**From New Scientist #1424, 4th October 1984** [link]

Professor Calendar (1930-82) was a noted mathematician. The almanacs published by him were well known. A census enumerator once asked him about the ages of his sons. The professor said: “All my three sons have the same date of birth and were born on the same day of the week”. He continued, “there are no twins and the date of today is the sum of their ages”. The enumerator, an able mathematician himself, replied: “Sir, I could not get their ages. Is one of them twice as old as his brother?”. The professor replied in the affirmative.

What was the age of the eldest son?

[enigma277]

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This Python program considers all possible dates between 1930 and 1982, and groups them by (day of the week, day of the month, month of the year). Then for each group it considers three of the dates, works out possible ages for those dates such that the sum of the ages is between 1 and 31, and groups the ages by their sum.

The sets of ages are then examined to find a sum that corresponds to more than one group, and exactly one of the groups contains an age that is twice that of another age in the group.

It runs in 306ms.

Solution:The eldest son is 16 years old.The middle son is 10, and the youngest son is 5 (exactly half the age of the middle son), so we know that the Census Enumerator visited on the 31st of the month.

There are five possible age sums that correspond to multiple ages – 23, 26, 28, 29, 31. But only when the sum is 31 is there a group where one sibling is twice the age of another.

The only other option for a visit on the 31st of the month is that the ages of the sons are (18, 12, 1), but in this scenario none of the siblings is twice the age of one of the other siblings. So the Census Enumerator is able to use his question to distinguish the two cases.

There are many possible dates that correspond to ages (16, 10, 5). For example, the birthdays could be on Sunday 24th May in 1953, 1959 and 1964, and the Census Enumerator visited on Halloween, Friday 31st October 1969.

In the UK, a census is taken in years that end in 1, generally at the end of March or beginning of April.

I think we can assume that the professor must have been at least 21 to have had three sons, none of them twins! But are the dates 1930-1982 significant?

Between 1901 and 2099 a given date recurs on the same day of the week after successively 5, 6, 11, 6 years: a cycle of 28 = 4×7 years.

Possible ages are 0 + 5 + 11 = 16, 0 + 6 + 11 = 17, 1 + 6 + 12 = 19, 1 + 7 + 12 = 20, 2 + 7 + 13 = 22, 1 + 6 + 17 = 24, 3 + 8 + 14 = 25, 3 + 9 + 14 = 26, 3 + 8 + 19 = 30 (no confusion); or 0 + 6 + 17 = 23, 2 + 8 + 13 = 23; 0 + 5 + 22 = 27, 2 + 7 + 18 = 27; 0 + 11 + 17 = 28, 4 + 9 + 15 = 28; 0 + 6 + 23 = 29, 4 + 10 + 15 = 29; 1 + 12 + 18 = 31, 5 + 10 + 16 = 31. Have I forgotten any?

If the enumerator had called on the 23rd or 27th or 28th or 29th he would have asked a different question, such as “is your youngest son less than a year old?”

It must have been 31 March 1971 or 1981.

I have to say that “the date of today” is strange wording.

I don’t think 1971 works, so if we’re limiting possible dates to those close to the official dates of the UK Census, that leaves Tuesday 31st March 1981 as the most likely date for a visit. The children would have been born on the same date in March 1965, 1971 and 1976. The Professor would be around 34, 40 and 45 at the time of these births.

The children being born on the same date in March 1945, 1951 and 1956 works for Census Enumerator visiting on Friday 31st March 1961. But the Professor would have to have been around 14 at the time of the birth of the first child, so let’s discard that solution.

The Wikipedia page on Census in the UK [ http://en.wikipedia.org/wiki/Census_in_the_United_Kingdom ] mentions that there was a trial census in 1966, so that makes a visit date of Thursday 31st March 1966 a possibility. The children could be born on the same date in January or February of 1950, 1956 and 1961, or April to December 1949, 1955, 1960. The Professor would have been around 19 when the first child was born, so this would work too.

This is basically the same as Jim’s solution.

Is there another solution? The 1961 census date was 23 April. The three boys could have been born in March or April in 1949 (age 12), 1955 (age 6) and 1960 (age 1), total ages 19, and the enumerator could have made his enquiries on 19 March or on 19 April. The prof would have been 19 or so when his firstborn appeared.

Ages of (12, 6, 1) can’t be a solution to the problem because it is the only possibility for an age sum of 19, so the enumerator would be able to work out the ages without further questioning.

Thank you Jim, for your patience too.