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A Tangled Tale — where story and sum entwine

A lyrical reading of Lewis Carroll's mathematical tales: themes, knots, and the quiet craft of reasoning woven into fiction.

By Zacharia Maganga — an homage to the past, skeptical of glib answers, and fond of method. · September 1, 2025

Introduction — a traditional whisper

Lewis Carroll (Charles L. Dodgson) fashioned A Tangled Tale as a sequence of short stories, each concealing a puzzle. The work insists: mathematics is not a dry ledger but a human occupation, ancestral and exacting. Below I unpack the central themes and sample the knots, showing how narrative and arithmetic braid themselves together.

Themes portrayed

The union of story and mathematics

Carroll demonstrates that equations may hide in conversation — tickets, inheritances, and journeys become problems to be solved. The lesson: reason lives in common life.

Playfulness and precision

The tales are playful yet demand rigor: wit will not substitute for correct method. Carroll teases the reader but rewards those who compute carefully.

The tangled nature of problems

Problems arrive knotted and social; the mathematician's art is to disentangle, patiently and strictly.

Sample knots (short retellings)

Knot I — The ticket puzzle (Arithmetic progression)

Two travellers argue about a railway fare and the way the sums split. Beneath the repartee is an arithmetic progression: if fares change in a regular step, what is the missing fare? The formal rendering often leads to a simple linear equation; solved carefully, the tangle gives way.

// symbolic form
// Suppose fares follow: f, f+d, f+2d
// Given a relation among sums: solve for f and d
// Example linear setup: 3f + 3d = S (known) => reduce to f and d

Knot VI — Her Radiancy (Combinatorics & probability)

A courtly scene hides a counting puzzle: seating, choices, and the chances that a particular arrangement arises. Carroll invites readers to enumerate possibilities — a small prelude to modern combinatorics.

Knot IX — The serpent with corners (Geometry)

A geometric sketch wrapped in narrative: distances, angles, and a traveler’s map. The geometry is simple, classical — measured reasoning wins over fanciful leaps.

The mathematics — a short worked idea

Here is the kind of step-by-step thought Carroll demands. Suppose a knot reduces to finding integer solutions to a pair of linear relations:

Find integers x,y such that:
  2x + 3y = 17
  5x - 2y = 4

Solve: multiply and eliminate, or use matrix form.

Solving quickly (elimination): multiply first equation by 2: 4x + 6y = 34. Multiply second by 3: 15x - 6y = 12. Add: 19x = 46 → x = 46/19 (not integer). The puzzle forces us to re-check assumptions — perhaps the knot allowed rational solutions, or perhaps we misread the narrative hint. Carroll's lessons: check assumptions; the story hides constraints.

Why the form matters

Carroll's device — story that contains a problem, followed by an answer page — trains patience. The reader becomes an active detective, moving between text and calculation. The traditional outlook praises this method: reason must be earned.

Further reading & sources

1. Lewis Carroll, A Tangled Tale (1885) — the primary text and origin of the knots.
2. Collected writings and biographies of Charles L. Dodgson — for historical context and notes on Carroll's life and mathematical tastes.

Notes: this post combines literary reading with basic mathematical sketches. Where rigorous citation or archival quotes are needed, consult a scholarly edition of Carroll's works or the original 1885 publication.