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Problem 436

Combinatorics Difficulty 2.7 Multiple choice CEMC Gauss (Grade 7) · Canada · 2012

Chris lies on Fridays, Saturdays and Sundays, but he tells the truth on all other days. Mark lies on Tuesdays, Wednesdays and Thursdays, but he tells the truth on all other days. On what day of the week would they both say: “Tomorrow, I will lie.”?

Pick one

Official solution

If either Chris or Mark says, “Tomorrow, I will lie." on a day that he tells a lie, then it actually means that tomorrow he will tell the truth (since he is lying).

This can only occur when he lies and then tells the truth on consecutive days.

For Chris, this only happens on Sunday, since he lies on Sunday but tells the truth on Monday.

For Mark, this only happens on Thursday, since he lies on Thursday but tells the truth on Friday.

Similarly, if either Chris or Mark says, “Tomorrow, I will lie." on a day that they tell the truth, then it means that tomorrow they will lie (since they are telling the truth).

This can only occur when they tell the truth and then lie on consecutive days.

For Chris, this only happens on Thursday, since he tells the truth on Thursday but lies on Friday.

For Mark, this only happens on Monday, since he tells the truth on Monday but lies on Tuesday.
Therefore, the only day of the week that they would both say, “Tomorrow, I will lie.", is Thursday.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.