2d+x-d => (3x+3d)/7 = 2x-3d => 24d = 11x
from 1 and 2
24d = 11 * 20 => d = 220/24
So, d = 55/6
Thus, the portions are:
a. x - 2d = x - 2*55/6 = 20 - 55/3 = 5/3
b. x - d = x - 55/6 = 20 - 55/6 = 65/6
c. =========================> 20
d. x + d = x + 55/6 = 20 + 55/6 = 175/6
e. x + 2d = x + 2*55/6 = 20 + 55/3 = 1155/3
Take a natural number. Use a prime root, which is the lowest natural number that comes within 1/10000 of PI?
Maybe we have defined Pi incorrectly all these years. Instead of a constant that multiplies the square of the radius to find the area, maybe Pi should have been defined as a constant that multiplies the radius first, before squaring the result. That constant would of course be the root of Pi, which happens to equal Gamma (½). Interesting that the area of a circle is half the diameter multiplied by Gamma (½) all squared. That Gamma function just keeps popping up everywhere.
12:26 I solved it using algebra. I was curious as to how they solved it at the time without algebra. But now that I look in wikipedia, it looks like they did have rudimentary algebra at the time! I have to say what threw me off at first was that I don't think I had ever encountered a progression problem with fractions. These type of problems were all using natural numbers when I was a student as far as I remember.
THIS IS SOO COOL! 🤩 Im always fascinated by ancient cultures and how they think 😊
Just paused to say I really hope they find something very nearly but not quite right so Matt can feel extra close.
Pi = 3.16? Call that a Parker pi
So how do you write "left as an exercise for the reader" in hieroglyphics?
The real solution to the bread problem is "make 20 more loaves of bread", that way you can have a whole number solution: 2, 13, 24, 35, 46, with a difference of 11.
Continu, you are on the right track!
In the bread problem. The difference is 55/6 and the smallest share is 10/6. I wonder if Egyptians had rulers in 1/6 ?
13:46 "Sekhat" sounds like "secant" to me.
Me too. But I looked up "secant" and the term (in its modern usage) was "First used by Danish mathematician Thomas Fincke in "Geometria Rotundi" (1583)". He based it on "Latin secantem (nominative secans) "a cutting," present participle of secare "to cut" (from PIE root *sek- "to cut")." (Quotes from etymonline, the on-line etymological dictionary). So I think the phonetic similarity is due to chance.