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Computer Assisted Mathematics: Tools and Tactics for Solving Hard Problems\
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Daniel Lichtblau
danl@wolfram.com\
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Wolfram Research, Inc.
October/November, 2006\
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"This talk was given at IMA on November 2, 2006. It is a minor variant of \
one given at the ",
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" Technology Conference in October 2006. Caveat: This was prepared using the \
version of ",
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" under development as of October 2006. A few examples will not work quite \
the same way in version 5."
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"In this talk I will present several problems that have caught my attention \
over the past few years. We will go over ",
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" formulations and solutions. Along the way we will meet with a branch\
\[Hyphen]and\[Hyphen]bound loop in its natural habitat, some rampaging Gr\
\[ODoubleDot]bner bases, a couple of tamed logic puzzles, and at least a \
dozen wild beasts (well...would you believe... a ",
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" developer wearing Halloween fangs?). You'll laugh. You'll cry. You'll \
write your holiday cards (be sure to spell my name correctly)."
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"As the purpose is to illustrate a few of the many ways in which ",
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bit of detail in selected examples. Do not let this deter you; there will be \
no exam, and it is the methods, not the problems, that are of importance. The \
examples are culled from problems I have seen on Usenet groups (primarily ",
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At a workshop in 2004 I was posed the following: Find a positive integer such \
that, when multiplied by 9, you get the same result as by moving the least \
significant digit to the most significant. The person asking is editor\
\[Hyphen]in\[Hyphen]chief of the Journal of Symbolic Computation. He told me \
the (smallest) result had 44 digits, and it had taken him substantial \
computational effort to find it.\
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We confirm below that the smallest solution has 44 digits. It can be shown to \
be the repeating block of the fraction 9/89.\
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"This is a topic of some apparent interest insofar as it shows up with \
regularity on the ",
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"I have written code to address this sort of question on a few occasions, \
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"There is a price to pay for getting the power of algebraic reduction. We \
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I do not propose to explain in any detail what these are. For our purposes \
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Various tactics are used to extend this idea to handling rational functions \
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.36