Cryptography Reference
In-Depth Information
on it, because complex mathematical problems have a peculiarity: once their
solutions are found, they often become much simpler. The following examples
show just how much simpler.
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You certainly know about Rubik's cube, which challenges you to turn
the layered pieces such that each of its six sides has a different color.
It took me two weeks of occasional trial and error to get my first two
layers in place. The next attempt succeeded after three days, then it
took only one — I had grasped the trick. I then felt I had to proceed
more systematically. Within a week, I found a sequence of 'pieces' and
composed a puzzle out of them. Later I handled the cube without training
(but using a crib) within five to ten minutes. I'm convinced that everybody
can do this.
•
A much more drastic example is the base problem in functional analysis.
The problem itself originates from mathematical basic research; I won't
explain it here. Anyway, it concerns an assumption expressed in the 1930s
which is relatively easy to formulate, as many hard problems are. For
decades, leading mathematicians had cut their teeth over it. Nobody was
able to prove it, until a Dutchman found a counterexample in the mid-
1970s: it was all wrong! The proof that this was a counterexample in the
first place was said to have been about 600 pages long — an inconceivable
mental achievement. I heard a lecture about this proof, cut down to 'only'
80 pages, in Warsaw. Coryphees in functional analysis I so much admired
shook their heads over the complexity of a single theorem. So I wasn't
really sad that I failed to understand most of it. Six months later, a Polish
mathematician told me that the proof had been cut down to less than five
pages and had become readable.
Such stories seem to repeat themselves more often than not in mathematics. The
so-called Hilbert problems were very popular at the end of the 19th century. I
remember that at least one of them had been solved by an 'outsider', a student
from former Leningrad.
So let's summarize:
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Even if great minds cannot solve a problem, an unknown person with
unconventional ideas may sometimes be successful.
•
Even if a solution initially appears outrageously complicated, it can some-
times be drastically simplified.
