# 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.

**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: