Mental arithmetic
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There's an artwork by Nikolay Bogdanov-Belsky called Mental Arithmetic. In the Public School of S.Rachinsky, 1895 - Nikolay Bogdanov-Belsky - WikiArt.org[^] The interesting part is the task on the blackboard. (102 + 112 + 122 + 132 + 142)/365 Like the Russian boys, you have no calculator and no paper. Upvotes for: 1. A good reasoned guess at the answer 2. The exact answer, with an explanation of how you got it by mental arithmetic. <edit> I should have expected you to brute force it in your heads. (You did do it without paper or calculator, right?) So from now on I will upvote elegant solutions!</edit>
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
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There's an artwork by Nikolay Bogdanov-Belsky called Mental Arithmetic. In the Public School of S.Rachinsky, 1895 - Nikolay Bogdanov-Belsky - WikiArt.org[^] The interesting part is the task on the blackboard. (102 + 112 + 122 + 132 + 142)/365 Like the Russian boys, you have no calculator and no paper. Upvotes for: 1. A good reasoned guess at the answer 2. The exact answer, with an explanation of how you got it by mental arithmetic. <edit> I should have expected you to brute force it in your heads. (You did do it without paper or calculator, right?) So from now on I will upvote elegant solutions!</edit>
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
Answer: 2 (ALL IN M'HEAD) I pretty much know those squares by heart. First 3 = 365 (already a hint): 1 Next two 169 + (200-4) = 365: 1 1 + 1 = 2
"The difference between genius and stupidity is that genius has its limits." - Albert Einstein
"If you are searching for perfection in others, then you seek disappointment. If you seek perfection in yourself, then you will find failure." - Balboos HaGadol Mar 2010
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There's an artwork by Nikolay Bogdanov-Belsky called Mental Arithmetic. In the Public School of S.Rachinsky, 1895 - Nikolay Bogdanov-Belsky - WikiArt.org[^] The interesting part is the task on the blackboard. (102 + 112 + 122 + 132 + 142)/365 Like the Russian boys, you have no calculator and no paper. Upvotes for: 1. A good reasoned guess at the answer 2. The exact answer, with an explanation of how you got it by mental arithmetic. <edit> I should have expected you to brute force it in your heads. (You did do it without paper or calculator, right?) So from now on I will upvote elegant solutions!</edit>
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
It appears to me the problem is summing the terms of numerator - not multiplying them. Isn't that a '+' between the terms? That's a considerably easier problem and one I can probably manage.
"They have a consciousness, they have a life, they have a soul! Damn you! Let the rabbits wear glasses! Save our brothers! Can I get an amen?"
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It appears to me the problem is summing the terms of numerator - not multiplying them. Isn't that a '+' between the terms? That's a considerably easier problem and one I can probably manage.
"They have a consciousness, they have a life, they have a soul! Damn you! Let the rabbits wear glasses! Save our brothers! Can I get an amen?"
:wtf: You're quite correct, fixed it.
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
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There's an artwork by Nikolay Bogdanov-Belsky called Mental Arithmetic. In the Public School of S.Rachinsky, 1895 - Nikolay Bogdanov-Belsky - WikiArt.org[^] The interesting part is the task on the blackboard. (102 + 112 + 122 + 132 + 142)/365 Like the Russian boys, you have no calculator and no paper. Upvotes for: 1. A good reasoned guess at the answer 2. The exact answer, with an explanation of how you got it by mental arithmetic. <edit> I should have expected you to brute force it in your heads. (You did do it without paper or calculator, right?) So from now on I will upvote elegant solutions!</edit>
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
About 160,000,000 - my mental arithmetic runs out of registers to remember digits in after 5 or 6 digits ... and long division starts using 'em up fast ... :laugh: But some of it is easy: 102*112*122*132*142 == 100 * 121 * 144 * 169 * 196 (10*10 is easy, and each square adds 2 more than the previous: 100 -> 121 Adds 21, 121 -> 144 adds 23, so the next two terms are 144 + 25 and 144 + 25 + 27) We were rote taught our "times tables" up to 12 by 12, so the first three are imprinted on my brain... long multiplication in your head is reasonably easy as long as you keep the decimal places straight. [edit] Oh, right, that's easier: 100 + 121 + 144 + 169 + 196 = 730 730 / 365 = 2 Easy peasy! [/edit]
"I have no idea what I did, but I'm taking full credit for it." - ThisOldTony "Common sense is so rare these days, it should be classified as a super power" - Random T-shirt AntiTwitter: @DalekDave is now a follower!
