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194 STELLING
we knew exactly what to do this time and there was no panic. It was handled exactly the
same way as the previous one. By the end of January 1989, the plane of order 10 was
dead a third and, hopefully, the final time. (Lam 1991: 316)
No more errors were found from this point onwards. But the fact that errors were found in
the first place opened up the problem of whether or not there had been errors that slipped
through the cracks, errors either due to faulty programming or undiscovered hardware
malfunctions. After discussing (and discarding) the possibilities of programming errors, Lam
says:
There is, moreover, the possibility of an undetected hardware failure. A common error
of this type is the random changing of bits in a computer memory, which could mean
the loss of a branch of a search tree. This is the worst kind of hardware error, because
we might lose solutions without realizing it. The CRAY-1A is reported to have such
errors at the rate of about one per one thousand hours of computing. At this rate, we
expect to encounter two to three errors! We did discover one such error by chance.
After a hardware problem, Patterson reran the 1000 A2’s just before the failure and the
statistics have changed for the A2 processed just prior to the malfunction. How should
one receive a “proof” that is almost guaranteed to contain several random errors?
Unfortunately, this is unavoidable in a computer-based proof—it is never absolute.
However, despite this reservation, we argued [...] that the possibility of hardware errors
leading us to a wrong conclusion is extremely small. (Lam 1991: 316)
How small exactly? The worst case scenario would be this: Suppose there is in fact a finite
projective plane of order 10. Then there are 24,675 codewords of weight 19, each giving rise to
an A2. 17 The overall number of A2’s, including the ones that do not lead to a positive result, is
somewhere around 500,000. Since we are trying to conserve computing time as much as
possible, the procedure begins by checking all A2’s for isomorphisms between them. Because
two isomorphic A2’s behave the same way, we only need to check one of them, and thus the
superfluous copies are eliminated at the start. So if all of these 24,675 A2’s are isomorphic,
then we really only check a single one of them, and therefore there is exactly one A2 in our list
that can successfully be extended to an order ten plane. The possibility of this specific A2
being affected by a memory error, given that two to three such errors happen during the
calculations, is still less than 10 -5 , or 0.001%. Far more likely, however, is that not all of the
A2’s that give rise to the supposed plane of order ten are isomorphic, in which case there
would be several distinct A2’s that would each have to have been affected by such a random
error, in which case “the probability of hardware errors affecting all of them is infinitesimal.”
(Lam 1991: 317)
Basically, the argument depends on the observation that if a plane of order ten exists, it
can be constructed from many different starting points. Random hardware failures are
unlikely to eliminate all of them. In other words, the fact that no one has yet
constructed one is a very strong indication that it does not exist. (Lam 1991: 317)
It should be noted that the authors of this result themselves were careful in the original
publication to point out the uncertainty behind their methods:
Because of the use of a computer, one should not consider these results as a “proof”, in
the traditional sense, that a plane of order 10 does not exist. They are experimental
results and there is always a possibility of mistakes. (Lam, Thiel & Swiercz 1989: 1120)
17
We are skipping over some details here. Readers interested in why all of a sudden we are talking about
weight 19 codewords, or how we arrived at that specific number of A2's, should consult Lam, Thiel &
Swiercz 1989.