Marcus's new book on Amazon here: amzn.to/3xrujmS (US) amzn.to/3jmBJD1 (UK) Marcus on the Numberphile Podcast: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-PVSkzNOXG1k.html And a Numberphile video about Gödel's Incompleteness Theorem: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-O4ndIDcDSGc.html
Why cant we calculate the perimeter of a oval? Whats the difference between an oval vs a rectangle with curved corners? Is it the same? I think ovals aren't real shapes. I think there irrational shapes.
Are you sure that the balance is properly constructed? That design often crops up in physics tests, and people often get the wrong answer in that they argue that it will automatically be horizontal if the weights on each side are equal.
“Normally if given a choice between doing something and nothing, I chose to do nothing. But I will do something if it helps someone else to do nothing. I’d work all night if it meant nothing got done.” - Ron Swanson The essence of this quote
As with all of these excellent interviews, Brady does an outstanding job of stimulating and directing the presenter in each case. That is not at all a common skill, and he does it with understated grace. He asks a clever question and gets out of the way for the presenter to answer, and lets him answer. And the graphics merge well. Very nice interview, and very well edited. Just excellent.
In electrical engineering, I was always so impressed with how much easier phasors and complex numbers make analysing AC circuits. You can either do a bunch of hard differential equations or you can just use algebra.
Same here. Complex variables was the one mathematics course that (almost) literally made my head explode. I had been exposed to transforms previously but none quite as practically useful as that one.
OMG YES! I did the exact same thing in electrical school. The way they taught us to solve AC circuits was basically by using phasors but decidedly _without_ complex numbers. I dropped out of electrical engineering but I've always loved the idea of imaginary numbers - at first it was honestly just because of how whimsical they sounded. I tried to show a few people how to use the complex mode on our calculators instead of having to make a table of orthogonal components every time but it didn't really catch on. Oh well, it was still pretty cool to feel like I had a sort-of shortcut and the semester in EE wasn't a total waste.
In fact, you can get to 81 with only 4 weights - 2, 6, 18, and 54 - if we assume that we only have to weigh exact integers. The key is that we can cheat with inequalities. For instance, we can weigh x like this x > 2 x + 2 < 6 Thus, x < 4 3 is the only integer between 2 and 4, so x=3. 81 can be counted as x > 2+6+18+54
Oh, that is a nice trick. But yeah, you're assuming not only that you want to get an exact integer but that you've been *given* an exact integer. The original problem allows you to weight out a specific integer amount of, say, sand by assuming equality. So you could answer the question "How many kilos of sand is in this bag? (By the way, it's an integer)" if the answer is 3, but you can't weigh out 3kg of sand like this.
Why do you liken the theorem to a shortcut? What was it a shortcut to? I think a better analogy is that a technique that enables you to do something easier than it was possible before is like a shortcut. So you might see a "shortcut" used in a proof. Or used in a later simpler proof of some theorem which previously only had a hard proof. But not the theorem itself.
@@rosiefay7283 because the theory is one line, whilst the proof is hundreds of pages. So, knowing the theory is true, allows you to use "the shortcut".
@@robertelessar Glad you enjoyed it. He has several books of such sayings, which he called Grooks. That one is on the first page of Grooks 1. Check out his Wikipedia article Piet_Hein_(scientist)
One of the earliest examples most people encounter of a mathematical shortcut is addition, which is a shortcut to counting - 7+5 means "start at 7 and count 5 more", which isn't too bad, but 700+500 would take you several minutes to count up (as well as needing some way to keep track of when you'd counted the 500 more), but if you know addition, you can work it out in seconds. And then multiplication is a shortcut to repeated addition in a similar way.
I was thinking about this the other day. There's the meme of mathematicians being bad at arithmetic. What if the people who go into math _are_ the people who are bad/lazy at arithmetic, so they looked for shortcuts? The shortcuts during learning, ironically, can lead to a much deeper understanding and appreciation for the math.
