For Argument Sake, There Are Ten Conga Dancers

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The following logic problem appeared last week
in the New York Times Wordplay column.

In a party there are truth-tellers and liars. At some point, 
they all form a huge conga line and start singing. 

The first one goes, “There’s at least one liar here!”
The second one goes, “There are at least two liars here!” 
And so on, until the end of the line. 
Who is lying? Who is telling the truth?
 
In a second version of the party, 
the first in line sings, “There’s exactly one liar here!”
The second goes, “There are exactly two liars here!” And so on. 
In this case, who is lying, and who is telling the truth?
 
In the third version, the first sings, “There’s at most one liar here!”
the second, “There are at most two liars here!” 
And so on. What then?
 
You may want to ignore the solutions below
until you come up with your own answers.

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CASE 1
If there are ten dancers, the first five are telling the truth.
The fifth person is correct in stating There are AT LEAST five liars here.

The last five are lying and the lie told by the sixth person is 
There are AT LEAST six liars here....

CASE 2
If there are ten dancers, nine of them are liars.
Only the ninth person is telling the truth.
That person is correct in stating There are EXACTLY nine liars here

But the tenth person, when saying There are EXACTLY  ten liars,
is telling a lie. Everyone prior to the second-to-last dancer
is equally lying.   


CASE 3
The most difficult of the scenarios because the statement
In a party there are truth-tellers and liars. 
At some point, they all form a huge conga line and start singing...
diverts you from the correct answer.
 

If there are ten dancers, none of them are liars.
All of them are telling the truth.

For example, when the seventh person says
There are AT MOST seven liars here,
the truth is being told because
zere is less than seven. 


The so-called "liars" on the dance floor, in this scenario
did their lying elsewhere!

***
The solutions posted here were created by two learned math teachers.
On their behalf, I will not claim the solutions to be
the copyrighted property of this website.

Further clarification is available upon request. 
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