This lovely puzzle (for upper-elementary and beyond) is from Nikolay Bogdanov-Belsky’s 1895 painting “Mental Calculation. In Public School of S. A. Rachinsky.” Pat Ballew posted it on his blog On This Day in Math, in honor of the 365th day of the year.

I love the expressions on the boys’ faces. So many different ways to manifest hard thinking!

Here’s the question:

No calculator allowed. But you can talk it over with a friend, as the boys on the right are doing.

You can even use scratch paper, if you like.

### Thinking About Square Numbers

And if you’d like a hint, you can figure out square numbers using this trick. Think of a square number made from rows of pennies.

Can you see how to make the next-bigger square?

Any square number, plus one more row and one more column, plus a penny for the corner, makes the next-bigger square.

So if you know that ten squared is one hundred, then:

… and so onward to your answer. If the Russian schoolboys could figure it out, then you can, too!

### Update

Simon Gregg (@Simon_Gregg) added this wonderful related puzzle for the new year:

Harder puzzle: Show that there are infinitely many numbers that can be written as both a sum of two consecutive perfect squares AND a sum of three consecutive perfect squares. (I worked this out with a couple of mathematicians back in 1995, when the three of us were attending a conference in Russia; we saw this very painting, and it led us to set ourselves the puzzle as a challenge.)

Wow, I think that one is probably above my pay grade! I’ll play with it, though, and see if I get anywhere. π

In the interest of commiserating with all math students and mathematicians who miss seeing something “obvious:” I didn’t get the strong connection between the original painting calculation and Jim’s puzzle until I spent some time working on the latter.

Another closely related puzzle: show that there are infinitely many squares that can be written as the sum of two consecutive squares.

Which leads to the obvious extension: for all natural numbers N, is it true that there are infinitely many natural numbers that can be written as both the sum of N squares and N+1 squares? For what it is worth, I don’t know the answer to this.

Nice! Your first one sounds like something I might be able to wrestle into submission. At least, it seems like a more approachable first step toward James’s puzzle. π

This also has a geometric interpretation. You can find (many but not all) of the Pythagorean triples in this fashion. Start looking at the sequence of squares. Every time you run into two whose difference is a square you’ve found another triple. The first occurs at 16 and 25 (3^2) then 144, 169 (5^2), and then 576, 625 (7^2). So you can restate the puzzle as prove there are infinitely many Pythagorean triples which have side lengths of n and n + 1.

Regarding my puzzle (show that there are infinitely many numbers that can be written as both a sum of two consecutive perfect squares and a sum of three consecutive perfect squares), see https://math.stackexchange.com/questions/952216/numbers-represented-as-two-different-sums-of-squares as well as oeis.org/A007667. A discussion of the problem can even be found in one of Martin Gardner’s books (see page 22 of “Time Travel and Other Mathematical Bewilderments”).

Regarding Joshua’s question (for all natural numbers N, is it true that there are infinitely many natural numbers that can be written as both the sum of N squares and N+1 squares?), the answer is Yes. In fact, for every N, there is a number that is both the sum of N+1 consecutive squares and the sum of the immediately following N consecutive squares; see http://oeis.org/A059255.