The people who are truly bad at math are those that are unable to think critically and apply the principles that they've learned. "Learned" being that they had an understanding of the reason why something works at some point. Without being to apply principles that they've learned and critically think in ways to connect these concepts together then it doesn't matter how many shortcuts are presented to them because they'll have no idea how to use it and when to use it.
@@nomathic7672 yeah, that's why I specified bad at arithmetic. There's also the people who are great at "manipulating equations" (aka math up to high school) but find out _math_ isn't for them when they encounter proofs in college. That was many of my fellow math majors.
@@rupen42 I definitely agree that arithmetic or "manipulating equations" is a very different skillset from high level math, but I'd argue that there's a much simpler and equally important reason why mathematicians aren't superb at arithmetic. They haven't had to do basic arithmetic in years. They're not lazy, they're just rusty. The little tricks and methods to quickly / accurately do arithmetic need to be practiced. If you spend 10 minutes a day doing arithmetic you'll stay sharp -- but they haven't.
I found way more interest once I learned the number theory behind the rules behind maths as opposed to just accepting them. They're all derived from some basic set if rules.
That's the big piece that a lot of educators unfortunately skip. My high school, luckily, basically taught all math as if we were inventing the methods ourselves and that helped a lot with understanding.
@@evanbelcher Agreed. Those who are taught mathematics’ core concepts, rules, & basic theory bf high school have far greater opportunity & opportunity to succeed. 🧮
Basically all of software engineering is built on shortcuts and abstractions. No programmer would be able to make anything if they had to worry about every detail of how a computer works, but since we can build programs that use previously written and tested libraries and APIs, all of that complexity goes away and you can focus on just the problem you want to solve. A bit like proven theorems in math.
I mean, you *could* write programs, from scratch, all the way down to the metal. Indeed, *somebody* wrote all those libraries & APIs. But if every programmer had to do that, it would be a pointless waste of time, it would involve far more debugging by each programmer, and there would be no compatibility between programs made by different programmers. These were all problems with early programming, *because* they hadn't agreed on libraries yet.
Until you're building software with 10000+ dependencies, and you have no idea whether or not they have security holes, or have been outright highjacked to inject security holes 😂
I got 1 3 9 27 and hence four weights is needed. Here's my thought: I starts from 1, obviously I need 1 weight. Now if I add one more weight, say x, I can cover 1, x, x+1, x-1 ，so naturally I choose x = 3 so that I can cover 1, 2, 3, 4. Now again, if I add one more weight y, I can cover 1, 2 , 3, 4, y±1,2,3,4 . so naturally I choose y = 9 so I can cover 1 to 13. Then again if I add one more z I can cover 1 to 13, z±13 and naturally z is 27 and I can cover everything up to 40. This method can go on and on.
Same as I got. I thought through them sort of.. Slower than that. Logically rather than mathematically. But once I saw the pattern, then it made sense. I'm also glad I wasn't the only one to pause the video for a few minutes and work it out!
@@viliml2763 and I thought about that too, but it was a decently safe assumption that turned out correct. I also thought, if there is a way to solve it with 4 different sized weights (i.e not 1,3,9,27), and the smallest wasn't 1,then you're using more material to make the weights, which presumably means they cost more. No, that's not part of the puzzle, but it's a fun little consideration. In fact, here's a question: is 1,3,9,27 the only 4-weight solution? And if not, what's the heaviest, or is there a heaviest?
Some teachers don't bother showing their students the beauty of the journey along the path of mathematics enabling their problem solving skills. Shortcuts are great once you've climed the mountain the hard way. People who take a rocket ship to the top can find that they are not acclimated to the climate and feel uncomfortable. If students aren't given the tools to derive shortcuts on their own they will always be dependent on teachers to hand them solutions rather than develop the solution through problem solving. Again shortcuts are great once the fundamentals have been mastered.
It was not “8 out of 10 cats” though. It was “8 out of 10 owners said their cat preferred it”. And after complaints to the Advertising Standards Authority, it was changed to “8 out of 10 owners *who expressed a preference* said their cat prefers it”. Says nothing about the owners who just said “eh, whatever” when they were asked, as they aren’t counted…
I suppose that varies from one country to another. Here in Brazil, Whiskas sued Friskies (Nestlé) because of the unsubstantiated slogan "8 out of 10 cats prefer Friskies" (oh, the irony). Later, Nestlé sued Masterfoods for the slogan "Cats prefer whiskas".
