hardmath

There is a party of 100 high-powered politicians. All of them are either honest or liars. You walk in knowing two things: - At least one of them is honest. - If you take any two politicians, at least one of them is a liar. From this information, can you know how many are liars and how many are honest?

Yes, from the information you know 1 is honest and 99 are liars. One of them is honest satisfying the first piece of information. Then if you take the honest man and any other politician, the other politician must be a liar to satisfy the second piece of information, 'If you take any two politicians, at least one of them is a liar.' So 99 are liars.

Why this works

This riddle cleverly uses logic and the conditions provided to arrive at the conclusion. The first condition states that at least one politician is honest, which we can denote as "H." The second condition indicates that in any pair of politicians there must be at least one liar, meaning if one is honest, the other must be a liar. By considering these two points together, we deduce that if we have one honest politician, then every other politician must be a liar to satisfy the second condition. Therefore, with one honest politician, the total must be 99 liars, making the ratio 1 honest to 99 liars. This logical deduction showcases how the constraints of the riddle lead to a specific, definitive answer.

Common Wrong Answers

“50 honest and 50 liars”

This guess assumes a balanced distribution of honest politicians and liars, which contradicts the second condition that states if any two politicians are chosen, at least one must be a liar. If there were 50 honest politicians, it would be possible to choose two honest politicians, violating that condition.

“99 honest and 1 liar”

This guess incorrectly assumes that the majority of the politicians are honest. However, it violates the second condition because if there are 99 honest politicians, it is possible to select two honest individuals, which would contradict the requirement that at least one must be a liar.

“100 honest”

This guess contradicts the first clue that states at least one of them is honest, but it also violates the second condition. If all were honest, then choosing any two would not satisfy the requirement that at least one must be a liar.

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