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Mathematica and trivial math problems that school children can solve.
*To*: mathgroup at smc.vnet.net
*Subject*: [mg129818] Mathematica and trivial math problems that school children can solve.
*From*: Andrzej Kozlowski <akozlowski at gmail.com>
*Date*: Sat, 16 Feb 2013 01:09:24 -0500 (EST)
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Recently I came across an article in a mathematics and physics journal
aimed at high school students. The main reason why it attracted my
attention was that its theme was computer algebra and, in particular,
Mathematica. The moral of the story was quite sensible: these programs
can do wonderful things but don't expect them to be smart, in the human
sense, anyway. The example that was given was apparently taken from the
Internet as a notorious "breaker" of computer CAS programs. This was the
illustration in Mathematica:
Reduce[(10 x + 2)^2009 == (2 x + 4)^2009, x, Reals]
Apparently all CAS programs used to get stuck on this, back in 2009,
that is. Well, it is now 2013 and computers are much faster so actually
we get:
Timing[Reduce[(10*x + 2)^2009 == (2*x + 4)^2009, x, Reals]]
{6.51, x == 1/4}
So, to produce an illustration suitable for the present day we add an
extra 0 inside the exponent:
Reduce[(10 x + 2)^20009 == (2 x + 4)^20009, x, Reals]
Mathematica spends far too long to do this than anyone would care to
wait, while a reasonably smart school high-schooler can do it in no time
at all. So should we be disappointed with Mathematica's performance
here? Well, the first thing I always want to say when someone comes up
with examples of this kind is: don't try to judge CAS programs by the
sort of things you can do very well yourself. Chances are they will come
up short (one reason I will explain below). The right way to judge CAS
program (and perhaps all computer programs) is by testing them on things
that we either can't do well or can't do at all, like solving general a
polynomial equations of some large degree.
Of course we can easily see the reason for the problem above. When
Mathematica is given a polynomial equation to solve, the first thing it
does is to expand the polynomials. In the overwhelming majority of cases
this will be the right thing to do and only in some very rare ones, like
the above, there is a better approach. Does it mean that Mathematica
does not "know" the other approach and that it could not be programmed
to use it? Of course not. In fact, by slightly changing the problem
above you can make Mathematica return the answer almost immediately:
Reduce[(10 x + 2)^(20009 Pi) == (2 x + 4)^(20009 Pi), x,Reals] // Timing
{0.301534, x == 1/4}
All we did above was make the power non-rational, which prevented
Mathematica from using the expansion method. Obviously, it would be
quite easy to change Reduce, so that it would first look for such
special cases before turning to the standard expansion-based algorithm.
But would this sort of thing be worth the time spent by the programmer,
even if in this case it is very short? Obviously there are lots of other
special cases, corresponding to various situations where for some reason
standard computer algebra algorithms do not work well or work slowly,
but which human beings have no difficulty with. Should they all be
implemented by "heuristic" techniques?
Of course this sort of thing (using heuristics when there are not good
algorithms) is done quite often. But it is reasonable to do so only if
there is a fairly high chance of this sort of situation arising in the
course of some naturally occurring computation that was not designed to
show that there a human being can beat Mathematica at certain things
(actually it is still very easy to show that and it will surely remain
so for many years to come). It is not worth spending time and effort
only to deprive people certain people of the satisfaction of posting
trivial examples of what they call "bugs" in Mathematica. Computers are
not intended to replace us (not yet, anyway) but to help us.
Andrzej Kozlowski
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