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
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.
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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 was about to "guess" 2 based on the same premise: first three and a "feeling" about the last two (:
------------------------------------------------ If you say that getting the money is the most important thing You will spend your life completely wasting your time You will be doing things you don't like doing In order to go on living That is, to go on doing things you don't like doing Which is stupid. - Alan Watts https://www.youtube.com/watch?v=-gXTZM\_uPMY
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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
We know that (a+b)²=a²+b²+2ab So: 11² = (10 + 1)² = 10² + 1² + 2x10x1 12² = (10 + 2)² = 10² + 2² + 2x10x2 13² = (10 + 3)² = 10² + 3² + 2x10x3 14² = (10 + 4)² = 10² + 4² + 2x10x4 So: 10² + 11² + 12² + 13² + 14² = 5x10² + (1² + 2² + 3² + 4²) + 2x10x(1 + 2 + 3 + 4) 1 + 2 + 3 + 4 = 10 1² + 2² + 3² + 4² = 1 + 4 + 9 + 16 So: 10² + 11² + 12² + 13² + 14² = 5x100 + 30 + 2x10x10 = 500 + 30 + 200 = 730 730/365 = 2
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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
Noticed that 10^2 + 11^2 + 12^2 = 365. Then 13^2 = 169, 14^2 = 196 - and those two summed are 365. So dividing 2 lots of 365 by 365 gives 2 as the answer. If you don't know your squares, use the identity (n+1)^2 = n^2 + n + (n+1) 10^2 = 100 (that's easy enough to remember!) For the next, add 21 (10+11). Then add 23, then 25, then 27 for the other squares. However - for the first three squares, we're adding 300 to 2*21 and 23 - 2*21=42, add 23 and you have 65, a total of 365. The last two, we have 200 + (23+25+25+27) + 2*21+23. The last bit we know is 65. The middle bit is clearly 100, so the last two squares sum to 365 too.
Java, Basic, who cares - it's all a bunch of tree-hugging hippy cr*p
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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
We know that (a - b)² = a² + b² - 2ab So: 10² + 14² = 10² + 14² - 2x10x14 + 2x10x14 = (10-14)² + 2x140 = 4² + 2x140 11² + 13² = 11² + 13² - 2x11x13 + 2x11x13 = (11-13)² + 2x143 = 2² + 2x(140+3) = 2² + 2x140 + 2x3 12² = 144 = 140 + 4 So: 10² + 11² + 12² + 13² + 14² = 4² + 2² + 2x140 + 2x140 + 2x3 + 140 + 4 = 16 + 4 + 5x140 + 6 + 4 = 20 + 700 + 10 = 730 And 730/365 = 2
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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 ((12-2)**2 + (12-1)**2 + 12**2 + (12+1)**2 + (12+2)**2) the -2ab from the first two cancel the +2ab from the second two. so 5 * 12**2 + 4 + 1 + 0 + 1 + 4 5 * 144 + 10 divide top and bottom by 5 146/73 (which is essentially what Joop said. But I scrupulously didn't cheat by looking at previous. The hardest part was not picking up a pencil!)
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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
10^2 + 11^2 + 12^2 + 13^2 + 14^2 = 10^2 + (10 + 1)^2 + (10 + 2)^2 + (10 + 3)^2 + (10 + 4)^2 = 5 x 10^2 + some junk... i.e. 500 + junk in that junk there is 2x10x1 + 2x10x2 + 2x10x3 + 2x10x4 = 2x10(1 + 2 + 3 + 4) = 2 x 10^2 = 200 that's 700 + what's left of the junk at this point it became obvious that either the result is 2 or it's best to use a calculator.
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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
Everyone should know the squares - at least for 1 to 20. After that, you just need to know that multiples of ten are (x * 10)^2 = x^2 * 100. You can then do the halfways [numbers ending in 5] (x * 10 + 5)^2 = ((2x + 1) * 10)^2 / 4 [looks a lot more complicated than it is]; thereafter, for numbers ending in 1, 2, 6, 7 you apply (x + 1)^2 = x^2 + 2x + 1 (or x^2 + (x + 1) + x) [do it twice for 2 and 7] and for numbers ending in 3, 4, 8, 9 you apply (x - 1)^2 = x^2 - 2x -=1 or x^2 - (x + 1) - x = x^2 - x - x - 1 [do it twice for 3 and 8] At least, that's what I use! And I assure you, once you've got the hang of them that are simple. Edit: It is also useful to memories powers of two and squares of prime numbers
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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.
