Mental Calculation: A Math Lesson In A Russian School In 1895

Mental Calculation: A Math Lesson In A Russian School In 1895

N. P. Bogdanov-Belsky painted this in 1895. Can you solve the problem presented before the disciples of this old Russian village school? Remember, this has to be mental calculation, no computer aid allowed. Here it is:

I have just one question: How did we ever win the Cold War?

Comments

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[Cindy K]

I'd seen it before so I knew the answer was 2. ;) Seriously, though, mental math is a huge boost to every day life. My dad taught me early and it stuck. I freak people out with it sometimes, especially with percentages, but other times I can't count the fingers on one hand. It's not something I turn on or off very well. It just kind of happens when the brain is firing properly.

I never really had any math higher than Algebra, though. I stuck with the soft sciences like politics.

[George]

I nailed it by using rules of algebra to split the equation into two parts...

10^2 + 11^2 + 12^2 is pretty to easy add up if you know your tables to 12. I realized it summed to 365. I used this to separate the equation into two parts knowing the above divided by 365 equals one.

A little more mental multiplication reveals that 13^2 + 14^2 also equals 365 and BOOM it becomes obvious the answer is 2.

It took me about 3 or 4 minutes to solve using brute force...I imagine there are fat more elegant ways to solve the expression.

[Debbie]

I have coached UIL number sense for many years. Another technique is to use the square of a sum. (10+0)^2+ (10+1)^2+ (10+2)^2+ (10+3)^2 + (10+4)^2
which for each binomial is a^2+2ab+b^2
(taught in Alg I & II)
hence
10^2+2(10*0)+0^2=100+0+0
10^2+2(10*1)+1^2=100+20+1
10^2+2(10*2)+2^2=100+40+4
10^2+2(10*3)+3^2=100+60+9
10^2+2(10*4)+4^2=100+80+16

notice this is
500+200+30 =730
I would factor
730 into 2*365
wa la
((2*365)/365)=2.

My best student could do 47 mental calculations in 10 minutes which won the UIL number sense contest.

America has fine math teachers and students

[Weantre]

Maths - Induction

( 14.15.29/6 - 9.10.19/6 )/365

[Frank Nofsinger]

The artist had a sense of humor. The teacher smiles while his peasant students grimace and groan. !0 sq is 100, five calcs later put it into less ergo about
750- Gentle teacher deosn't want to embarass.
The ability to approximate is how we won the cold war and most of the others.

[Guy]

I'm pretty sure the book was High Speed Math, Self Taught by Lester Meyers.

Per comments here: http://www.amazon.com/Speed-Mathematics-Secret-Skills-Calculation/dp/0471467316/ref=pd_sim_b_1 , there was a 1961 edition - probably what I had.

One trick I remember is how multiply something by 25: Add 2 zeros to the end (multiply by 100) then divide by 4.
This works because 100/4 = 25.

[Sotosoroto]

To do it in my head, I simplified it even more than Guy above:

500/365 = 2.

[Tom McMahon]

Guy, was it the Trachtenberg system?

[Guy]

At first glance it looks impossible, but it helps to split it up:
((100 + 100 + 100 + 100 + 100) + (0 + 21 + 44 + 69 + 96))/365
=
((500 + (21 + 69) + (44 + 96))/365
(500 + 90 + 140)/2
= 730/365
= 2

I used to have a book, I think it was printed in the 50's, all about these kind of math-in-your-head tricks.

[james wigderson]

They put the Commies in charge of the schools, which is probably a good argument against ever allowing a rematch since, well, look who're running our schools now.

[TerryN]

Little more than 1/2 a pi?