Norway Math Olympiad Question | You should be able to solve this! 

Much simpler to just calculate 2^18 first, then minus 1. I don't see why 511*513 is any easier to calculate than 512*512.

As a computer enthusiast I remember powers of 2 up to 2^16 at least which is 65536. Then I multiply it by 4 and subtract 1.

This cannot possibly be a math Olympiad question. This is way too easy.

If you remeber that 2^10=1024, 2^18 = 1024x1024/4. The division first will bring ...6 (divide 24 only by 4), then 6 multiplied by 1024 will bring 4 as the last digit. Can be done in the head like 5 seconds.

This is cool; using the mathematical principles that we already know in unorthodox ways is a good way to strengthen one's skills

I did this in my head a little differently, but it could be done on paper the same way. Binary numbers are easy to work with and from computer experience, I know that 2^8 = 256, so 2^9 = 512. So, 512 is squared by (500 + 12) x(500 + 12), we know that (a + b)^2 = a^2 + 2ab + b^2. The rest is trivial, 500^2 = 250,000. 2 x 500 x 12 = 1,000 x 12 = 12,000, and 12 x 12 = 144. 250 + 12 = 262, so we have262,000 + 144 - 1 = 262,143. If you are not familiar with binary numbers, it is a simple matter to count up, 2, 4, 8,, 16, 32, 64, 128, 256, 512.

Though there are many ways to arrive at a solution, this one really beat around the bush lol

I'd write 2^18 = (2^3)^2^3 =(8^2)^3 = 64^3 = 64 x 64 x 64 = 262,144 ----> then subtract one unit, resulting to 262,143 :)

you can use congruence of numbers to solve this also a to the n and b to n identity

I prefer (2^10)^2 / 4 = (1000^2+2*24*1000+24^2)/4 = 1,000,000/4+12000+12^2 -> 262143 (minus 1)

Thanks, but It isnt olimpic way to do It. The better way I Saw the resolution is: 2^18 = 2*9 * 2*9 = 512*512 = (500+12)(500+12) = 500² + 2*12*500 + 12² = 250.000 + 12.000 + 144. Subtracting -1, we have 262.143 without a lot of multiplications

2^18=4^9=64^3=64*64*64=262,144 That minus 1=262,143 Multiplication is cool, it doesn't bite you.. Try it for small numbers. 😉

Mam... It is difficult to understand.please explain easily.

Is this the intellectual level of Norvegiens? In five second you solve it with the calculator.

If this was really an Olympiad question, the level is decaying... 😅

((500X500) + 2(500X12) + (12X12))-1

I’m a programmer, so I already have all the powers of 2 burned into my brain. So just minus one.

I literally multiplied 2 by itself 18 times and found the answer faster than her 💀💀

The correct answer, in binary, is: 111111111111111111.

Just wanted to share a different approach: 2^18 - 1 = 2^10 x 2^2 x 2^6 - 1 = 1024 x 4 x 64 - 1 = 4096 x 64 = 262 144 - 1 = 262 143

Is there a reason why we would know what 2^9 is and not know what 2^18 is? That is really the essence of this problem.

2 to the 18th is 2 to the 16th times 4 (or 2 squared). Anyone with a computer brain knows 2 to the 16th is 65536 (good old computer memory and bits). So times 4 is 262144, subtract 1 = 262143. Everyone should be able to do this in the modern world in their head.

1:02 "As we know, 2 to the power of 9 is equal to 512" fym we know I don't know are we supposed to???? Anyways serious commendments for having that knowledge

Children of the 8 bit era certainly know all the powers of 2 up to 16; So if you already know that 2^16 is 65536 you may as well go for the direct route. Either double it twice, or multiple by 4 (which ever you find easiest) and subtract 1.

