## 21 February 2010

### Math puzzle

The painting above reportedly dates from the 1800s.
It is put on display in Russian most prominent galleries “Tretiakovka” so anyone can check it there by themselves. It shows a scene from a study process in a village school back then. There is a teacher and a bunch of children. Then all attention goes to the blackboard: It has some task for the kids that seem to puzzle them, but that’s just a temporary confusion.  They all look to be ready for the challenge. The name of this art masterpiece is something like “Oral maths test in village school”. It gets what it means - no calcs involved, just mind skills of counting, even without any piece of paper and pencil. So now the question - can any contemporary city school kid do that? I am not sure.
I can't vouch for any of the above, but the Tretyakov Gallery has an immense collection of Russian fine arts, and I see no reason for the painting not to be valid.

But I can vouch for the math puzzle.  You can click to enlarge the painting, but it's probably easier to reproduce it here:
And it CAN be done in one's head without too much difficulty.

Found at English Russia.

Update:  I am recurrently astounded by the breadth of knowledge of TYWKIWDBI readers.  Today Hexmaster identified the painting and the artist - "The painting is called "Counting in their heads". It was made by Nikolay Bogdanov-Belsky in 1895" - and provided a link to the painting's entry in the Russian Wikipedi: The painting on the Russian Wikipedia.

1. 2! The answer is 2! *Does celebratory dance* I just sat and stared at my computer screen for probably 3 minutes doing that.

2. The "shortcut" way to do it is to start by asking "why divide by 365?" It's not a calendar puzzle, so there must be some other reason...

Then when you look at the sum of squares you see they are consecutive squares. Those numbers won't be quite linear in relation to each other, but over this short range will be almost so. The lowest square is 100, the highest is 196, so the average will be a bit under 150. If so, the total would be under 750.

Or figure the middle square (144) is the average square. x5 = 720.

Use either of those numbers with the 365 divisor and guesstimate that the dividend is probably 730, with a quotient of 2 as the answer to the puzzle.

But adding the five squares in your head is the proper way. Not many people can concentrate that well.

Congratulations.

3. Cute! Looks difficult, but is trivially easy.

4. A slightly easier way is:
10^2+11^2+12^2+13^2+14^2=
(12-2)^2+(12-1)^2+12^2+(12+1)^2+(12+2)^2

Now, the pluses and minuses from expanding the squares cancel out, and we get:
5*12^2+1+4+1+4=5*(12^2+2)=5*146=5*2*73

Still, not really that much of a shortcut

5. I don't think the kids are "puzzled," I think they're doing the math in their heads--in particular the two in the foreground.

The boy to the teacher's right, it looks like, has already figured it out and is whispering the answer to the teacher. The one to the teacher's left was probably the first to get the answer.

They're all intelligent-looking boys, including the one in front with the dirty, ragged clothing. It wasn't accidental that the artist made him the central focus: Even our poorest kids are smart.

I love their expressions, especially that of the ragged kid.

The painting's style and ethos reminds me strongly of Norman Rockwell.

6. You are right. The answer came pretty quickly. I added the first three squares mentally, saw 365, and was able to guess (correctly, as it turned out after I checked my work) the second set of squares would also add up to 365. 2 it is!

7. My instant initial estimate(guess) was 2. Then I spent a few minutes(my 71 year old brains are slow)on it and found the correct answer to be 2.

8. Swift, I also was reminded of Norman Rockwell, but it almost certainly must be a Russian artist. If anyone knows, please chime in. I haven't been able to locate the painting elsewhere on the web.

9. Minnesotastan--I don't doubt for a second the artist was Russian. It's just that the similarity in styles--and the similar use of the homely scene to convey a positive message about the society--is rather startling.

Rockwell started painting professionally in the 1910s, for Boys Life, per Wikipedia. It's hard to tell when the Russian work was painted, but what can be seen of the teacher's outfit suggests late 1800s. If so, the two painters were not that far from being contemporaries.

I just looked through all the paintings on the Tretyakovka Gallery site from 1800 to 1900 and didn't find it...!

10. I had fun figuring it out in my head last night.

11. I had no problem..... doing this with a calculator :D

12. I've been thinking this over. Doesn't it look like too much of a coincidence that the total of squares of numbers from twice the number of fingers on one hand to twice the number of days in the week adds up to twice the number of days in the year?

I think it does! I'm sure this is clear evidence of a conspiracy! I don't know what the purpose is, or who drives it, but conspiracy it is! Has to be!

(Off to fold tinfoil for a hat...)

Oh, by the way, I forgot these: !!!11!!

13. You can sign up to do the SAT question of the day - http://sat.collegeboard.com/practice/sat-question-of-the-day

every third day there's a "math" question. The other day it was multiply

1/2*2/3*3/4*4/5*5/6*6/7

which kind of reminds me of the question in that painting except of course it's mean to be solved by high school juniors - of whom 1/3 of the kids motivated enough to solve the SAT question of the day got it wrong.

sight

14. When I was in school, that would have been a question for about 6th graders.

15. One of many solutions for this, and probably the fastest one for most people, is the one that Chris gave before, but instead 5*146, or even 5*2*73 i would calculate 5*150-5*4=750-20=730, and that way its easy to see that it IS a shortcut.

The key here is to see that "symmetry".

I remember having this in before high school, and as you see this is very good eguation to show different techniques of solving that kind of equations, so I guess thats why it was popular enough to end up beeing painted ^^

16. Hi, and thanks for finding a nice problem in a great picture. In return, I decided to help you with the identification.

The painting is called "Counting in their heads". It was made by Nikolay Bogdanov-Belsky in 1895.

http://en.wikipedia.org/wiki/Nikolay_Bogdanov-Belsky

The painting on the Russian Wikipedia

17. ...Or even "Counting in their heads, in the class of S. A. Rachinsky."

I realised that the name of the teacher couldn't really be left out since he was indeed a real teacher. He had been professor in Moscow before returning to the village of Tatev, where he founded a school. I guess his habit of giving his boys difficult problems was for real, too.

This is true art.

18. To make the addition easier, clump the terms:

10^2 + 11^2 + 12^2 + 13^2 + 14^2
= (10^2 + 11^2 + 12^2) + (13^2 + 14^2)
= (100 + 121 + 144) + (169 + 144)
= 365 + 365
= 730

For me, my mind works better if I cluster information. So as I added along the squares, I separated the first 365 out.

Then, just square 13 and 14, voila :)

19. Your second "144" should be "196." Obviously just a typo since you understood the math.

20. There is a lovely pattern with sequential (consecutive) numbers that shows
3^2 + 4^2 = 5^2 (the well-known Pythagorean Triad), and
10^2 + 11^2 + 13^2 = 14^2 + 15^2 = 365

Believe it or not, there are an infinite number of such patterns!
For example, the next ones are 21^ + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2
and 36^2 + 37^2 + 38^2 + 39^2 +40^2 = 41^2 + 42^2 + 43^2 + 44^2

The beautiful example in Bogdanov-Belsky's painting can therefore be thought of as (365 + 365)/365 = 2

Wounderful to see young minds engaging with such arithmetic!