In art, you will generally learn things like the human body or other complex shapes as a series of simple circles and rectangles. You basically draw a cardboard tube mannequin and then start filling in details on top of that. There are lots of other shortcuts to draw attention to a particular place, make the picture stand out more, etc.
One of the best channels to learn mathematics in a fun way, this channel is really a " GEM " ! We wish we could make such high quality content one day and influence as many people as you do today ! This channel is one of the best examples which proves that all subjects are equal but maths is 100 times better than them any day .
The grocer with 4 weights can measure heavier integral weights too, all up to 80 Kg! Doubling the values in the video, the weights {2, 6, 18, 54} allos even integral weighings up to 80Kg. Odd amounts can be weighed as x > n and x < n + 2. With 3 weights, the grocer can measure integral weights up to 26 Kg this way. So the original problem still requires 4.
13:07 the TSP is only NP-complete for the general case. There are actually clever algorithms to solve it in polynomial time if the graph is embedded in a set number of dimensions, like cities in a map.
The shortcut perspective is very interesting. The work does need to be done upfront though with the proof but once that's solid, you can take the shortcut. It reminds me of how you have to put in the work upfront in other areas to be able to use the shortcut, like practicing an instrument as stated in the video. There's just different levels of "work upfront" for these different areas. I'm a Computer Scientist so our work upfront is coding something that we can then use a billion times to shave off some time that would be spent doing something manually. The whole of computing is standing on each others shortcuts and building these complicated systems from "stacked" components that others have written to save us the trouble of having to write the code ourselves. We try to write as little new code as reasonably possible. It's an interesting perspective. I love the video.
For the weights solution given at the end, that is a number representation called balanaced ternary. The traditional set of weights of powers of two represents the number in binary - each weight represents the place values. the weight being on the scale represents that place being '1' in the binary number, and the weight being off the scale represents that placebeing '0' in the binary number. Balanced ternary has three digits '1', '0', and '-1', and each place value is a power of 3. (typically some other symbol is used to mean '-1' so you don't have minus signs in the middle of the number). Again the weights represent the place values, on is '1' and off is '0'. And additionally, the weight being on the opposing side of the scale represents a '-1' for that place.
Thing is, though, if there wasn't a program to automate it, the task would have taken much longer. Or you might not have been able to spare the time and effort, so the task wouldn't have got done at all. It's a 10-second task only as a result of your 10 minutes of programming work.
I had watched a lecture from Marcus only 2 days ago. This popping up in my subscription was such a pleasant surprise. Such an interesting topic, thanks for a great video as always (:
Often this is what makes programming computers fun as well; you can either perform a boring mundane task over and over, -or- write a small snippet of code to do it for you. Sometimes writing the script takes longer than just doing the job, but you get more than the result in return, you get they joy of figuring out how to write the script!
I agree. And another thing: often you have a choice between something which is easy to write, where it's easy to see that it does the job, and something else which is subtler. Some people would condemn the former as brute force, and prefer a programmer to implement the latter because it's cleverer and perhaps does less work. But if the time saved in running the thing doesn't repay your effort in doing the harder programming, it's a false economy.
My initial thought for the weights was powers of 2 and just use binary. Then placing them on either side (duh!) was mentioned. My current answer is 4 weights, using powers of 3. This works, and works very neatly. Not sure if it's optimal, but I'll be very annoyed at myself if I don't get it.
yes, I also think that its powers of 3. And the choice of 40 as the maximum of the given range of numbers which should be covered is probably also a hint that the "4 weights using powers of 3"-solution is intended, because it fits so well to the "1 to 40"-range (as with this solution, all integer values from 1 to 40 can be measured, but not 41)
Winner. I also started "somewhat" on binary but on recognition of both sides I began exhaustively going 1,3; but then instead of jumping to 9 I lamely only went 5, which handled 5, 5+1, 5+3-1, 5+3, 5+3+1. Then the next needed would be 10. So 1,3,5,10,20,40. Owww! Glad I read your post. I have a new perspective on cubing now. It had no intuitive resonance to me previously, just being a "stronger variant of squaring." Thanks for the inspiration! P.S. Mother YT has been brutally deleting my innocuous messages, capriciously but relentlessly, so don't reload the page or I may be gone after you do!
