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Saturday, July 26, 2008

William's Whereabouts

William lives in a street with house-numbers 8 up to 100. Lisa wants to know at which number William lives.

She asks him: "Is your number larger than 50?"
William answers, but lies.
Upon this, Lisa asks: "Is your number a multiple of 4?"
William answers, but lies again.
Then Lisa asks: "Is your number a square?"
William answers truthfully.
Upon this, Lisa says: "I know your number if you tell me whether the first digit is a 3."
William answers, but now we don't know whether he lies or speaks the truth.

Thereupon, Lisa says at which number she thinks William lives, but (of course) she is wrong.

The Question: What is Williams real house-number?


Answer: Note that Lisa does not know that William sometimes lies. Lisa reasons as if William speaks the truth. Because Lisa says after her third question, that she knows his number if he tells her whether the first digit is a 3, we can conclude that after her first three questions, Lisa still needs to choose between two numbers, one of which starts with a 3. A number that starts with a 3 must, in this case, be smaller than 50, so William's (lied) answer to Lisa's first question was "No". Now there are four possibilities:

Only the combination "number is a multiple of 4" and "number is a square" results in two numbers, of which one starts with a 3. William's (lied) answer to Lisa's second question therefore was "Yes", and William's (true) answer to Lisa's third question was also "Yes".

In reality, William's number is larger than 50, not a multiple of 4, and a square. Of the squares larger than 50 and at most 100 (these are 64, 81, and 100), this only holds for 81.

Conclusion: William's real house-number is 81.

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