A Tangled Tale · Chapters
Chapters explained
Chapter companions for A Tangled Tale by Lewis Carroll.
Chapter 1
Epigraph
A brief Latin epigraph introduces the book, inviting the reader to accept it as it is.
Chapter 2
To My Pupil
A poetic dedication praising a pupil's mastery of arithmetic and encouraging them to pursue lasting fame in mathematics, even surpassing Euclid.
Chapter 3
A Tangled Tale
The chapter presents a series of mathematical puzzles and logical conundrums wrapped in whimsical narratives, featuring characters like Clara, Balbus, and Mad Mathesis as they engage in playful yet challenging problems involving trains, percentages, and displacement.
Chapter 4
Knot I: Excelsior
Two knights descend a mountain at different speeds, discussing their pace and the time of day. The younger knight struggles with arithmetic as the elder challenges him to guess the time they stood on the peak, promising to reveal the total distance traveled between three and nine o'clock.
Chapter 5
Knot II: Eligible Apartments
Hugh and Lambert search for lodgings in Little Mendip with their tutor Balbus, encountering a series of landladies offering single back rooms with cabbages. They also puzzle over a riddle from their father about a dinner party guest list involving complex family relationships.
Chapter 6
Knot III: Mad Mathesis
Clara and her aunt Mad Mathesis visit the Charing Cross Metropolitan Station and engage in two train-meeting matches. The first match involves counting trains met while traveling opposite directions around a circular route; Clara loses. In the second match, they start counting only after their trains first meet, and Clara feels optimistic about winning.
Chapter 7
Knot IV: The Dead Reckoning
Two travelers, having completed a holiday trip and shed their knightly disguises, are aboard a ship near the Equator. They observe native fishermen from Mhruxi carrying money in sacks, which are accidentally swept overboard when the jib is hoisted. The Captain attempts to compensate the fishermen but is stymied by a puzzle: the sacks were weighed in pairs, yielding five measurements that cannot be directly resolved into individual weights.
Chapter 8
Knot V: Oughts and Crosses
Clara and her aunt Mad Mathesis engage in a match of marking pictures at an art gallery, using crosses for good marks and oughts for bad marks under complex scoring conditions. Clara struggles with the rules and her choices, eventually encountering two kind old ladies who mistake her for being mad.
Chapter 9
Knot VI: Her Radiancy
Balbus and Norman arrive in Kgovjni and are taken to the palace, where they meet the Governor and are presented to Her Radiancy. The Governor explains that they must judge a competition among three scarf-makers—Lolo, Mimi, and Zuzu—who differ in rate of work, weight, and warmth. The travellers are then locked in a dungeon to perform the calculation, with the threat of bamboo beatings if they delay.
Chapter 10
Knot VII: Petty Cash
Clara struggles to remember the cost of her luncheon while her aunt Mad Mathesis insists on recording it in her petty cash tablets. They encounter two old sisters at the train station who share the same predicament, leading to a humorous and absurd situation where Clara cannot recall the amount she paid.
Chapter 11
Knot VIII: De Omnibus Rebus
The Governor escorts Balbus and Norman out of the Imperial presence, explaining the grurmstipths (omnibuses) that run to the harbour. They witness the Commander-in-Chief struggling to arrange twenty-four pigs in four sties according to Her Radiancy's peculiar rule. After parting with the Governor, the travellers miss the sea-bound omnibus and walk, leading to a discussion about when the next omnibus will overtake them.
Chapter 12
Knot IX: A Serpent with Corners
Hugh and Lambert are on the beach at Little Mendip, where Hugh experiments with buckets to test the principle of displacement. He is puzzled when the displaced water seems less than the bucket's bulk. At dinner, Hugh presents his difficulty to Balbus, who sets them a geometry problem about a flower-garden with a serpentine gravel-walk. Balbus then retreats to his room to ponder Hugh's mechanical paradox, writing a satirical essay on immersion and displacement before falling asleep.
Chapter 13
Knot X: Chelsea Buns
Clara and her aunt Mad Mathesis encounter a Chelsea bun seller while discussing a morbid arithmetic problem about wounded soldiers. Later, they join Clara's father, his three sons, and Balbus for a tense family gathering where the father announces a critical occasion involving his sons' ages and a bag of guineas, setting up a puzzle for them to solve.
Chapter 14
Appendix
Alice expresses a desire to help untangle a knot, setting a playful and curious tone for the appendix.
Chapter 15
Answers to Knot I
The chapter presents the solution to a puzzle about two travelers walking on level ground, up a hill, and back down, calculating the total distance walked and the time of reaching the hilltop. It includes a detailed solution, a list of correct and incorrect answers from various solvers, and a poetic rendition by Simple Susan and Money Spinner.
Chapter 16
Answers to Knot II
This chapter provides solutions to two problems from Knot II: the Dinner Party Problem and the Lodgings Problem. The Dinner Party Problem asks how many guests the Governor of Kgovjni must invite given a set of relationships, with the answer being one. The Lodgings Problem involves finding which of four numbered houses minimizes the sum of distances to the other three, with the answer being No. 9. The chapter includes commentary on various submitted answers, highlighting errors and praising correct solutions.
Chapter 17
§1: The Dinner Party
The Governor of Kgovjni attempts to host a small dinner party, inviting four relatives who turn out to be the same person. The chapter presents the puzzle, discusses various answers received, and lists the class of correct solvers.
Chapter 18
§2: The Lodgings
Balbus presents a mathematical problem about a square with 20 doors on each side, numbered consecutively from a corner. The task is to determine which of four numbered doors (9, 25, 52, 73) minimizes the sum of distances to the other three. The answer is No. 9. The chapter then reviews 25 submitted solutions, criticizing common errors and praising correct analytical approaches.
