Source:
http://mathnexus.wwu.edu/Archive/problem/detail.asp?ID=134
Consider this list
of twelve statements:

Precisely
one of these statements is false. 

Precisely
two of these statements are false. 

Precisely
three of these statements are false. 

Precisely
four of these statements are false. 

Precisely
five of these statements are false. 

Precisely
six of these statements are false. 

Precisely
seven of these statements are false. 

Precisely
eight of these statements are false. 

Precisely
nine of these statements are false. 

Precisely
ten of these statements are false. 

Precisely
eleven of these statements are false. 

All twelve
of these statements are false. 
Which statements
are true? Explain.
Which statements
are false? Explain.
Any statements
that could be true or false? Explain.

Source:
James Tanton's "A Dozen Questions About a Dozen," Math Horizons, April
2007, pp. 1216. 
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Drawing
from tom on 8/21/2009 
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Hint: Consider
a smaller problem:

Precisely
one of these statements is false. 

Exactly
two of these statements are false. 
Does this help?
Can you transfer your reasoning
to the full
set of twelve statements?
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.
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Finger pointing down
from darrell94590 on 1/2/2006 
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..
THE SOLUTION
Solution Commentary:
First,
since all of the statements contradict
each other, it is
impossible for two statements to be true.
But, if all twelve
statements are false, then the last statement must be correct, which leads
to a contradiction.
Thus, the only option
is for eleven statements to be false and one
statement to be true...so
now, which one is the true statement? 