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Everyone should know the squares - at least for 1 to 20. After that, you just need to know that multiples of ten are (x * 10)^2 = x^2 * 100. You can then do the halfways [numbers ending in 5] (x * 10 + 5)^2 = ((2x + 1) * 10)^2 / 4 [looks a lot more complicated than it is]; thereafter, for numbers ending in 1, 2, 6, 7 you apply (x + 1)^2 = x^2 + 2x + 1 (or x^2 + (x + 1) + x) [do it twice for 2 and 7] and for numbers ending in 3, 4, 8, 9 you apply (x - 1)^2 = x^2 - 2x -=1 or x^2 - (x + 1) - x = x^2 - x - x - 1 [do it twice for 3 and 8] At least, that's what I use! And I assure you, once you've got the hang of them that are simple. Edit: It is also useful to memories powers of two and squares of prime numbers
For those I don't know I use a binomial expansion in my head to make it easier. Not as quick as memorization (look up tables are always rather fast - even for computers). So, if given 27 * 82 it would become (30-3)*(80+2) Numbers to juggle mentally: +2400, -6, -240, +60 Corresponding to the outer two terms and the cross terms. If you do it now and then it remains pretty efficient - but if you've not done it for a year or two it take some cobweb sweeping to set one's storage back to efficient levels. 2214
"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
OK... I failed to do it in my head, but worked it out in text below then checked it in Excel ... and still was wrong so made I the needed corrections (2 errors compounded) to my text So... I'm dumber than a 19th century schoolboy, but it was a fun exercise any way. I used to do all my math in my head before we had pocket calculators (yes I'm an old fart) I need to do more of this kind of thing to get that back. 0 times 10 is 0 plus 100 is 100 1 times 11 is 11 plus 110 is 121 plus 100 is 221 2 times 12 is 24 plus 120 is 144 plus 221 is 365 3 times 13 is 39 plus 130 is 169 plus 365 is 4 carry the 1 and 2 plus 1 for 3 carry the 1 and 3 plus 1 plus 1 is 534 4 times 14 is 56 plus 140 is 196 plus 534 is 0 carry the one and 2 plus 1 for 3 carry the one and 5 plus 1 plus 1 is 7 for 730
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I did it almost the same way, but decided not to multiply 144 by 5, since I knew I was going to divide by 5, since 365 is dividable by 5. So:
((12-2)²+(12-1)²+12²+(12+1)²+(12+2)²)/365 = (144+(2*(2²+1²))/5)/(365/5) = (144+10/5)/73=146/73 = 2
(I want MathML.)
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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
To solve mentally with no use of paper, I need to imagine 5 squares, made of stones laid out on a table. we have first square that is made of 10x10 red stones. second square is made of 10x10 red stones plus 1x10 green stones at top and 10x1 green stones at right, then to fill the square we have a 1x1 square of blue stones at top right corner. third square is made of 10x10 red stones plus 2x10 green stones at top and 10x2 green stones at right, we fill the square with 2x2 stones. etc... red stones are 500 (5x10x10) green stones are 200 (20 on second square, 40 on third, 60 on fourth, 80 on fifth) blue stones (squares of 1, 2, 3, 4) are 1+4+9+16 = 30 total 730 stones. 730 / 365 = 2.
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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
The difference of two consecutive squares is the higher number * 2 - 1. So 11^2 = 10^2 + (11 * 2) - 1 = 100 + 22 - 1 = 121. Therefore the answer is [(100 * 5) + (11*2-1)*4 + (12*2-1)*3 + (13*2-1)*2 + (14*2-1)] / 365 = 2. My gut answer is that the answer was probably an integer with 2 being likely.
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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.
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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
e looks like the same kind of sequence.