Knights and Farmers
At a round table are sitting 2017 people. Each of them is either a knight or a farmer. Knights always speak the truth while farmers always lie. Everybody says, "One of my neighbors (left or right) is a knight and the other is a farmer."
How many knights are sitting at the table?
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1 solution
For a Knight to make the statement it is required that the Knight to sit next to exactly 1 Knight and 1 Farmer.
For a Farmer to make the statement it is required that the Farmer to sit next to 2 Knights (1 Knight on each side) or 2 Farmers (1 Farmer on each side).
If there are any Knights on the table, the Farmer who sit next to the Knight has to be sandwiched by the knights (Otherwise the Farmer sits next to 1 Knight and 1 Farmer and the statement is not a lie). This forms a repeating pattern of FKKFKK... and for a round table the number of people sitting at the table would be divisible by 3.
However 2017 is not divisible by 3 (2017 is prime). Therefore it is impossible for any Knights to be at the table while the statement is still valid.