# Answers to Knot VII explained — A Tangled Tale

> Chapter companion for A Tangled Tale by Lewis Carroll.

## What happens

The chapter presents the solution to a mathematical problem involving the cost of lemonade, sandwiches, and biscuits, and discusses the methods used by various competitors. It includes a class list of solvers, critiques of incorrect or incomplete solutions, and a digression on logic and reasoning.

## Themes in this chapter

### Mathematical Reasoning

The chapter provides a detailed algebraic solution to a system of equations and discusses methods for solving problems with multiple unknowns.

### Humor and Absurdity

Includes playful commentary on competitors' mistakes, such as the 'dreamy remark' and the absurdity of assuming sandwiches cost nothing.

### Social Critique

Critiques the reasoning of various solvers and the nature of competition, with a digression on logic and arguing with a lady.

## Character check-ins

### Clara

Referenced in the context of her luncheon cost problem.

### Balbus

Mentioned for his general principle about ascertaining the cost of a luncheon.

### Simple Susan

Thanked for kind words of sympathy.

### Old King Cole

Listed in the class sections for his solution.

### Hecla

Shared highest honours for using a method certain to produce the answer or prove impossibility.

### Martreb

Shared highest honours for the same reason as Hecla.

## Key lines

> If it is all his own, he will make a good algebraist in the time to come.

If Little Jack's solution is original, he shows promise as a future algebraist.

_Praise for a competitor's algebraic reasoning._

> The two Kings are fearfully deliberate! I suppose walking quick, or taking shortcuts, is inconsistent with kingly dignity.

The two kings' solutions are extremely slow, perhaps because haste is beneath their royal status.

_Humorous critique of the slow pace of certain solvers._

> In logical language, in order to disprove a 'universal affirmative,' it is enough to prove its contradictory, which is a 'particular negative.'

To disprove a statement that applies to all cases, one only needs to show a single counterexample.

_Explanation of logical reasoning used to challenge Balbus's principle._

## Links

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