TITLE: Can we reduce the number of states of a Turing Machine?
QUESTION [9 upvotes]: My friend claims that one could reduce the number of states of a given turning  machine by somehow blowing up the tape alphabet. He does not have any algorithm though. He only has the intuition.
But I say it's not possible. Else one could arbitrarily keep decreasing the states via the same algorithm and arrive at some constant sized machine.
Who is right?

REPLY [7 votes]: Yes it is possible to reduce the number of states a Turing machine uses to decide a problem $X$ by increasing the number of symbols.
However it gets very tricky when you are close to the minimum number of states/symbols needed to solve  $X$
Here is a nice survey paper about the efforts to find the minimum number of states and symbols Turing machines need for universality. 
http://portal.acm.org/citation.cfm?id=1498068
"The complexity of small universal Turing machines: A survey"
Damien Woods and Turlough Neary,
Theoretical Computer Science archive
Volume 410 Issue 4-5, February, 2009