## A past puzzle

By David Shaw, 26 August 2020

**Difficulty: Taxing**

Here’s a beautiful artwork from 1895, titled *Mental calculation. In public school of S. A. Rachinsky*, painted by Nikolay Bogdanov-Belsky.

Can you solve the puzzle the kids are pondering?

If you can’t quite see the equation, here it is:

**Scroll down or click for a hint, or the answer!**

## Brainteaser hint

• Although the kids in the painting are trying to do it in their heads, you can always use a pencil!

• Remember that teachers often like questions with nice round answers.

• 10^{2} means 10 x 10. Calculate these first!

## Brainteaser answer

The answer is 2.

Here’s the working:

*For more brainteasers and puzzles for kids, subscribe to Double Helix magazine!*

4 September, 2020 at 4:27 pm

For doing mental arithmetic it’s handy to know a few arithmetic shortcuts.

This problem is easier to do in your head if you know that (x + y)(x+y) = xx + 2xy + yy

If you set x=10 and y to 0, 1, 2, 3, 4

then you can write the sum 10×10 + 11×11 + 12×12 + 13×13 + 14×14

as

10×10 + (10+1)x(10+1) + (10+2)x(10+2) + (10+3)x(10+3) + (10+4)x(10+4) =

100

+ 100 + 20 + 1

+ 100 + 40 + 4

+ 100 + 60 + 9

+ 100 + 80 + 16

= 5×100 + 20+40+60+80 + 1+4+9+16

= 500 + 200 + 30

=730

I can do that in my head much more easily than I can calculate squares and add up three-digit numbers easily.