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Earlier today I set you these three problems about birthdays.
1. Philippe [geddit?] was born on 29 February 2016. His parents decided to celebrate his first birthday 365 days later.
On what date did they celebrate his first birthday?
Solution 28 February 2017.
My guess is that most of you got this question wrong. Certainly, my intuition when I first saw this problem was that the answer was March 1. If Feb 29 is the day after Feb 28 in leap years, it seems right that March 1 is the day after Feb 28 in non-leap years. This reasoning is erroneous.
Think about it like this. In normal years, one day after Feb 28 is March 1, two days after Feb 28 is March 2, three days after Feb 28 is March 3, and so on until 365 days after Feb 28 is Feb 28.
In leap years, one day after Feb 29 is March 1, two days after Feb 29 is March 2, three days after Feb 29 is March 3, and so on. Clearly, 365 days after Feb 29 is also Feb 28!
2. My great grandmother was born on the first Sunday of the year. Her seventh birthday was also on a Sunday.
In which year was my great grandmother born?
Solution The hint here was that in the set up to the question I said that leap years come around ‘about’ once every four years. A more accurate description is that the come around exactly every four years apart from years divisible by 100 (but not 400). So, 2000 was a leap year, but 1900 wasn’t.
In a normal year, if a date falls on a Sunday one year, the following year that date falls on a Monday. So, for a date to fall on a Sunday one year, and a Sunday seven years later, we need a run of seven years with zero leap years. The last time this happened with the first Sunday of the year (i.e. one in January) was 1897-1904. So my great grandmother was born in 1897.
3. The mother is 21 years older than her daughter. In six years she will be five times older than her daughter.
Where is the father?
Solution With the mother!
Let’s rewrite the statement of the question algebraically. Let M = age of mother now, and D = age of daughter now. Then
-
D + 21 = M
-
5(D + 6) = M + 6
Substituting the first one in the second:
5D + 30 = D + 21 + 6
which is
4D = –3, or D = –3/4 years. In other words the daughter is aged -9 months. She is being conceived!
I hope you enjoyed today’s problems. I’ll be back in two weeks.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
My latest book is So You Think You’ve Got Problems.
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