### Solutions to Assignment #1

Please refer to Assignment #1 for the tale and the questions about it.
To answer the bonus question ("Who wrote this tale? Where did it first appear?") first, the tale was part of A Tangled Tale by Lewis Carroll (Charles Lutwidge Dodgson), which was first serialized in a magazine. (Which one? When did it appear?) Here is Carroll's own solution:

### § 3. THE SONS' AGES.

*Problem*.---"At first, two of the ages are together equal to the third. A few years afterwards, two of them are together double of the third. When the number of years since the first occasion is two-thirds of the sum of the ages on that occasion, one age is 21. What are the other two?"
*Answer*.---"15 and 18"

*Solution*.-Let the ages at first be `x`, `y`, (`x`+`y`). Now if `a`+`b` = 2`c`, then (`a`-`n`) + (`b`-`n`) = 2(`c`-`n`), whatever be the value of of `n`. Hence the second relationship, if *ever* true, was *always* true. Hence it was true at first. But it cannot be that `x` and `y` are together double of (`x`+ `y`). Hence it must be true of (`x`+ `y`) together with `x` or `y`; and it does not matter which we take. We assume, then, (`x`+ `y`) + `x` = 2`y`, *i.e.* `y` = 2`x`. Hence the three ages were, at first, `x`, 2`x`, 3`x`; and the number of years, since that time is two-thirds of 6`x`, `i.e.` is 4`x`. Hence the present ages are 5`x`, 6`x`, 7`x`. The ages are clearly *integers*, since this is only "the year when one of my sons comes of age." Hence 7`x`=21, `x`=3, and the other ages are 15, 18.

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