I'm disbelieving of this being a Harvard Interview Question, but that is the title given to a question circulating on social media. It runs as follows:
| "7 men have 7 wives. Each man and each wife have 7 children. What's the total number of people?" |
|---|
It's ambiguous and that's probably why it has gone viral, with people offering a variety of answers and opinions and even the occasional quip, such as "Were they going to St Ives?".
So it's not the maths that is interesting but the impetus to debate, discuss wrangle, argue. disagree and tut about the ambiguity.
Still, to get back to the boring old maths, I thought I would ask:
If 7 is replaced by n, what is the smallest answer that we could put forward a case for, and what is the largest?
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A naïve interpretation is that there are a total of men and wives, with one wife per man. Each of couples has children, so the total number of people is .
But we can go smaller than this! If we allow there to be more than one generation, some of the children can also be men and wives. For the first generation, we have 1 man and 1 wife with children. If is even, then (permitting incestuous marriages) we have couples in the second generation, leading to grandchildren which can form couples, which is always greater than the couples we have left to handle. So the number of great-grandchildren is , for a total of people.
In the case of odd , we instead have couples among children, for a total of grandchildren and great-grandchildren, so the total number of people is still . Finally, for we do not have enough children to produce any grandchildren, for a total of 2 parents and 1 child, which is still .
For the largest answer, I would argue that we have men with wives each, for a total of wives. Now, considering that "each man and each wife" does not specify whose wife is involved, it is a valid interpretation that each man has children with all the wives, leading to couples each having children. In this case, the total number of people is .
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Phew that's impressive, it never occurred to me that some of the children could also be amongst the parents! (Perhaps this is a Harvard Interview question after all 🤣!)
So to take the smallest case, how would that pan out with a concrete number, say with n = 7 as per the original question?
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In the smallest case with n=7, we start with 2 people, the original parents. Since n=7, this couple has 7 children.
In the second generation, 6 of our 7 children form 3 couples, with 1 child left over, and each has another 7 children. So there are 21 grandchildren in the third generation.
Now we have had 1 couple in the first generation, and 3 in the second generation, and we need a total of 7 couples. So 6 of the 21 grandchildren form another 3 couples to complete the set, and they each have another 7 children, so there are 21 great-grandchildren.
The total number of people is then 2 parents + 7 children + 21 grandchildren + 21 great-grandchildren = 51 = .