One day in middle school, I was playing with perfect squares 0, 1, 4, 9, 16, 25, 36, 49, 64, 81…

Something neat pops out! A bunch of 9-ish things!

One number was 9

One number was 36, and 3+6=9

One number was 81, and 8+1=9

Adding the digits in each number together until we have all single digits (digital roots) like

81: 8+1=9

64: 6+4=10 and then 10: 1+0=1

We get a pattern:

1 4 9 7 7 9 4 1 9 1 4 9 7 7 9 4 1 1 1 4 9 7 7 9 4 1 9

(1 4 9 7)-(7 9 4 1) 9 (1 4 9 7)-(7 9 4 1) 9 (1 4 9 7)-(7 9 4 1) 9

A beautiful periodicity!

You can actually catch part of this pattern in a common thing! When you replace the numbers in a typical multiplication table with their digital roots, this pattern makes up the diagonal. The Vedic Square is an abbreviated multiplication table (9×9) and so the first 8 elements of the pattern fall along its diagonal.

Also, how weird that 0 isn’t included, since 0^{2}=0?

I have to be missing part of the picture to not see why – *do you know?*

I haven’t played with more than the visual stuff yet, but I suspect we might find another neat thing based on how that looks if we take out further the series by individual summation steps, where if 82^{2} = 6724, then

step 1: 6+7+2+4=19

step 2: 1+9=10

step 3: 1+0=1

The workbook I played with this in is here 🙂

*(see another beautiful periodicity in the Perfect Squares Last Digit Sequence)*

1, 4, 9, 7, 7, 9, 4, 1, 9