Hacker News Reader

Comment by jimhefferon | Hacker News Reader

The caption "The equation 1⁰² + 1¹² + 1²² = 1³² + 1⁴², whose answer is that both sides equal 365, was immortalized in a different form in this 1895 painting: “Mental Arithmetic. In the Public School of S. Rachinsky.” (NIKOLAY BOGDANOV-BELSKY)" has funky exponents. Medium requires that I make an account to say that, so I am saying it here.

View on HN

I think the equation on the board is ( 10^2 + 11^2 + 12^2 +13^2 + 14^2 ) / 365 which solves for 2.

The students are expected solve this with "Mental Arithmetic". Try that in your head!

Thanks. I cannot see that on my screen. The powers of 1 caught my eye, though.

Happy Pi-Day!

Let a=12 then (a-2)^2 + (a-1)^2 + a^2 + (a+1)^2 + (a+2)^2 /365

and a binomial a+b squared yields a^2 + 2ab + b^2 so

a^2 + 4 + a^2+1 + a^2 + a^2+1 + a^2 + 4

add up all those a^2 's to get 5a^2 + 10

replace a by 12 and 720 + 10

       yields   730

Divide by 365

That's what the sharp kid in the foreground is doing at the moment of the painting.

Assuming one's memorized a table of squares already.

10^2 = 100

11^2 + 13^2 = 121 + 169 = 290

12^2 + 14^2 = 144 + 196 = 340

100 + 290 + 340 = 730

730 / 365 = 2

You can also "walk it up" if you haven't memorized them:

      10^2*5
    + (10+11)*4
    + (11+12)*3
    + (12+13)*2
    + (13+14)
    ------------

       500
     +  84
     +  69
     +  50
     +  27
     -----
       730

It's a bit of detail to juggle in your head but not too bad if you hang on to your current subtotal and remember where you are in the sequence to compute the next addend, and you know it's easier to add multi-digit numbers left-to-right than-right to-left.

You only carry from the ones place twice.

That formatting error is repeated in at least one other place too.

On the subject of that painting, can anyone tell me what the kids are wearing on their feet?

Muckluks. Or the 1890 Russian version

These are bast shoes [0], known in Russian as lápti, which were the typical footwear in villages until the 20th century. They are strapped to the leg so that they don't fall off.

[0]: https://en.wikipedia.org/wiki/Bast_shoe

Reply on news.ycombinator.com