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
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
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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
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.
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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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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.
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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
Five consecutive squares always average to the middle square plus 2. 146 * 5 = 730. Answer = 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
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.
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Five consecutive squares always average to the middle square plus 2. 146 * 5 = 730. Answer = 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
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.
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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
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)