The weight of any block can be multiplied by 3 numbers (-1,0,1), which means that the trinary system is the best one to represent the weights, for example if we want to represent the weight 22(in decimal) we first write it in trinary as 0211(in trinary) (i.e. 2*9+1*3+1*1=22) but because we have negative weights we can always replace any 2 by -1 and adding 1 to the next weight and, i.e. 0211 -> 1(-1)11 -> 1*27-1*9+1*3+1*1=22. another example: 25(in decimal) = 0221(in trinary) -> 1(-1)21 -> 1(-1+1)(-1)1 -> 10(-1)1 (1*27-1*3+1 = 25). again this is the correct answer because the weights have 3 states, however if the weights only have 2 states (1,0) the binary system is the best one.
London Underground Map, the dangers of topology: Circa 1987, in London, referring to the famous Underground map, I planned my route from where I was to where I wanted to be. After three trains (three legs), taking almost an hour, I emerged into the sunlight from the destination station. I immediately recognized from the surroundings that I'd only gone about a city block, less than 300m. I was just around the corner from where I'd started. I could have walked the distance in just a few minutes.
Wait, which stations were those, if you happen to remember? On the positive side, at least these days most navigation apps would hopefully spot that for you and suggest walking... in theory!
Done that myself on visiting London, ended up about quarter of a mile from where I started. Should of asked a local and quite literally walked round a corner instead.
I remember that when my 2nd grade teacher introduced herself she said "I like tricks", those tricks while not as useful with the math I do nowadays I still remember those tricks.
There are certainly shortcuts in science too. Finding the "right" way to measure something can give you access that's otherwise impossible or very difficult. The Wu experiment for parity violation is a great example of that. Often something as simple as measuring the difference of two quantities rather than their absolute size is a great shortcut, allowing many systematic errors to cancel to nearly zero.
I use the term "insights" rather than shortcuts. Spend your time finding the patterns, symmetries, constraints, etc. and then the solution to a problem is often obvious. And the same insights can be reused in other problems.
Shortcuts for athletics sounds very similar to finding a shortcut for NP. Once you’re skilled at one area or physicality you’ll have a huge leg up on looking into another area than someone that’s just starting fresh.
Calling this ahead of time: you need 5 weights (1,3,9,27,27) since every integer from 1 to 40 has a guaranteed ternary expansion and 2*3^n for the nth place is 3^(n+1)-3^n which is equivalent to adding an extra 3^(n+1) weight on one side of the scale and a 3^n weight on the other side.
Thanks for leaving this up. Too many people would've been too embarrassed, but this looks like a mistake I easily could have made, and is the closest (bar one) to a correct answer I've seen here.
@@france8607 ternary means "with base 3". Usually we write numbers with base ten, i.e in the so called "decimal" system, e.g. 121 = 1*10^2 + 2*10^1 + 1. In the ternary system, 121 would be the representation for 1*3^2 + 2*3^1 + 1 = 1*9 + 2*3 + 1 = 16.
The sum of the first N integers is N(N+1)/2, or N/2 times the sum of the first and last term, or N times the average of the first and last term. So f(100) is 5050. Here's another shortcut, which seems trivial from general principles, but still a handy application of that shortcut: 51+52+53+54+...+70 is f(70) minus f(50). So it's 2485 - 1275 = 1210. OR...just take then number of terms, 20, times the average term, 60.5 :) Or, like Gauss, 10 times (51+70).
1:28 Seeing as you need only cover every integer, you could use 2, 6, 18,... And then your stated upper bound of 40 is not felicitous (try 26 or 80 instead). The reason is that you can handle e.g. 9 by observing that it is heavier than 8 but lighter than 10.
