Diophantus' childhood ended at $14$, he grew a beard at $21$, married at $33$, and had a son at $38$. Diophantus' son died at $42$, when Diophantus himself was $80$, and so Diophantus died four years later when he was $84$. Checks out!
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Also $18$ is divisible by each of $2,3,9$; so the $1$ st son gets $9$ camels, the $2$ nd son gets $6$ camels, and the third son gets $2$ camels. Miraculously , we get $9 + 6 + 2 = 17$ camels , hence the extra camel that was brought before can be returned back to the owner.
The only way to get the 13/27 answer is to make the unjustified unreasonable assumption that Dave is boy-centric & Tuesday-centric: if he has two sons born on Tue and Sun he will mention Tue; if he has a son & daughter both born on Tue he will mention the son, etc.
In case this is the correct solution: Why does the probability change when the father specifies the birthday of a son? (does it actually change? A lot of answers/posts stated that the statement does matter) What I mean is: It is clear that (in case he has a son) his son is born on some day of the week.
The father then randomly handed each son one of the two envelopes with a probability of $0.5$. After both sons opened their envelopes, his father privately asked each son whether he wanted to switch his envelops with the one his brother had.
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