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!

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