I do not see how this is more simple than 512x512-1. We all know 2^16=65536, so double it twice in a row and substract 1. More over 513x511 is quite obvious, 1x513 is obvious and everybody knows 13x5=65

Prefiro multiplicar 512 por 512 e subtrair da unidade Em duas linhas o problema é resolvido

as a programmer, I have remembered many 2^x values without calculating them, for example, 2^24 is 16777216 and 2^16 is 65536. Oh yes, 2^18 is 262144, so if it is minused by 1😊

Since there are many similar problems, most people probably know how to calculate them easily without making mistakes. You need not stick to the factorization formula. There are also concrete examples in this comment section. As a solution method, (1) decompose them into round numbers. That is, 2¹⁰ =1024=1000+24, 2⁹ =512=500+12, 2⁸ =256=250+6. (2) subtraction should be avoided, and addition should not carry over (no overlap of the same digits). It is unavoidable that there is a single overlap in this problem.

This is how we make an easy mathematics question into a complicated one

This is click bait or padding on the back making you believe that you can understand an Olympiad problem. Seriously a square minus b square is 6th grade regular curriculum.

we might know 2^18 = 2^(6*3) as being the number of RGB colors you can encode when each Red, Green, Blue color component has 6bits resolution (64 levels) and that number is 262144 ... at least people should know 2^10 = 1024 and 2^16 = 65536 ;)

I have more options which is easier and faster than her. She is using longest and complecated method.🙁

This one is hilarious. When I saw it I thought, surely you just solve it using arithmetic? But hey if there's a trick to working with exponents of 2 I've love to know it. No - you go through a convoluted process which gets you nowhere, then just solve using arithmetic anyway.

We could just do 513x511 and it would be done so much faster ;-;

Duh, it's 111 111 111 111 111 111. Hey, you didn't specify the number base.

With the base numberof 2, it is easy to double the first at least10 - 12 times. And with pen and paper, you have no problem reaching 18. This is easily grinded out without fancy math shortcuts :)

Why all the complicated brackets and splitting, adding, subtracting, changing powers? I don't understand any of it. Why doesn't she just go 2*2 18 times and the take one away? I got the answer that way by about 90 seconds and just needed my fingers to keep track of the multiples and my head to multiply by 2. Why make it so hard?

It took me awhile, but I finally realized that this was quite an easy problem, and I thought that I might be able to do it in my head by simply doubling 18 times. Well, I didn't get that far on my first three attempts. but I now know that I can do it...eventually. How far did I get? Thirteen or fourteen times, but that was good enough for me. I now know that I can solve the problem in my head if I try long enough.

Hey. Just take the tenth power of 2 i.e. 1024, multiply it by itself, and divide the result by 4.... subtract 1 from the resulting value to get 262143

Well ... first question should be, what kind of result is expected. Because 2^18-1 is pretty nice. 11111..11 binary is also good. It seems, that normal writing multiplying 513*511 is also trivial. And if I use pretty known fact, that 2^16=65536 ? Two addition and almost done.

De tête, la réponse est : 111 111 111 111 111 111 (en base 2). Beaucoup plus simple 😅

In 2^n, for n = 1, unit place digit is 2 For n = 2, unit palce is 4 For n = 3, unit place is 8 For n = 4, unit place is 6 This cycle of 2, 4, 8, 6 will repeat for every power of form 4m+1, 4m+2, 4m+3, 4m+4. So if n = 18 it is of form 4m+2 so units place is 4, and subtracting 1 from it will give 3.

lol in India, we use an identity to do 511*513, it's basically, (x+a)(x+b)=x^2+(a+b)x+ab this eases out the calculation so you can do (500+11)(500+13) 500^2+ (11+13)500+ (11)(13) very easy calculations, you can do them in mind 250,000 + 24*500 + 143 = 250,000+12,000+143 =262,143 (we also have a trick to multiply numbers with 11 and rest of the calculations were pretty simple).

I love Math, I love your channel and I love you. Many blessings for you, from the heart of God. ❤

This is where I learn my maths...

