Puzzle Of The Day: 89
Truth Day
Philip lies a lot.
[1] He tells the truth on only one of the days of the week.
[2] The days of the week in order are:
Sunday, Monday,Tuesday,Wednesday,Thursday,Friday,Saturday,Sunday,Monday,etc.
[3] One day he said: “I lie on Mondays and Tuesdays”.
[4] The next day he said: “Today is either Thursday, Saturday, or Sunday”.
[5] The next day he said: “I lie on Wednesdays and Fridays”.
On which day of the week does Philip tell the truth? Suppose his declaration in [3] is the only true one. Then, from his false declaration in [5], he made the declaration in [3] on a Wednesday or a Friday. Then, from [2], his false declaration in [4] was made on a Thursday or a Saturday. So his declaration in [5] is the only true one. Then, from his false declaration in [3], he made the declaration in [5] on a Monday or a Tuesday. Then, from [2], his false declaration in [4] was made on a Sunday or a Monday. His declaration in [4] would be true. So his declaration in [4] was made on a Monday. Then from [2], his true declaration in [5] was made on a Tuesday and Philip tells the truth on Tuesday.
From [1], Philip’s declarations in [3] and [5] cannot both be false; otherwise he tells the truth on more than one of the days of the week. From [1] and the fact that the declarations in [3] through [5] were made on consecutive days, Philip’s declarations in [3] and [5] cannot both be true; otherwise he tells the truth on more than one of the days of the week. So either Philip’s declaration in [3] is the only true one or his declaration in [5] is the only true one.
This situation is impossible.
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