"A client of mine," said a lawyer, "was on the point of death when his wife was about to present him with a child. I drew up his will, in which he settled two-thirds of his estate upon his son (if it should happen to be a boy) and one-third on the mother. But if the child should be a girl, then two-thirds of the estate should go to the mother and one-third to the daughter. As a matter of fact, after his death twins were born—a boy and a girl. A very nice point then arose. How was the estate to be equitably divided among the three in the closest possible accordance with the spirit of the dead man's will?"

Can anyone please advise.

[This puzzle comes from H E Dudeney's Amusements in Mathematics, published in 1917.]

  • 3
    $\begingroup$ There was a comment at the end of the puzzle text saying it was doing the social-media rounds. I've edited in the actual origin of the puzzle. $\endgroup$ – Gareth McCaughan Nov 25 '17 at 12:54
  • $\begingroup$ @Gareth Mc Caughan thanks for providing information of actual source and edition $\endgroup$ – Mea Culpa Nay Nov 25 '17 at 13:13
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    $\begingroup$ FWIW, this is probably from ancient Rome (or even before that) - I've read about it a couple of times in various puzzle books, and themathmompuzzles.blogspot.com/2011/06/roman-puzzle.html & redhotpawn.com/forum/posers-and-puzzles/roman-twins.146605 seem to agree with that. $\endgroup$ – user13963 Nov 25 '17 at 16:38
  • $\begingroup$ I'm not at all surprised to hear that it's older. But the particular wording here is straight out of Dudeney. $\endgroup$ – Gareth McCaughan Nov 28 '17 at 21:32

I would say that it's a simple matter of proportions.

The father values the son at twice as much as his wife, and his wife at twice as much as his daughter. Therefore, the son is valued at four times as much as his daughter. So, with all three in the equation, the son should get 4/7ths, the mother 2/7ths, and the daughter 1/7th.

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    $\begingroup$ For what it's worth, (1) I agree that this is the most natural solution and (2) it is in fact the one given in Dudeney's book. $\endgroup$ – Gareth McCaughan Nov 25 '17 at 13:00

I'd suggest a different approach on evaluating fairness.

A son was to get 2/3 (66.666...%), a daughter was to get 1/3 (33.333...%), and the mother was to get either 1/3 or 2/3, so setting 1/2 (50.0%) as her expected value seems appropriate to me, not needing more or wilder assumptions than going by bilateral proportions IMHO. - Obviously, that cannot work with them being three now, as it would add up to 150%. If we now say that they all should get less by the same proportional loss, each would need to get 1/3 less than their expected sum and thus the son would receive 4/9 (44.444...%), the mother 3/9 (33.333...%) and the daughter 2/9 (22.222...%).

To me, that seems a more natural approach to interpret the proportions of the last will.

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