Now, to chunk it in head, multiply (1000 x 256) + (2 x 10 x 256) + (4 x 256) - 1 If you practice chunking, you can do complex math REALLY fast in your head. You’ll also start to be able to memorize really long strings of numbers…like pi! As a math teacher, my students are often dazzled by this; it’s really VERY easy!

Experts in computer hardware in 1990s to early 2000s should know 2^18=262,144 that's the amount of 256MB memory in KB that would show up during start up of a computer of Pentium II/III or Athlon era.

She has been trying to find the result of the 513*511 operation for exactly 2 minutes since 1:17 seconds of the video. If she writes these two numbers one under the other and multiplies them, she can get the result in 20 seconds.

It's a lot easier to solve if you just know your powers of 2 up to 18, as any decent computer programmer does.

Already we knew 2 power of 9 is 512..then 512* 512 is 262144..finally subtract 1 then we will get 262143

elegant but Id rather just calculate 2^18 and then -1 at most, do 512x512 -1

262,143. I just doubled 2 seventeen times then subtracted 1. Did it in my head.

Some nice tricks, but what about, in this case, just calculating the thing? 2^18 = 2^10*2^8=1024*256=256000 + 24*256. The latter is 25*2^8-256=100*2^6-256=6144. So we get 262144; subtract one and you get 262143, with the "25 and we can make that 100, it's a power of 2" happening in one's head (if you don't get that, you can also easily calculate 1024*256 simply on paper). You would, of course, know the first 10 powers of 2 by heart (they teach that in school, just like the first 20 squares)... but then the solution represented also relies on the fact that you now 2^9=512 by heart. But then, squeezing in a binomian formular was rather beautiful. (I do mean that.)

i just love how apparently 2 to the ninth power is common knowledge.

See if you play 2048 you’ll already have all powers of 2 memorized up until 2^11, so just take 2^9 as 512, square it, and subtract 1

Am I the only person that just did the math in my head and calculated the same result? I understand the method behind that, but you can easily just calculate the result in less than 1 minute.

U know what 2 to the power 6 =64 and now find 64 to the power 3 and subtract 1. Easy....

U make it more harder than normal solution 😂😂

Any old school computer scientist can answer it without by without even drawing on paper. 2^18 is close to 2^16, which is the limit of array size in an 16 bit OS. So its just 4x65536 - 1.

How many people did not understand the beauty of the presented solution! 😢 Thank you for what you are doing! That was pure mind pleasure! ❤

Wouldn’t it be easier to multiply 512*512 using grade school math and subtract 1?

maths which is the only thing can be done by various methods ❤

THANKS FOR NICE SHARING

Зная степени двойки(а в Российских школах дети это знают, так же как и таблицу умножения) вычислить этот можно за минуту. 2^10*2^8 - 1= 1024*256 - 1= 262143. Привет из России!

This solution method is like using a chainsaw to cut bread...

Very helpful

I just simplified in my head to 512^2 - 1. Pretty easy to multiply 512 x 512 on paper and subtract 1.

It is surprising that back in the 80-90s of the last century, almost all schoolchildren (at least in my country) in grades 6-7 were able to multiply 513 by 511 without a calculator and expansion. I think that in 20 years, schoolchildren will not be able to even multiply 500 by 500 without gadgets.

Instead of going for FOIl 1:43 method.. just multiply 513*511 = 262143

I wish you were my maths teacher when I was little! I would have definitely developed interest on Maths. You make the problems look so easy.

rly stupid solution in XXI century. It's almost well know fact that 2^16 is 65536 (256*256) and this you need only multiply by 4 and minus 1 at the end.

A very nice solution! Thanks.

For those who are used to binary and computers, simply do 1024 (2^10) x 256 (2^10) = 262144, and subtract 1, or start at 65536 (2^16) and double it twice.

You can simply do a difference of squares : 2^18 - 1 = (2^9 + 1) (2^9 - 1) = 513 X 511 = 262 143.

