# §3: The Sons' Ages explained — A Tangled Tale

> Chapter companion for A Tangled Tale by Lewis Carroll.

## What happens

This chapter presents a mathematical problem about three sons' ages, where the relationships between their ages at two different times lead to a solution. The narrator critiques various incorrect or incomplete solutions submitted by readers, praising only a few for their logical rigor. The chapter concludes with a farewell from the author, reflecting on the end of the series.

## Themes in this chapter

### Mathematical Reasoning

The chapter centers on solving an algebraic age problem, emphasizing logical deduction and critique of flawed reasoning.

### Humor and Absurdity

The narrator uses playful, sarcastic commentary to mock incorrect solutions, comparing solvers to animals like Old Hen and Old Cat.

### Social Critique

The critique of solvers' methods reflects on intellectual rigor and the pitfalls of assumption-based reasoning.

## Character check-ins

### Jane E.

Submitted an incomplete solution, calculating present ages but leaving the second occasion unnoticed.

### Dinah Mite

Correctly determined the relationship between ages at first but made a tentative assumption, resulting in a merely 'honourable' classification.

### Algernon Bray

Solved the problem correctly but noted that fractional ages would lead to infinite answers, earning highest honours.

### Simple Susan

Used a partly tentative method to find five possible sets of ages and eliminated four, classified as 'Clumsily Right'.

### White Sugar

Detected an oversight in the problem regarding the son's exact age on the day of coming of age, providing a second solution.

## Key lines

> 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?

Initially, the sum of two ages equals the third. Later, the sum of two ages is twice the third. The time elapsed since the first occasion is two-thirds of the total ages at that time, and one son is now 21. Find the other two ages.

_This is the core problem statement of the chapter._

> I take this opportunity of thanking those who have sent, along with their answers to the Tenth Knot, regrets that there are no more Knots to come, or petitions that I should recall my resolution to bring them to an end.

The author thanks readers who expressed sadness that the series is ending or asked him to continue.

_This reflects the author's farewell and the end of the puzzle series._

## Links

- HTML: https://www.betterreads.online/discover/a_tangled_tale_se/chapters/3-the-sons-ages
- Book: https://www.betterreads.online/discover/a_tangled_tale_se
- All chapters: https://www.betterreads.online/discover/a_tangled_tale_se/chapters