The painting above, done in 1895, is of Russian peasant boys doing a mental arithmetic puzzle. As you can see, one of them has the answer and is whispering it into the teacher’s ear.

The puzzle, shown on the blackboard, is:

(10^{2} +11^{2} +12^{2} +13^{2} +14^{2}) / 365

Like the Russian boys, you have no calculator and no paper. Prize for:

1. A good reasoned guess at the answer

2. The exact answer, with an explanation of how you got it by mental arithmetic.

Vinh Chu, Tariq Hall, Sharif Quansah, Dion Miller, and Mugisha Uwarigiye solved this, and won prizes.

Solutions

Thanks to Pierre Benard for suggesting the following method:

10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2}

= (12-2)^{2} + (12-1)^{2} + 12^{2} + (12+1)^{2} + (12+2)^{2}

Hence: Numerator = 5 * 12^{2} + 10 (all double products cancel by symmetry – only remains “two 4 and two 1” to be added to the “main part”)

Hence: twice the Numerator = 10 * 12^{2} + 20 = 1460 (this twice is simpler to mentally compute for a boy because involving 10 times something)

Hence Numerator = 730 (mental calculus)

Hence the reply is 2 (because 730 is twice 365)

This method is quite “graphical” at the start, so that a brute force computation is elegantly avoided, and the initial process can easily fit inside the head of a (clever) boy.

Then, once the first step is achieved, completing the result does not require “memory storage” since it occurs in a sequential way, each result being hold for the next step.

We may logically think that this puzzle came just after the Professor’s lesson about developing (a+b)^{2} and (a-b)^{2}. Maybe they explicitly learned that

(a+b)^{2} + (a-b)^{2} = 2 a^{2} + 2 b^{2}

Other methods:

**Good guess:** 2

10^{2} is 100, so the numerator (top number) of the division sum is bigger than 500.

14^{2} is 196. You don’t know that offhand? You should at least know 16^{2}=256, because it comes up so much in computer science, and know 12^{2}=144 from primary school, and so be able to guess that 14^{2} is less than 200. In any case, the numerator is less than 1000.

Therefore the answer is bigger than 1 and less than 3.

It’s a good guess that the answer is a whole number. The teacher would hardly ask the question if the answer wasn’t something neat. Since 365 = 5×73, the answer is either a whole number or a fraction with 1/365, 1/73, or 1/5 in it – not neat.

So a good guess is 2.

**Exact answer:** 2

**Way 1**: If your mental arithmetic is good enough, you can do it by brute force. Add 10^{2}+11^{2}+12^{2}+13^{2}+14^{2} = 100+121+144+169+196 in your head, and get 730.

**Way 2**: One way round a problem that looks too complicated is to *generalise*. In this case, replace mental arithmetic by mental algebra.

(k-2)^{2}+(k-1)^{2}+k^{2}+(k+1)^{2}+(k+2)^{2}=5k^{2} + 4 + 1 + 1 + 4

=5k^{2}+10

=730 if k=12

Having done this, we could work out e.g. 20^{2}+21^{2}+22^{2}+23^{2} +24^{2}=2430 in a few seconds.

**Way 3**: Another way round a problem that looks too complicated is to do a simpler version of the same problem, and then build up from that to the more complicated problem.

Working out 10^{2}+11^{2}+12^{2}+13^{2}+14^{2} in your head is too hard? Start instead with

0^{2}+1^{2}+2^{2}+3^{2}+4^{2}

That’s 30.

Then work up to 10^{2}+11^{2}+12^{2}+13^{2}+14^{2} step by step.

First step: 1^{2}+2^{2}+3^{2}+4^{2}+5^{2}

To get to that from 30, we lose 0^{2} but add (0+5)^{2}, so overall we add 2×5×0+5^{2}

Then to get to 2^{2}+3^{2}+4^{2}+5^{2}+6^{2} we lose 1^{2} but add (1+5)^{2}, so overall we add 2×5×1+5^{2}

To get all the way up to 10^{2}+11^{2}+12^{2}+13^{2}+14^{2} we add:

2×5×(0+1+2+…+9) and 10×5^{2}

1+2+…+9=45 and 10×5^{2}=250

Therefore

10^{2}+11^{2}+12^{2}+13^{2}+14^{2}=30+10×45+250=730

**Way 4**: We could also use some tricks to make the brute-force method a bit easier. For example:

10^{2}+11^{2}+12^{2}=365 (you should be able to work that out in your head from primary-school times tables), so the answer is exactly 2 if:

10^{2}+11^{2}+12^{2}=13^{2}+14^{2}

We can use difference-of-two-squares here.

10^{2}+11^{2}+12^{2}=13^{2}+14^{2}

⇔ 10^{2}=14^{2}−11^{2}+13^{2}−12^{2}

⇔ 100=(14+11)×3+(13+12)

⇔ 100=75+25 ▇