Mental arithmetic
-
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
-
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.)
-
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.
-
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.
-
(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.
-
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.
-
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!)
Member 4317199 (Paddy) wrote:
The hardest part was not picking up a pencil
Yeah, even when you realize there's an easier way it's still not that easy to do in the head.
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
Member 11577008 wrote:
OK... I failed to do it in my head, but worked it out in text below then checked it in Excel
Upvote for honesty.
Member 11577008 wrote:
So... I'm dumber than a 19th century schoolboy
Don't count on it, from what I can see in the picture there is one kid that whispers something in the ear of the teacher, the rest are still thinking.
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.
Visual solving, I like it.
Wrong is evil and must be defeated. - Jeff Ello Never stop dreaming - Freddie Kruger
-
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
Easiest i could think of was to simplify the numerator to (12-2)^2+(12-1)^2+12^2+(12+1)^2+(12+2)^2 And then it becomes a more manageable 5*12^2 + 2*1^2 + 2*2^2 At which point most folks would find it easy to compute the numerator to 730. Some may even factor out the 5.
-
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
This is how I mentally solved it. Well, I used my fingers, also. 10^2 + (10 + 1)^2 + ... + (10 + 4)^2 taking into account (a + b)^2 = a^2 + 2ab + b^2, we have 10^2 appears 5 times: 500 the 2ab term gives: 2 * (10*1 + 10*2 + 10*3 + 10*4) = 20 * (1 + 2 + 3 + 4) = 200; 700, up to now then, the sum of the squares: 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30. numerator sums 730. denominator: 350 + 15, times two gives 700 + 30; answer: 2
-
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
Brute force. All in head, no paper or anything else. Wrote the post after doing it. 10x10 = 100 11x10 = 110+11 = 121 (don't remember this one) 12x12 = 144 subtotal = 365 13x10 = 130+39 = 169 (don't remember this one) 14x10 = 140+56 = 196 (don't remember this one) subtotal = 365 total = 730 divide by 365 = 2 Add one more pair of numbers and I might not have been able to do it. I dropped the 169 on the floor once before adding the second pair. I never learned the complete multiplication tables as I could do the above sort of math quickly enough to get by.
-
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!
Referring to my previous two solutions, here is a third and better way to do the original problem: 14^2 - 2^2 = 12*16 (difference of squares) 13^2 - 1^2 = 12*14 12^2 - 0^2 = 12*12 11^2 - (-1)^2 = 12*10 10^2 - (-2)^2 = 12*8 Adding the five lines: (sum of squares from 10^2 to 14^2) = 5*12^2 + (sum of squares from (-2)^2 to 2^2) This allows three more general problems to be investigated: PROBLEM 1: Find all sums of five consecutive squares divisible by 365, and find the resulting quotients. PROBLEM 2: Find all sums of (2n+1) consecutive squares divisible by 365, and find the resulting quotients. PROBLEM 3: This year 2020 is a leap year. Replace 365 = 5*73 by 366 = 6*61, then by any fixed whole number.
-
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
Five consecutive squares always average to the middle square plus 2. 146 * 5 = 730. Answer = 2
-
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 used mostly addition for the entire problem in my head. Step 1: You add a zero to the end of each term & sum the them. 100 + 110 + 120 + 130 = 600 Step 2: you multiply each term by its second digit and sum the results. 0*0=0 1*11 = 11 + 2*12 = 24 + 3*13 = 69 + 4*14 = 56 + total = 160 Step 3: the value of the expression in parenthesis is the sum of step 1 and step 2 600+160 = 730 Step 4: 730/365 = 2 All this is possible to do in your head. Though at my age my declining memory required me to redo some of these steps several times to get the answer. A calculator is much faster. This was an interesting exercise though.
-
Five consecutive squares always average to the middle square plus 2. 146 * 5 = 730. Answer = 2
-
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
Yeah, 2. Multiply, add, divide. I've seen enough Vedic math ways to do things that I know my brain-dead solution could be improved. I never compare myself with any student from any time other than to say, "I'm still learning." Imagine running into a completely unschooled guy who comes up with this for an approximation of pi? Srinivasa Ramanujan - Wikipedia[^] Say WHAT!?
Dang! My '58 Renault Dauphine has another flat tire.
-
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
-
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'd say 2 too. 10^2=100 14^2=196 more or less the average of extreme values is (100+196)/2~150 150*5=750 750 must be not so far from the sum of the powers but 750/365~ 2.something than the reply most probably is 2 (5 secs the whole thing)