Written: June 6, 2008.

Arthur Benjamin
is deservedly famous
for his feats of mental calculations, that appeal both to mathematicians
and to general audiences. But unlike many calculating prodigies, who often are
*idiot savants*, Art is also a top-notch mathematician, and is as successful at
his "day job", as a professor of mathematics at Harvey Mudd College, as he is in his
"night job" of doing mental arithmetics. Unfotunately, as I will point out
in this opinion, some of his talent is wasted. The same can be said about many
other mathematicians, and I decided to "pick" him- as a **case study**- partly
because of his celebrity, and partly because that I am sure that he can take some criticism.

Of course, Art is a real phenomenon, not only in his "mental arithmetics" but
his general mathematical persona.
His inimitable *style* of writing, teaching, and lecturing, conveys his
contagious enthusiasm for doing mathematics, and shows, by example, that
math should and *could* be fun.

When Art is doing one of his *mathemagics* shows, he does not explicitly state:

"All the calculations that I am doing in my head could be done even faster with electronic calculators, and the mathematical content of addition, multiplication, square-root extractions etc. is today, and has been for the last ten thousand years, purely routine. Doing sums or products in one's head is a challenging human sport, but it is not serious mathematics research, and hence should be labeledfor entertainment only."

Of course, it is hardly necessary, since it is so obvious, and one can't accuse
him of intellectual dishonesty for not posting the above disclaimer.
Everyone knows this, even the most "general" public.
But, it is far less obvious, to many people, including Art himself, that many
of the "theorems" given
"elegant combinatorial proofs" in his attractive book, with Jenny Quinn,
Proofs that Really Count,
and many of his proofs in his articles with collaborators and students
(see his website), are also, by **today's** knowledge, just as routine as, say, 11x14=154.
While he is welcome to give a three-page "insightful" direct combinatorial proof
of, for example,
Σ_{k} binomial(2n,k) F_{2k}=5^{n}F_{2n} ,
I believe that intellectual honesty requires that he should state,
**at the very beginning of the article**, that this proof is "for entertainment only", since
this identity-and tons of others like it- are today routinely provable by computers
(and often even by humans).
In fact, for this particular identity (and for many similar ones), as I show in pages 10 and 11 of my
article,
it suffices, in order to give a **completely rigorous proof**, to
check the identity for **only** the first four special cases, n=0,n=1,n=2,n=3 .

Many years ago, I once said, in a talk,

"A Direct Combinatorial Proof is like Sex. If it is good, then it is great. If it is bad, then it is still better than nothing".I am no longer so sure. I still think that Direct Combinatorial Proofs are like Sex, except that

Furthermore, there is a uniform method, described in the article, by Phillip Matchett Wood and myself, A Translation Method for Finding Combinatorial Bijections, that uniformly translates "ugly algebraic proofs" to "gorgeous combinatorial proofs".

One example of a great waste of time (in my opinion) is the
article, by Benjamin, Alex
Eustis and Sean Plott who go to great lengths in order to prove the above- mentioned identity.
( Σ_{k} binomial(2n,k) F_{2k}=5^{n}F_{2n} , ).
But the "Ig Nobel" should go to the article
Combinatorially Composing Chebyshev Polynomials,
by Art Benjamin and Daniel Walton. They take **eleven** pages to find an "insightful"
combinatorial proof of the composition formula for Chebyshev polynomials

T_{m} (T_{n}(x))=T_{mn}(x), where
T_{n}(x) are the Chebyshev polynomials, defined by
T_{0}(x)=1,T_{1}(x)=x, and for n ≥2,
T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x) .

This is an **utter triviality** that can be proved in two lines. First observe that

T_{n}((z+1/z)/2)=(z^{n}+z^{-n})/2 ,
since both sides satisfy the recurrence
A_{n}=(z+1/z) A_{n-1}-A_{n-2} ,
(and the initial values at n=0,1 match).
Hence

T_{m} (T_{n}((z+1/z))/2)=
T_{m} ((z^{n}+ z^{-n})/2)
=((z^{n})^{m}+ (z^{n})^{-m})/2 =
(z^{mn}+z^{-mn})/2=T_{mn}((z+1/z)/2)
QED.

Now don't get me wrong. There are lots of gorgeous proofs in the
above-mentioned Benjamin-Quinn book, and the numerous papers by Benjamin and
his coauthors (those that take less than one page).
But those proofs that take more than one page are "too much of a good thing",
and not worth the trouble. One should not fall in love with the hammer.
In mitigation, one should admire the ingenuity of the authors (some of them are unergraduates),
and in the sense of sharpening one's proof-muscles, and as *etudes* in that
genre, there are not really a waste. Also, if one gets hooked on direct combinatorial
proofs, and loves them for their own sake, then by all means, it is a harmless pastime to
design them and to read them, but one should be **honest** and explicitly state, at
the *very first sentence* and also in the **abstract**:

"We give an elegant, albeit long, direct combinatorial proof to the following result that can be quickly and/or routinely proved by plain algebra."

Opinions of Doron Zeilberger