One hundred logicians make one hundred statements about themselves. Only one can survive scrutiny.
The Hundred Numbered Logicians
Riddle statement
In a room there are one hundred logicians.
The first one says:
“There is exactly 1 liar here.”
The second one says:
“There are exactly 2 liars here.”
The third says:
“There are exactly 3 liars here.”
And so on, until the last one, who says:
“There are exactly 100 liars here.”
Every logician is either a truth-teller, who always tells the truth, or a liar, who always lies.
How many truth-tellers are there?
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Solution
The statements are mutually incompatible: two different exact numbers of liars cannot both be correct.
Therefore, at most one of the hundred logicians can be a truth-teller.
Now suppose there is exactly one truth-teller. Then there are 99 liars.
In that case, the true statement is the one made by the logician who says, “There are exactly 99 liars here.” Everyone else is lying, so this is perfectly consistent.
Could they all be liars? No. If all one hundred were liars, then there would be exactly 100 liars, so the last logician would be telling the truth, a contradiction.
So the only consistent possibility is that there are 99 liars and 1 truth-teller.
So there is exactly one truth-teller: the 99th logician.