Professor your appearance has changed drastically. Throughout the video I was scratching my head where have I seen your videos, only to discover that you made three part series on measurement on BBC.
Here’s a one line solution-> Let the no of min weights be n+1. The min number of weights imply that to get 40, all the weights should be exhausted and be used exactly once. That means 40 base10 = 11..1 base n where the number in base n has n+1 ones. It turns out 40 base 10 = 1111 in base n=3 since 40 = 3^3 + 3^2 + 3^1 + 3^0. Hence ans is n+1 = 3+1 = 4. The weights are 1,3,9,and 27.
I'm impressed that you got through twenty minutes on math and shortcuts without once using the word "heuristic". And for the record, the story about young Gauss is a personal favorite of mine, along with the one where at the age of three he corrected his father's payroll calculations ("Vater, die Rechnung ist falsch".)
Man, I thought I was so smart thinking "Aha! you want to add stuff up with the least amount of numbers? Just use powers of 2! (exclamation, not factorial) The answer is 5! (exclamation, not factorial)" Seeing the solution knocked me down a couple of pegs but reminded me to see the 'full problem' before simplifying it in my head.
Something about this really hit close to me. I am a developer and my main job is to find the correct shortcut for a problem. I really loved solving a problem my way and this video really bring back some good memories.
Rules of thumb...are shortcuts. For example in chemistry, "like dissolves like" is a rule. If you're wondering if salt will dissolve in butane, the answer is no because salt is polar and butane is non-polar.
I'd love a full Numberphile dive on P vs NP. It's such a broad problem and one that I still feel like I don't fully grip. The idea that if you prove ONE of those problems has a simple solution means ALL of them fall with it breaks my brain a little.
1, 3, 9, 29. I think. Firstly, by adding a weight to the other side, you are in essence subtracting that number from the original side. So, from 1 and 3, you can get 1, 2 (3-1), 3, and 4 (3+1). Then, you can subtract or add that from the next number along, which in that case is 9. 9-(3+1) is 5, which is the next number along, after all. Then, we see a pattern, which is that the next number is always 2x the previous numbers combined +1, since your new number, when subtracted by all the others, should have one more than the sum of all the smaller numbers. Thus, (1+3+9)x2+1= 29. Edit: Ufgufgiagfi looked back and realised that I messed up and thought (9+3+1) was 14. I would have gotten it right were it not for basic addition.
The lesson from Alan Turing is that all computation is addition. Addition is Turing complete. So if you are ever doing a calculation that is not obviously addition. its a short cut to some underlying addition.
Double-sided balance >> TRINARY SYSTEM. 1, 3, 9, 27kg. You can put weight in side opposite the unknown, or on the same side. You can add, subtract, or do nothing. 3^4 combinations are possible, every one measuring a unique weight, but about half of these measure negative weight. So you can measure from -40KG to 40KG (81 combinations). If you knew that the unknown was always an integer number of KG, you could limit the resolution of the test weights to 2Kg, permitting weighing up to 80Kg before you need to add another weight to the set.
The lesson I heard here is abstraction. Abstract the problem into a simpler model and solve that instead. Abstract pieces of that abstracted problem. Divide it up as small as you can, into its atoms, and remove those that aren't necessary to the next level up. I use this idea all the time as a software engineer. The most fun I have in my job is when I'm confronted with a difficult problem and told to fix it. I break it down into the data I have available and the data that's needed at the output. Then I can start finding patterns in the transformations to get from A to B and combining like terms to simplify the overall issue into a much more manageable one. In the best cases, I come out with something that not only solves the problem at hand, but solves other, seemingly unrelated problems as well and problems that had yet to come up. Extensibility and reusability is at the core of my development style.
In the natural sciences, there is shortcut strategy in common with mathematics - generalising. That is, once we have solved the specific case (for particular subatomic particles or molecules or species), can and how can that be generalised to help us solve other problems that share some similarities? This then builds towards developing predictive theoretical frameworks in whatever particular subfield you are working in. Mathematical biology is full of this.
