Home > Riddles > The hundred logical numerados
One of those pieces where the collective self-description closes at a unique and quite unexpected point.
In a room there is a hundred logicians. The first one says:
“There is exactly 1 liar here.” The second says:
“There are exactly 2 liars here.” The third says:
“There are exactly 3 liars here.” And so on, until the last one, which says:
“There are exactly 100 liars here.” Every logician is either truthful, and always tells the truth, or a liar, and always lies.
How many truthful people are there?
The statements are incompatible with each other: two different quantities cannot be true at the same time. Therefore, at most one of the hundred logicians can be truthful.
Let's see what happens if there is exactly one truth-teller. Then there would be 99 liars.
In that case, the true statement would be that of the logician who says: “There are exactly 99 liars here.” And everyone else would be lying, which fits perfectly. Could they all be liars?
No. If the hundred were liars, then there would be exactly 100 liars, and the last one would be telling the truth, contradiction.
So the only consistent possibility is that there are 99 liars and 1 truth-teller.
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