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

         2

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.

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

Happy Pi-Day!