So we "know" that 2^9=512, but we cannot find out 2^18=512^2 so we need to write a dissertation to calculate 2^18-1

Как то очень примитивная задача, это для какого возраста? Тем более что сейчас многие знают что 10**8 = 256 или что 10**10 = 1024. Поэтому можно просто в лоб 1024*256-1

2^20 is 1048576 , then divide by 4 and subtract one...

Wouldn't it be easier to multiply directly 512 * 512 = 256,000 + 5,120 + 1,024 = 262,144 and just to subtract 1?

It's more complicated and longer. If i know that 2 to 9th is 512, just do the 512*512, minus 1.

a^2 - b^2. But the -1 was not squared. 🤷‍♂️

Much simpler to calculate 2*2*2*... iteratively in your head. Especially if you know that 1 Mb = 2^20 = 1048576 bytes

Wow...all this, instead of saying that 18=3x6. Then 2^18 = (2^6)^3 = 64^3

Who works in IT knows that 2^20 is 1048576. So divide by two twice and you have result :-). It would be 256*1024.

1024×256 -1 . I'll calculate this quicker then the slow method in video.

Why not just 512 x 512 - 1. Quicker than doing the additional factorisation

I just know - from years of doing computer science work - that 2^10 is 1024 and 2^8 is 256. 1024 * 256 is really easy, because 1, 2, and 4 are just doublings of 256 and adding them in different 10s places. then subtract 1.

Probably the simplest and shortest way: 2^18-1=(2^9)^2-1=(512)^2-1 Applying Vedic Math: 512*512=524*5*100+144=262144 (which can mentally be calculated in about 15 sec) Therefore: 2^18-1=262144-1=262143

"As we know 2^9=512" So u should know also that 2^18=262,144.... Pointless

Easier decomposing 2^18-1 as a difference of squares, then as a difference of cubes times a sum of cubes . At the end: 63 times 73 times 57= 262,143

Based on the rules of exponents 1 = 2^0. So, my next step was 2^18 - 2^0 equaling 2^18, which is incorrect. Why does this method not wok?

I tired everything like G.P series Limits (calculus) Logarithms Binomial At the end all the results were in the power of 2 some even exceeding 2^18 😅 so its better to learn some power table as 2^10 always helps

This seems like a very primitive task, what age is this for? Moreover, now many people know that 10**8 = 256 or that 10**10 = 1024. Therefore, you can just straight up 1024*256-1

Answer is 3FFFF (in hexadecimal) 🤭

2^10 x 2^8 - 1 = 1024 x 256 - 1 = 262144 - 1 = 262143 OR 2^12 x 2^6 - 1 = 4096 x 64 - 1 = 262144 - 1 = 262143 I would think like that, because I'm used to calculate until 2^11, so 2^10 and 2^12 are easy. If I didn't feel confident multiplying 2 numbers with more than 3 algarisms, I can separate 18 in 12 and 6, because 2^6 is the last potence of 2 with 2 algarisms, so it's easier to calculate 4096 x 64. I wouldn't use factoration because it's not necessary, and my first thought was 1 = 2^0.

I was hoping for some intelligent method

2 up too 18 is 524288, 2 minus 1 is half of 524288 is 262144

How do you know that 2^9=512? Did you use calculator? So you can calc 2^18 as well. Do you remember it? But some people do not, and some people remember (myself included) that 2^18=262144

2^18 = 2^20 / 4 = 1048576 / 4 = 524288 / 2 = 262144. Actually, multiplying by 2 is very simple, so if you have a good trained memory, you can multiply by 2 in memory. Also if there is no rule that requires to provide answer in base 10, you can just write that (2^18 - 1) base 10 = (1000000000000000000 - 1) base 2 = 111111111111111111 base 2 😊

What is the meaning of this🤨 just use the calculator men😂😎😏

That's a complicated way to find $3FFFF 😆