Written: April 1 (!), 2013

It is ironical that mathematicians, who claim to be so enamored with *absolute truth*-
hence their obsession with the quaint notion of *rigorous* proof- are pretty good at
hiding the truth from their peers, and the general educated public, when they think (erroneously!)
that the revealed facts would either tarnish the image of their demi-gods (like Gauss), or
cast doubts on their rigid dogmas.

I am talking about the very plausible rumors that a noted French mathematician (whose name I am withholding
for obvious reasons, so let's call him *monsieur le professeur X*, or Pr. X, for short),
found, in his attic, a diary that must have been
written by the great Gauss. According to this diary, Gauss had a love affair with Sophie Germain,
and together they had a "natural daughter" (*fille naturelle*), that
was put up for adoption. Pr. X, who recently discovered the
diary in his attic, is apparently the great-grandson of that daughter-born-out-of-wedlock, and
as a corollary, the great-great-grandson of two of the greatest mathematical giants of 19th-century math,
(of **both** sexes!)

But the mores have changed since the times of Carl Friedrich and Sophie, and by today's standards, it is hardly a "scandal" to have a child out of wedlock, even if you are mathematical giant. So it is surprising that Pr. X is withholding such an important document, ostensibly not to "spoil" the image of his illustrious (biological) ancestors.

But I have a conjecture about the *true* reason why this ("illegimate") scion is not making
this priceless document public. According to rumors, Gauss describes, in this newly found diary,
the *true* method that
lead him, when he was a mere 7-years-old lad, to figure out, in less than a minute, that

1+2+ ... +100=5050 .

In 2006, the great Amer. Sci. computing columnist, Brian Hayes, has already debunked the conventional belief that Gauss did it by the "method of bridges",

(1+100)+(2+99)+ ... + (49+52) +(50+51)= (101)(50)=5050 ,

or, almost equivalently, by adding it to its reverse

1+ 2+...+ 100 + 100+ 99+ ...1 = 101+101+...+101=(101)(100) ,

and then dividing by 2. Brian Hayes speculated about some alternative methods, none of them turned out to be the right one.

According to the diary, written by Gauss shortly before he died (and apparently entrusted to his
"bastard" daughter that he must have kept in touch with), he described how he *actually*
figured out the right answer.

He reasoned as follows:

1=1=(1)(2)/2 ,

1+2=3=(2)(3)/2 ,

1+2+3=6=(3)(2)=(3)(4)/2 .

Then, being the bright lad that he was, he detected a pattern, and made a conjecture

1+2+...+n=n(n+1)/2 ,

**without** proving it by "complete mathematical induction", (or by the "method of bridges" that
was attributed to him by later unreliable historians like E.T. Bell), after all, he was
only seven-years-old!, he "plugged-in" n=100, and got the **right answer**.

We all know that in addition to being a great prover, Gauss was also a great experimental mathematician,
conjecturing the prime number theorem almost a hundred years before it was proved "rigorously".
The fact that his amazing feat, when he was seven-years-old, was done "only" via
experimental math, using "incomplete induction", (and **not** the method of bridges [or, worse, the tedious "proof by complete mathematical
induction" that we indoctrinate our "introduction to Proofs" students with]), only, *enhances*
my image of Gauss as a true visionary, and Pr. X should not be embarrassed. *Not bad* for
a seven-years-old child.

I am sure that the diary contains lots of other gems, both personal, and mathematical, and Pr. X owes it to us to make it public! And don't be embarrassed, Pr. X. Even though Gauss's argument is empirical, it is fully rigorous! As Gauss himself should have realized by the time he was eleven, since both

1+2+ ...+n and n(n+1)/2

are polynomials of degree 2, once you check that they are equal for **three** different values,
they *must* be **identically equal**, so Gauss had a fully rigorous proof (without knowing it, at the time).

And if you ask me, seven-years-old Gauss's *"empirical"* approach is **far superior**
to the "clever", but ad-hoc, "method of bridges", erroneously attributed to him by later historians.
For example, if you try, to do

1^{2}+2^{2}+3^{2}+...+ 100^{2} ,

the method of bridges fails miserably, but Gauss's "experimental" method works!

EmptySum=0=0(1)(1)/6 ,

1^{2}=1=(1)(2)(3)/6 ,

1^{2}+2^{2}=5=(2)(3)(5)/6 ,

1^{2}+2^{2}+3^{2}=14=(3)(4)(7)/6 .

Now detect a pattern

1^{2}+2^{2}+3^{2}+ ...+ n^{2}=n(n+1)(2n+1)/6 ,

and, *bingo* ,

1^{2}+2^{2}+3^{2}+ ...+ 100^{2}=(100)(101)(201)/6= 338350 ,

and once again, this is a fully rigorous proof! (why?)

So Pr. X, PLEASE reconsider. Of course, the original copy should remain with you, but all you need to do
is go to a scanner, and then upload it to your website (if it exists). If you don't have a scanner, then ask someone to make
a photocopy, and I'll be more than glad to do the scanning, and post it in my own website, of course
with due credit to you. If you feel that some parts are too "private", then you may copy only the mathematical
parts, but please, you owe it to us to have *open access*
to the gems of the "Prince of Mathematics". Please! Bitte Schön!!, S'il vous plait!!!

Opinions of Doron Zeilberger