Maths prize: Russian peasant boys (1 December 2015)

365

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:
(102 +112 +122 +132 +142) / 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.


Solution

Good guess: 2

102 is 100, so the numerator (top number) of the division sum is bigger than 500.

142 is 196. You don’t know that offhand? You should at least know 162=256, because it comes up so much in computer science, and know 122=144 from primary school, and so be able to guess that 142 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 102+112+122+132+142 = 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.

k2+(k+1)2+(k+2)2+(k+3)2+(k+4)2=5k2+2(1+2+3+4)k+(12+22+32+42)
=5k2+20k+30
=730 if k=10

Having done this, we could work out 202+212+222+232 +242=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 102+112+122+132+142 in your head is too hard? Start instead with

02+12+22+32+42

That’s 30.

Then work up to 102+112+122+132+142 step by step.

First step: 12+22+32+42+52

To get to that from 30, we lose 02 but add (0+5)2, so overall we add 2×5×0+52

Then to get to 22+32+42+52+62 we lose 12 but add (1+5)2, so overall we add 2×5×1+52

To get all the way up to 102+112+122+132+142 we add:

2×5×(0+1+2+…+9) and 10×52

1+2+…+9=45 and 10×52=250

Therefore

102+112+122+132+142=30+10×45+250=730

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

132+142=365, so the answer is exactly 2 if
102+112+122=132+142

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

102+112+122=132+142
⇔ 102=142−112+132−122
⇔ 100=(14+11)×3+(13+12)
⇔ 100=75+25  ▇