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The puzzle is as follows:

In 2012, which began on a Sunday, Marina was happy because her birthday was on a Saturday and she celebrated it with her friends. We know, Marina was born in a year where there were more Fridays and Saturdays than other days of the week, and that the day of the week she was born was the twentieth time that appeared in that year. If in 2012 she turned between 20 to 29 years old, in what date she was born?

The choices given are:

  1. May 19th
  2. May 20th
  3. May 16th
  4. May 18th

I found this puzzle in my book Logic challenges. From the way it is presented it seems to have been adapted from either a reprinted version from Martin Gardner's Puzzle Carnival book of the late 1970s or Raymond Smullyan's What's the name of this book.

I don't know what's the intended meaning when it mentions that a pair of days appear more times in a year.

I assume that this situation happens when these days are contiguous such as in this case Friday and Saturday. It appears that when this happens such year begins with January 1st being Friday, and such year is not a leap year.

This hidden piece of information is not very obvious to me and I don't know if there is any way to prove it. Neither I know if I am in the right path.

I'm not sure how to use that the day of the week she was born appeared the twentieth time.

Can someone help me here? I'm looking for a step-by-step explanation of how to get to the answer.

According to my book the official answer is choice 1, or May 19th. But I have no idea how to get there.

How can I solve this without much fuss? More importantly, is my observation about the dates appearing more times in a calendar correct?

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One key observation is that

364 is divisible by 7

So if there are more Fridays and Saturdays than other days of the week in the year Marina was born then it must be that

There were 366 days in the year and the two extra ones were Friday and Saturday.
In particular, we would find that the first two days were Friday and Saturday and the last two days were Friday and Saturday

Finally

2012 is a leap year, begins on a Sunday and Marina celebrates her birthday on a Saturday.
Given that her birth year is also a leap year and starts on a Friday then Marina must be born on Thursday.
We know that Thursday 7th was the first Thursday. Hence, the 20th Thursday was May 19th

Now you are not asked for her birth year but given that she is between 20-29 years old in 2012 we find that

She must have been born in 1988 as this is the only year in the range that satisfies all the criteria.

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Actually, this can be solved even without using information about "Fridays and Saturdays".

We know that Marina's birthday was the 20th occurrence of some weekday during her birth year, so that means that she was born between days 134 and 140 (we use consecutive day-of-the-year numbering), since on day 133 (or earlier) each of the weekday had appeared at most 133/7=19 times since the beginning of the year, and since day 141 there will be 21st and later occurrences.
If she was born in a leap year, the same would hold for 2012; if not, the day-of-the-year numbering will shift by 1 (starting from March) due to the added February 29th. So, in any case, her birthday in 2012 would fall between days 134 and 141.
On the other hand, she celebrates her birthday on Saturday when the year had started on Sunday. That means that her birthday would fall on a day which number is divisible by 7 (since all the Saturdays in 2012 fall on 7th, 14th etc. days). So, the only number between 134 and 141 that is divisible by 7 is 140, that means that on 2012, Marina celebrates her birthday on the 140th day of the year, which is May 19th.

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  • $\begingroup$ And you also don't need 20th (divisible by 7 in 2012 is enough), assuming the correct answer is given. $\endgroup$ – Retudin Apr 21 at 6:39
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    $\begingroup$ @Retudin We need 20th to show that her birthday is indeed in mid-May, and not in the other time of the year (i.e. assuming no choices were given). If we know that one of the choices must be correct, it's sufficient just to check which of the choices falls on Saturday in 2012. $\endgroup$ – trolley813 Apr 21 at 6:43
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    $\begingroup$ @trolley813 - Exactly where I went. Pull up a 2012 calendar, look at May, done. As long as the year isn't required in the answer, this is a quick and accurate shortcut. $\endgroup$ – Joel Rondeau Apr 21 at 15:56
  • $\begingroup$ @JoelRondeau Actually, we don't even need a calendar since we can easily reconstruct it from the condition that 2012 started on Sunday. $\endgroup$ – trolley813 Apr 21 at 18:43
  • $\begingroup$ @trolley813 Sure you can, but I'm lazy $\endgroup$ – Joel Rondeau Apr 22 at 2:56

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