Math Explorations: 'Quadratic Friendships'

We've been reading aloud Malba Tahan's The Man Who Counted on DS' request. We decided we'd read about three chapters a day, trying to figure out some of the mathematical riddles for ourselves if he was interested. On our second day of reading, DS came across a mention of quadratic friendship between the square of 13 and the square of 16.


Taking note that the square of 16 is 256, here's the part of the story that really appealed to DS:

"The digits of the number 256 add up to 13. The square of 13 is 169. The digits of 169 add up to 16. As a result, the numbers 13 and 16 have a curious relation, which we could call a quadratic friendship. If numbers could speak, we might overhear the following dialogue: Sixteen says to Thirteen (DS loves the use of personification in stories, by the way), 'I wish to offer you an homage to our friendship. My square is 256, and the sum of its digits is 13.' And Thirteen would reply, 'Thank you for your kindness, dear friend. I wish to answer in the same coin. My square is 169, and the sum of its digits is 16.'" (Chapter 6, page 33)


And so began our quest to find out what would happen if we squared numbers 1 to 20, added their digits, squared those, added digits again and so on. It was a very, very interesting exercise and DS insisted on mapping the results in a grid of five numbers across and four down. We discussed how the pattern might look like if he'd mapped it differently too and later if he wanted to, we (his Dad joined in after returning from work) suggested he could add as many numbers as he wanted to (fingers crossed!). So this is what we found:

The numbers 1, 8, 10, 17 result in 1
[e.g. 8²=64, 6+4=10, 10²=100, 1+0+0=1]

The numbers 2, 4, 5, 7, 11, 13, 14, 16 and 20 result in 13 or 16 depending on your patience level (I was so amazed by this!).
[e.g. 11²=121, 1+2+1=4, 4²=16, 1+6=7, 7²=49, 4+9=13, 13²= 169, 1+6+9=16, 16²=256, 2+5+6=13, 13²=169...]

and, the numbers 3, 6, 9, 12, 15, 18 result in 9.


If you color the results in three different colors and chart them in a grid, you'll see pretty interesting patterns. They're not necessarily striking patterns and you do have to look at your chart carefully to identify them but they are definitely very interesting. I wouldn't have even imagined how interesting if we hadn't tried it!

Totally cool!