# Mathematical Musings

## The Francis 1-2-3 Theorem

If you want to see the actual mathematics here, you need to have (or to download now) the Adobe Acrobat Reader. (Its free).

What I call the Francis 1-2-3 Theorem (which I probably shouldn't because someone else has probably discovered this at some time or another) originally came from playing with my calculator in high school. I noticed that if you took a number like 123, or 234, or 456, etc. and added its reverse (321, 432, 654, etc.) to it, and then divided this sum by the sum of the individual digits in the number you started with (1+2+3=6, etc.), you always get the number 74. I thought this was interesting, and I went ahead and formulated a proof for it--somewhat disappointed to find that the proof was a simple ordinary algebraic exercise.

It wasn't until a few years ago that I picked it up again (after more mindless fiddling on a calculator) and went on to prove the theorem for numbers of any size (special thanks to my brother Greg for helping me with some of the tricky math), with the less remarkable conclusion that the derived quotient does not depend on a unique value of the number, but only on the general magnitude (10, 100's, 1000's,...,etc.

#### View the Francis 1-2-3 Theorem Proof

Update! As if the Francis 1-2-3 Theorem wasn't enough to shake the pillars of heaven, new insight has been brought to my attention by my trusty researcher, Greg Francis. A breakfast spent with a calculator showed him that incrementing by 1 is only a special-case of the more general theorem. In fact, the digits in the number need only be spaced by a constant k (so for the 1-2-3 Theorem, k = 1). I have provided the proof for this more general theorem below: