Wow! I couldn’t talk you out of it, huh? It takes very little math knowledge to do this, but actually figuring out why/how it works takes some advanced skillz.

You’ll need a calculator, paper and pencil, or a spreadsheet program.

Pick a number. Write it at the top of the page (or the top of the spreadsheet). Any number. If you’re using a calculator you may want to make things easier for yourself by picking an integer between 1 and 10, but if you’re using a spreadsheet or you don’t mind some tedious work, any real number will do. It doesn’t have to be positive; it doesn’t have to be an integer. You can use zero or pi or -10,000,000. OK?

Now pick a second number and write it immediately under the first number. The only restriction is that they can’t both be zero. (Either one can be zero, but not both.) Other than that, any two reals will do. Again, pick a small integer if you’ll be doing this by hand.

Now, write the sum of the first two numbers under the second number. So now you have a column of three numbers. Keep extending the column by writing underneath the bottom number the sum of the last two. So the fifth number is the sum of the third and the forth, and the sixth is the sum of the forth and fifth, and so on. Extend the column until it is at least ten numbers tall. More is fine.

Now divide the last number by the second-to-last number. Make sure you’re doing this in the right order: you’re dividing a bigger number by a smaller number so the answer you get should be greater than one. Multiply that result by two. Subtract 1 from that. Finally, square that number. What you’ve got now is very very close to an integer.

At this point if you’re doing this trick to a friend, you produce that integer in a dazzling way that is sure to impress him or at least win you a free drink. You can produce a playing card with that integer from a deck, or you can open a sealed envelope to reveal the integer written on a piece of paper, or you can rent a plane to write that integer in the sky. If you need other ideas of how to get a big finish to a magic trick when you know how it’s going to end from the start, see any Penn & Teller book. (Their card tricks always use the three of clubs.)

Anyway, the integer you’ve ended up with is 5.

Tell me why/how this works. Hint: it has to do with an Italian mathematician who thought a lot about rabbits.
 

Yeah, but the integer is irrelevant, because it's the original number that counts. This part:

"Multiply that result by two. Subtract 1 from that. Finally, square that number."

is simply a way of turning the number you get when dividing the last number in the series by the penultimate one into 5. Since that number is always going to be approximately (more closely the further you carry out the operation) 1.618034...

Isn't this the magic number that the "Da Vinci Code" guy refers to? I haven't read "The Code" yet, but Mother Nomad asked me about the "magic number" once, and I believe this was it. Meaning, your Paisan has to be Da Vinci?

I forget the logic, and I haven't had my second cup o' caffeine yet, and I have to cook Thanksgiving stuff, so I'll leave it to somebody else... but, it strikes me that the number (a natural ratio) occurs in a number of natural phenomena. Ratio of the lengths of certain body parts was one my mother referred me to. But, there were many more.

When you consider that a body part is merely the extension of a series of cells, you can imagine that the pattern of that series looks similar to the progression you've described. So it must be with many patterns in nature. There is probably an underlying natural mathematical/physical law that drives it... or perhaps the math drives the law. I imagine that the principle could be found as commonly in astrophysics as it could be in microbiology (although I'm just swatting theories now).

The more I talk about this, it seems that an old math class is calling out of my memory vaults. Fi? Or, to put it another way, Φ? At least I think that's the symbol. Seems that I remember there being another magical number in addition to pi (π), and that this was it. But, I could never remember the specifics... just that there was another number.

Okay, now I want to go read about this, and remember it, without the pressure of cramming for a final exam. Maybe I'll log something useful in my gray matter this time. Thanks for waking me up.
 

On a pseudophilisophical level, it's like genetics. Each new number is the result of its parentage. In this case, the parentage is always incestual, ultimately "purifying" the genetic makeup of each new offspring, and ultimately yielding a set of clones.

Okay, on to the second cup o' tea now. I'm getting weird when I should be getting wired.
 

Nomad: Good job. (sqrt(5)+1)/2 = 1.618034... = phi or Φ (pronounced fee). It is the golden ratio or the golden number, and is the number that your mom referrs to in The DaVinci Code. I haven't read it, but from what I hear, much of the mythology about phi that has sprung up, like DaVinci using it in his art, is probably bogus.

The Italian mathematician is Fibonnaci. He described series (called Fibonnaci series) in which each member is the sum of the previous two. (He was thinking of the growth of a population of rabbits.) The ratio of subsequent Fibonnaci series as the series gets longer approaches Φ, regardless of how the series starts.

A good book about Φ is Mario Livio's The Golden Ratio. The story of phi, the world's most astonishing number. It very nicely separates the real math from the cult-like mythology about phi in art and esthetics that has no basis in fact. Φ does, indeed, appear many places in math and in nature, but it is imagined in even more places, where it has no place.
 

I wonder if phi is the root of the words finite and infinte. Perhaps not, but it would be appropriate if it was.