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Answer: 2 (ALL IN M'HEAD) I pretty much know those squares by heart. First 3 = 365 (already a hint): 1 Next two 169 + (200-4) = 365: 1 1 + 1 = 2
"The difference between genius and stupidity is that genius has its limits." - Albert Einstein
"If you are searching for perfection in others, then you seek disappointment. If you seek perfection in yourself, then you will find failure." - Balboos HaGadol Mar 2010
This
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About 160,000,000 - my mental arithmetic runs out of registers to remember digits in after 5 or 6 digits ... and long division starts using 'em up fast ... :laugh: But some of it is easy: 102*112*122*132*142 == 100 * 121 * 144 * 169 * 196 (10*10 is easy, and each square adds 2 more than the previous: 100 -> 121 Adds 21, 121 -> 144 adds 23, so the next two terms are 144 + 25 and 144 + 25 + 27) We were rote taught our "times tables" up to 12 by 12, so the first three are imprinted on my brain... long multiplication in your head is reasonably easy as long as you keep the decimal places straight. [edit] Oh, right, that's easier: 100 + 121 + 144 + 169 + 196 = 730 730 / 365 = 2 Easy peasy! [/edit]
"I have no idea what I did, but I'm taking full credit for it." - ThisOldTony "Common sense is so rare these days, it should be classified as a super power" - Random T-shirt AntiTwitter: @DalekDave is now a follower!
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There's an artwork by Nikolay Bogdanov-Belsky called Mental Arithmetic. In the Public School of S.Rachinsky, 1895 - Nikolay Bogdanov-Belsky - WikiArt.org[^] The interesting part is the task on the blackboard. (102 + 112 + 122 + 132 + 142)/365 Like the Russian boys, you have no calculator and no paper. Upvotes for: 1. A good reasoned guess at the answer 2. The exact answer, with an explanation of how you got it by mental arithmetic. <edit> I should have expected you to brute force it in your heads. (You did do it without paper or calculator, right?) So from now on I will upvote elegant solutions!</edit>
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
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Answer: 2 (ALL IN M'HEAD) I pretty much know those squares by heart. First 3 = 365 (already a hint): 1 Next two 169 + (200-4) = 365: 1 1 + 1 = 2
"The difference between genius and stupidity is that genius has its limits." - Albert Einstein
"If you are searching for perfection in others, then you seek disappointment. If you seek perfection in yourself, then you will find failure." - Balboos HaGadol Mar 2010
W∴ Balboos, GHB wrote:
I pretty much know those squares by heart.
Obviously, why did I expect anything else. :)
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
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There's an artwork by Nikolay Bogdanov-Belsky called Mental Arithmetic. In the Public School of S.Rachinsky, 1895 - Nikolay Bogdanov-Belsky - WikiArt.org[^] The interesting part is the task on the blackboard. (102 + 112 + 122 + 132 + 142)/365 Like the Russian boys, you have no calculator and no paper. Upvotes for: 1. A good reasoned guess at the answer 2. The exact answer, with an explanation of how you got it by mental arithmetic. <edit> I should have expected you to brute force it in your heads. (You did do it without paper or calculator, right?) So from now on I will upvote elegant solutions!</edit>
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
I did not use a calculator or a notepad. :laugh: I've been working in JavaScript for too long. This is how I visualized it:
result = 0.0; for (i = 10; i < 15; i++) { result += Math.pow(i, 2); } result = result / 365;The question reminded me of writing my first basic programs to solve high school geometry and advanced math homework problems. (class of '85)
"Go forth into the source" - Neal Morse
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Answer: 2 (ALL IN M'HEAD) I pretty much know those squares by heart. First 3 = 365 (already a hint): 1 Next two 169 + (200-4) = 365: 1 1 + 1 = 2
"The difference between genius and stupidity is that genius has its limits." - Albert Einstein
"If you are searching for perfection in others, then you seek disappointment. If you seek perfection in yourself, then you will find failure." - Balboos HaGadol Mar 2010
I did it the same way. But then, I'm an accountant, so.... Kind of a nerd as-is.
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10^2 = 1 0*2 0^2 = 100 11^2 = 1 1*2 1^2 = 121 12^2 = 1 2*2 2^2 = 144 13^2 = 1 3*2 3^2 = 169 14^2 = 1 4*2 4^2 = 196 (the last term (16) carries the one over, so 4*2+1 = 9) Add those up: 365 + 365 (365 + 365)/365 = (365/365) ((1 + 1)/1) = 1 * 2 = 2
Elegant way to do it. :thumbsup:
M.D.V. ;) If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about? Help me to understand what I'm saying, and I'll explain it better to you Rating helpful answers is nice, but saying thanks can be even nicer.
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There's an artwork by Nikolay Bogdanov-Belsky called Mental Arithmetic. In the Public School of S.Rachinsky, 1895 - Nikolay Bogdanov-Belsky - WikiArt.org[^] The interesting part is the task on the blackboard. (102 + 112 + 122 + 132 + 142)/365 Like the Russian boys, you have no calculator and no paper. Upvotes for: 1. A good reasoned guess at the answer 2. The exact answer, with an explanation of how you got it by mental arithmetic. <edit> I should have expected you to brute force it in your heads. (You did do it without paper or calculator, right?) So from now on I will upvote elegant solutions!</edit>
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
(12-2)²+(12-1)²+12²+(12+1)²+(12+2)² using (a+b)² = a²+2ab+b² will cancel those 2ab. Hence remains 5*12² + 2*4 + 2*1 = 5*146 = 10*73 = 730. Divided by 365 = 2. So the exercise is indeed for the application of (a+b)²+(a-b)² = 2a²+2b².
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(12-2)²+(12-1)²+12²+(12+1)²+(12+2)² using (a+b)² = a²+2ab+b² will cancel those 2ab. Hence remains 5*12² + 2*4 + 2*1 = 5*146 = 10*73 = 730. Divided by 365 = 2. So the exercise is indeed for the application of (a+b)²+(a-b)² = 2a²+2b².
Yeah! This was my solution as well.
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
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There's an artwork by Nikolay Bogdanov-Belsky called Mental Arithmetic. In the Public School of S.Rachinsky, 1895 - Nikolay Bogdanov-Belsky - WikiArt.org[^] The interesting part is the task on the blackboard. (102 + 112 + 122 + 132 + 142)/365 Like the Russian boys, you have no calculator and no paper. Upvotes for: 1. A good reasoned guess at the answer 2. The exact answer, with an explanation of how you got it by mental arithmetic. <edit> I should have expected you to brute force it in your heads. (You did do it without paper or calculator, right?) So from now on I will upvote elegant solutions!</edit>
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
1. Each square is approximately 20 more than the previous, so 5x100+20+40+60+80=700, estimating a correction for the approximation and assuming a whole number solution as it's a mental arithmetic problem then the total is 730 and the answer is 2. 2. What everyone else said.
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There's an artwork by Nikolay Bogdanov-Belsky called Mental Arithmetic. In the Public School of S.Rachinsky, 1895 - Nikolay Bogdanov-Belsky - WikiArt.org[^] The interesting part is the task on the blackboard. (102 + 112 + 122 + 132 + 142)/365 Like the Russian boys, you have no calculator and no paper. Upvotes for: 1. A good reasoned guess at the answer 2. The exact answer, with an explanation of how you got it by mental arithmetic. <edit> I should have expected you to brute force it in your heads. (You did do it without paper or calculator, right?) So from now on I will upvote elegant solutions!</edit>
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6 => (14*15*29 - 9*10*19)/6*365 => 30*(7*29 - 3*19)/6*365 => (7*29 - 3*19)/73 => 146/73 => 2 ;-)
Ariel Serrano Informatica Ambientale S.r.l. (www.iambientale.it) Via Teodosio, 13, 20131, MI Milan, Italy.
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1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6 => (14*15*29 - 9*10*19)/6*365 => 30*(7*29 - 3*19)/6*365 => (7*29 - 3*19)/73 => 146/73 => 2 ;-)
Ariel Serrano Informatica Ambientale S.r.l. (www.iambientale.it) Via Teodosio, 13, 20131, MI Milan, Italy.
I even forgot this formula existed. I also wouldn't have done it in my head. :)
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
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Yeah! This was my solution as well.
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
Though one mistake, (a+b)² + (a-b)² = 2a² + 2b². Good we are still as intelligent as then (?). Probably the same trick the teacher would demonstrate. It would be interesting if some mathematician historian would check whether such tricks were indeed collected for instruction - of numerical math.
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There's an artwork by Nikolay Bogdanov-Belsky called Mental Arithmetic. In the Public School of S.Rachinsky, 1895 - Nikolay Bogdanov-Belsky - WikiArt.org[^] The interesting part is the task on the blackboard. (102 + 112 + 122 + 132 + 142)/365 Like the Russian boys, you have no calculator and no paper. Upvotes for: 1. A good reasoned guess at the answer 2. The exact answer, with an explanation of how you got it by mental arithmetic. <edit> I should have expected you to brute force it in your heads. (You did do it without paper or calculator, right?) So from now on I will upvote elegant solutions!</edit>
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
Use squares of binomials: 10^2 = (12 - 2)^2 = 12^2 - 4*12 + 4 14^2 = (12 + 2)^2 = 12^2 + 4*12 + 4 11^2 = (12 - 1)^2 = 12^2 - 2*12 + 1 13^2 = (12 + 1)^2 = 12^2 + 2*12 + 1 12^2 = 12^2 Add them up, sum = 5*(12^2) + 5*2 = 5 * 146 Denominator = 365 = 5 * 73 Hence ratio = 146/73 = 2 The difference of squares is quicker: (14^2 - 12^2) + (10^2 -12^2) = 26*2 - 22*2 = 4*2 (13^2 - 12^2) + (11^2 -12^2) = 25*1 - 23*1 = 2*1 Hence 10^2 + 11^2 + 12^2 + 13^2 + 14^2 = 5*12^2 + 5*2 = 5 * 146 But this year is a leap year!
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Answer: 2 (ALL IN M'HEAD) I pretty much know those squares by heart. First 3 = 365 (already a hint): 1 Next two 169 + (200-4) = 365: 1 1 + 1 = 2
"The difference between genius and stupidity is that genius has its limits." - Albert Einstein
"If you are searching for perfection in others, then you seek disappointment. If you seek perfection in yourself, then you will find failure." - Balboos HaGadol Mar 2010
Exactly the same thought process, and I also know the squares in my head - up to 16 anyway. Above that there's a bit of mental arithmetic required.