Chapter 19
Answers to Knot III
This chapter presents the solution to a railway meeting problem involving two travellers going opposite ways around a circular railway, with trains starting every 15 minutes in each direction. The answer for the first part is 19 trains met by each traveller; for the second part, the easterly traveller meets 12 and the westerly 8. The solution uses a least common multiple of 360 units to divide the railway and calculate meeting points. The chapter also reviews 45 submitted answers, categorizing them as right, wrong, or half-right, and includes a class list of successful solvers.
Chapter 20
Answers to Knot IV
The chapter presents the solution to a mathematical problem involving five sacks with given pairwise and triple weights, solved via arithmetic. The author then categorizes and comments on the ninety-seven answers received, praising concise solutions and critiquing guesswork or overly complex methods, while also naming and classifying competitors.
Chapter 21
Answers to Knot V
This chapter presents the solution to a puzzle about marking pictures with crosses and oughts, aiming to minimize the number of pictures marked while maximizing the total marks. The answer is 10 pictures with 29 marks, arranged in a specific pattern. The author reviews submitted answers, criticizing those that ignore the conditions, and provides a class list of solvers.
Chapter 22
Answers to Knot VI
This chapter provides solutions to two problems from Knot VI. The first problem involves A and B increasing their wealth from £1,000 to £60,000 by using zeros to transform numbers at the Bank of England. The second problem compares the merits of three scarf-makers (L, M, Z) based on rapidity, lightness, and warmth, demonstrating that the correct method is to multiply proportional marks rather than add them. The chapter also includes a class list of solvers and commentary on common errors.
Chapter 23
Answers to Knot VII
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.
Chapter 24
Answers to Correspondents
The author responds to correspondents regarding Knots II and VI, clarifying assumptions about house numbering and defending the mathematical reasoning behind the Double Rule of Three problems. He addresses criticisms of his tone, explaining that his 'trenchant expressions' were meant in jest, and distinguishes between severe language used in earnest and playful banter.
Chapter 25
Answers to Knot VIII
This chapter provides solutions to two puzzles: the Pigs Problem, which involves placing pigs in sties to satisfy a numerical condition, and the Grurmstipths Problem, an omnibus overtaking puzzle solved using relative speed and distance calculations. The chapter includes commentary on incorrect answers from correspondents.
Chapter 26
§1: The Pigs
The chapter presents a puzzle about placing twenty-four pigs in four sties such that, when going round and round, the number in each sty is nearer to ten than the number in the last. The answer given is 8, 10, 0, 6. Only two correspondents comment: Balbus notes it cannot be solved mathematically or by verbal quibble, and Nolens Volens suggests changing the direction of travel and carrying pigs in front.
Chapter 27
§2: The Grurmstipths
This chapter presents a mathematical problem involving omnibuses and a traveller, exploring the relative speeds and times of meeting and overtaking. The solution is derived using algebraic reasoning, and several submitted answers are critiqued for their errors.
Chapter 28
Answers to Knot IX
This chapter provides solutions and commentary on three problems: the Buckets Problem, Balbus' Essay Problem, and the Garden Problem. The author evaluates various submitted answers, highlighting errors and correct reasoning, and presents the final class lists for each problem.
Chapter 29
§1: The Buckets
The chapter presents a puzzle about a small bucket floating in a larger one, challenging Lardner's statement that a solid displaces an equal bulk of fluid. Various correspondents offer flawed explanations, and the author provides the correct interpretation: 'displaces' means occupying space that could be filled with water without altering surroundings. No class-list is published due to insufficient correct answers.
Chapter 30
§2: Balbus’ Essay
Balbus poses a puzzle about a solid immersed in water, where the water rises through an infinite series of decreasing distances (2 inches, 1 inch, 1/2 inch, etc.), and he concludes the water will rise without limit. The solution explains that the series never reaches 4 inches, as each term leaves a remainder equal to the last term. Three answers are received, with Tympanum, Old King Cole, and Vindex offering correct identifications of the fallacy.
Chapter 31
§3: The Garden
A mathematical problem about an oblong garden with a spiral gravel walk is posed and solved. The solution involves setting up a quadratic equation based on the area of the garden equaling the length of the walk. The chapter then reviews twelve submitted answers, pointing out errors in reasoning and highlighting correct solutions from various solvers.
Chapter 32
Answers to Knot X
This chapter presents solutions to two mathematical problems: the Chelsea Pensioners problem (finding the minimum percentage of soldiers who lost all four body parts) and the Sons' Ages problem (determining the ages of three sons given relational conditions over time). The author critiques various incorrect or incomplete answers submitted by correspondents, using playful and satirical commentary, and awards class rankings to the solvers.
Chapter 33
§1: The Chelsea Pensioners
This chapter presents a mathematical problem about the minimum percentage of Chelsea pensioners who must have lost all four body parts (eye, ear, arm, leg) given the percentages who lost each. The solution is 10%, derived by summing the percentages and distributing the total wounds. The chapter also reviews various incorrect answers submitted by competitors, critiquing their flawed reasoning.
Chapter 34
§2: Change of Day
The narrator postpones a geographical problem indefinitely, citing a lack of statistics and personal confusion, and questions their own ability to judge others when uncertain of their own standing.
Chapter 35
§3: The Sons' Ages
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.
Chapter 36
Endnotes
The chapter contains brief endnotes clarifying the meaning of the Pigeon-English term 'Maskee' and acknowledging a deceased friend as the source of the preceding essay.
Chapter 37
List of Illustrations
This chapter lists the illustrations included in the book, specifically mentioning a musical score for 'Chelsea Buns,' a genealogy diagram for a dinner party, and a diagram of a square with its doors.
