Product Placement

The discussion from two days ago led to another nice observation from Marilyn Burns.

Michael Jerrell suggesting thinking about it as an array. That makes it pretty clear that the products can only be different by 11 if the factors add up to 10.

I had never thought of that before... but that meant that if you decomposed the other way, then you could turn the product into a series.

So 4x6 = 3x5+3+5+1 = 3x5+9 = 2x4+7+9 = 1x3+5+7+9 = 3+5+7+9.

Hmmm, so 5x8 = 4x7+12 = 3x6+10+12 = ... = 1x4 +6+8+10+12.

Which means
MxN = (M-1)x(N-1) +M+N+1 = 1x(difference of M &N+1)+ (that +2)+...+(M+N-1)

But that's an arithmetic sequence.

So something like 23x28=(28-23+1=6)+8+10+...+46+48+50. These are easy to sum - ask any 5 year old Gauss. 23x28 = (50+6)x(# of terms)/2. So how many terms? Whichever is smaller determines, since you're going from 28-23+1 to 28+23-1. So 28-22 to 28+22, difference of 44 by 2s. 22 times up plus the first term... that's ... 23 terms. Hmm. Of course, since we're saying 23x28 = (56)x(something)/2.

Of course! First + last = (N-M+1)+(N+M-1) = 2N. So (first+last)*(# of terms)/2 = (2 x larger) (smaller)/2.

No new product, which I was hoping for when I started, but a nice connection from products to series.

Note if M=N, this is the classic squares are the sum of odds. Since N-N+1=1, N^2= 1+3+5+...+(2N+1), making this a good generalization, too.

Thanks, Marilyn and Michael!