The Whiskas ad actually claimed that 8 out of 10 _owners_ said their cats preferred Whiskas. Actually, in their market research, most owners said their cats didn't care, and only 8 out of 10 who expressed a preference said their cats preferred Whiskas.
If you add 1-10, you get 55. If you add 11-20, you get 155. Do this 10 times and you get 55, 155, 255, etc. So its 10(55) + (10-1)(100) = 550 + 4500 = 5050. Gauss's shortcut was way better, but I like mine too
The answer to the weight problem seems very closely related to the Towers of Hanoi puzzle, based on the way they were used to count up to 40 at the end of the video. Is this just a quirk of how the animation was done?
I see the visual similarity but can't think of any similarity in even how to analyse these two problems. But if you can find it, you can probably publish it! =D
maybe if you have not on a scale as a third pile you can draw it similar to the tower of hanoi. Maybe there is a connection between steps of 1 are possible(cointing to 40) and you dont have to pick up more than 1 at a time(tower of hanoi)
yes, that works ... but it is not the optimal solution ("we are looking for the *smallest* number of weights...") , because there exists also a solution with 4 weights
Seeing the enemy army, and instead of counting every soldier you count how many rows and columns and just multiply... or the practical person will instead ballpark that there's a lot of enemy soldiers with a glance and get out of there instead of counting ;)
It's interesting that Marcus's book is entitled *Thinking* Better, but his attempt at improving on the cello revolved around muscle memory. No wonder that didn't work! The shortcut to developing music skill is *thinking* music and developing a vocabulary of Tonal Patterns and Rhythm Patterns. We call this audiation. No one is too old to do this. But unfortunately most music teachers aren't versed in how to teach this. I'm working on changing that. No, this shortcut won't change your instrumental skills overnight. But it will change them over time. Whereas the *playing-is-muscle-memory* approach usually results in frustration or boredom, and quitting.
Best I could do was: 2, 3, 4, 10, 30 In the spirit of the Parker Square I gave it a go. Not the best possible answer, but it’s mine and I’m proud of it!
The field of complex numbers is not necessarily breaking the rule of the reals, it's just R[x]/(x^2+1) so you don't even need the imaginary unit i, one can simply take the set of all real polynomials mod x^2 + 1 as your field of complex numbers
I’ve been putting off watching the video so I can figure out the answer: 4 weights for 40 pounds! The weights are 1,3,9,27. In general, the largest integer we can make with n weights (assuming we can also make all of the integers between 1 and n-1) is the sum of the powers of 3 from 0 to n-1. I have an easy induction proof for this too!
Polynomial time complexity with a big problem size while quicker than exponential for the same problem, is not always quick, for example n^4 such as checking 1000^2 small image patches from a square image against another image of the same size results in a trillion checks. And exponential often means you can use smaller problem sizes and answer more profound questions, for example checking the 2^40 programs that are up to 40 instructions long, with a sensible timeout, can require roughly a trillion runs.
Using an extra trick, we can use 4 weights to measure up to 80kg of bananas. Just double each power of 3 to get: 2, 6, 18, 54. Since we know we're only measuring "whole number units" (1:11), we can solve for an even number weight as before and we can solve for an odd number weight through elimination: If 4 < x < 6, then x = 5.
Something that used to confuse me: People always say the traveling salesman problem is in NP, but I never understood how you could quickly check that a given route is the shortest. Isn't that just has hard as the original problem since you need to check it against every other route? It turns out that finding the shortest route is _not_ in NP as far as we know. What _is_ in NP is a different version of the problem where the goal is just to find a route with length less than a given number.
Thank you. Alarm bell started to ring in my head when he said that. Also be overstated what follows from the 'at most to odd vertices' to the bridge problem. The argument he gave shows this condition is necessary, but not that it is sufficient.
This is also how I like to think about theoretical mathematics. More often than not the more applied the general problem formulation is the more theoretical value it contains.
