Multiplication pattern

[milosch]

First off, it may have been more years since I was in school than many of you have been alive. Anyway, a long time. But, I have always had number patterns running through my head, although not to the extent of a John Nash.

So, recently, I started working on multiplication in my head. I started with 8x8, then 7x9, 6x10, and so on. I noticed that the difference in the product scaled down in an odd pattern of 1, 3, 5, etc. I'm sure there must be a name for this. It appears to work for numbers of any size, and I made a small spreadsheet to prove it to myself. So, nothing earth-shattering. Just a curiosity.

 

[Dr.Peterson]

I don't think it's significant enough to have a name; but it is easy enough to explain with algebra, and it doesn't surprise me at all. It is fun to explore, though.

Suppose the number you start with is n (in your example, n=8). Then your successive products are n^2, (n-1)(n+1), (n-2)(n+2), and so on.

The "kth" of these, starting with k=0 for the original square, is (n-k)(n+k). But algebraically, (n-k)(n+k) = n^2 - k^2; that is, it is a perfect square less than the original square.

So the differences from one of your numbers to the next are just the differences between successive squares. It is a well-known fact that these differences are successive odd numbers: 1-0 = 1, 4-1 = 3, 9-4 = 5, ... .

To make it more complicated but get directly to your result, the difference between two successive numbers in your sequence is (n-k)(n+k) - (n-(k+1))(n+(k+1)). If you do the algebra, this simplifies down to (n^2 - k^2) - (n^2 - k^2 - 2k - 1) = 2k + 1. This is just the kth odd number, as you found.

So this is a consequence of something that does have a name: the difference of squares.

 

[milosch]

Thanks, Doc. It's been more than 40 years since I took algebra but it looks familiar. I have since looked this up.