Unlucky Friday

Number Theory Level 2

A certain month has 5 Fridays, but the first and the last day of the month are not Fridays.

What is the 13th day of the month?

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Chung Kevin by Chung Kevin , 45, Australia

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2 solutions

If a month has 5 Fridays then the numerical dates of the first and fifth Friday differ by 28. If the month in question has only 30 days then the possible numerical dates for the first and fifth Fridays are either ( 1 , 29 ) (1,29) or ( 2 , 30 ) (2,30) . But as we are given that the first and last days of the month are not Fridays we see that we must be dealing with a month with 31 days, in which the numerical dates for the first and fifth Fridays are ( 2 , 30 ) (2,30) , (as the other options for a 31-day month, namely ( 1 , 29 ) (1,29) and ( 3 , 31 ) (3,31) do not qualify).

So with Friday being the 2nd day of the month, the next Friday will be the 9th day, and 4 days after this, namely T u e s d a y \boxed{Tuesday} , will be the lucky 13th.

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good question

ppk k - 4 years, 3 months ago

There is a single answer to the question, therefore the month starts on a Wednesday, so 13th has to be a Tuesday.

Tony Durran - 4 years, 3 months ago

I would be careful with the word "and" in the original problem. "The first AND the last day are not fridays." This says that the first can be on a Friday as long as the last isn't. Vice versa. Then the last case where they both are not on a Friday. Be careful with logic operators.

Getcha Water - 4 years, 3 months ago

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I take "the first and the last day of the month are not Fridays" to be logically equivalent to "neither the first nor the last day can be a Friday", which is the crucial element to finding a unique solution to the problem. My sense is that how you are interpreting it is "either the first or the last day of the month is not a Friday", where the "or" is inclusive, giving rise to the three cases you mention, whereas my interpretation just leads to the last of your options.

Brian Charlesworth - 4 years, 3 months ago

I got a heavy headache, when trying to think the answer.,, well now i know... Thanks anyway

Yusri Amilkassim - 2 years, 4 months ago
Tony Durran
Mar 3, 2017

There is a single answer to the question, therefore the month starts on a Wednesday, so 13th has to be a Tuesday.

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