Henry Ernest Dudeney/Modern Puzzles/Arithmetical and Algebraical Problems/Age and Kinship Puzzles

Henry Ernest Dudeney: Modern Puzzles: Arithmetical and Algebraical Problems

If you add the square of Tom's age to the age of Mary,

the sum is $62$;

but if you add the square of Mary's age to the age of Tom,

the result is $176$.

Can you say what are the ages of Tom and Mary?

Mrs. Wilson had three children, Edgar, James and John.

Their combined ages were half of hers.

Five years later, during which time Ethel was born, Mrs. Wilson's age equalled the total of all her children's ages.

Ten years more have passed, Daisy appearing during that interval.

At the latter event Edgar was as old as John and Ethel together.

The combined ages of all the children are now double Mrs. Wilson's age, which is, in fact, only equal to that of Edgar and James together.

Edgar's age also equals that of the two daughters.

Can you find all their ages?

Augustus de Morgan, the mathematician, who died in $1871$, used to boast that he was $x$ years old in the year $x^2$.

My living friend, Jasper Jenkins, wishing to improve on this, tells me he was $a^2 + b^2$ in $a^4 + b^4$;

that he was $2 m$ in the year $2 m^2$;

and that he was $3 n$ years old in the year $3 n^4$.

Can you give the years in which De Morgan and Jenkins were respectively born?

Two gentlemen with an eccentric approach to philosophy were pinned down by your investigative reporter.

They wished to riddle my mathematical understanding.

"Our two ages combined," said the first, "is $44$."

"Don't be silly," said the other, "it's $1280$."

They looked at me and said, "You see, we didn't tell you how we were combining them."

It was clear to me that the first number was their difference and the second was their product.

Now, how old were these two gentlemen?