My dad taught me there were 49 sets of 1+99=100, 2+98=100, and so on, then you add the 50 and 100 that are left out. But the sets of 101 are a fun shortcut too.
It is also said that laziness is the main virtue of software engineers - and since algorithms are basically math, that fits pretty well. The other two are impatience and hubris, btw.
in software lambda calculus proved so much of a shortcut that it's essentially replacing most forms of type inheritance that wasn't already displaced by moving to object composition where practical and newer programming languages have much less rich object-oriented features as a result too bc it's just not necessary
I hope that Professor du Sautoy will do another BBC Radio podcast series, or perhaps another BBC TV documentary, about Maths or History of Maths. The previous ones were very enjoyable and informative. Cheers Professor !! Thanks Brady. Stay well.
I started learning programming in order to automate my work, and 15 years later I do automation for a living and work harder than ever. It's surprising how much of a motivator laziness can be.
I figured out what Gauss did before ever hearing about him. Not as simple as his answer but came up with (n/2)+(n^2/2). All while sitting on the toilet thinking about the 12 days of Christmas song and wondering how many gifts are given. I felt like a genius in that moment when the formula worked.
When I studied Applied Math in 1970 it seemed to me that its purpose was to find the scenarios within domains of hard problems that could be solved with tricks so the solutions could be applied to Engineering. Because it was infeasible to manually calculate enough terms of the Taylor series to get a usable result (so still not precise enough for Marcus to call this a shortcut). But then computers rapidly pushed back the frontier of what real world problems could be solved. I switched to CS. At that time timesharing computers already existed that could do symbolic manipulation of groups so even college students could suddenly do much harder Pure Math problems. In a sense, the shortcut there was the use of a digital computer.
REAL STORY: As i was a kid, and i eat at my uncles house, he always gave me a coat under my plate. On this coat was every number, from 1 to 10 and on every number was small elephants, climbing this numbers. one elephant on the number 1, two elephants on the number 2 and so on... i always counted the elephants. it was 55. i thinked, and asked my self, why it is 55. i saw what Gauss saw: a pattern. 1+9, 2+8, 3+7, 4+6 plus 10, and the 5 was the last one. thats why 55 :D
A mathematician was preparing a lecture, and saw the caretaker counting the lights in the ceiling: "11, 12, 13,..." "It's 48" "How did you do that?" "Well, it's 6 this way, 8 that way, 6*8=48" "Oh, sure. But I need to know exactly. 14, 15,..."
With 4 weights it is even possible to go to 80 kgs: Step 1: Solve the problem for each number divisible by 2. Analogous to the solution in the video, you will need the following weights: 2, 6, 18, 54. Step 2: Whenever you weigh something that is more than n-1 and less than n+1 you know it must be n. 🙂
Let's examine Brady's bridge problem, shown at 6:00. The West bank shall be labelled A, the North B, and the South C. Starting from the West, we have island D, then E North of F, G North of H North of J, and K. (Plot of land, Bridges): (A,3),(B,6),(C,2),(D,2),(E,4),(F,3),(G,4,),(H,2),(J,2),(K,2) As there are exactly two plots of land, or nodes, with odd bridges, or edges, there is a way to cross all bridges exactly once starting at A and ending at F, or vice versa. One such path is described as follows: A,B,C,D,A,E,F,E,C,G,C,K,G,H,K,F. Q.E.D.
When I first started to learn calculus, I really didn't like it, because when you calculate derivatives you have to divide by some hypothetical number that's "infinitely close to zero" without being zero, and I felt like that was cheating. When I finally realized that there's no rule against cheating as long as you can explain yourself, my appreciation for math increased dramatically.
There is actually a way in which you can make this whole "dividing by something infinitely small" business very precise, which is typically not taught in most math courses as it requires quite a bit of background work. One can prove that we can extend the real numbers by so-called "non-standart" reals which are closer to zero than any "standard" real and thus infinitely small in that sense.
The resulting theory is called non-standard analysis and a great example of a shortcut in mathematics. It requires quite a bit of work to set it up, but once you have, it makes many proofs of classical analysis results a lot shorter and easier.
