A Tangled Tale · Chapter
§2: The Lodgings explained
Balbus presents a mathematical problem about a square with 20 doors on each side, numbered consecutively from a corner.
What happens
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.
Themes in this chapter
Mathematical Reasoning
The entire chapter revolves around solving a geometric distance minimization problem and critiquing the logical reasoning of various solvers.
Competition and Achievement
The review of solutions and classification into 'right,' 'partly right,' and '0' highlights a competitive, achievement-oriented framing.
Characters to notice
- Balbus
Poses the lodgings problem and reviews submitted solutions.
- Dinah Mite
Submitted an incorrect solution, insisting lodgers keep to the pavement.
- Mad Hatter
Submitted a partly-right solution jointly with Rags and Tatters.
- Matthew Matticks
Provided a neat synthetic proof identifying No. 9 as the correct house.
- Bradshaw of the Future
Praised for a full analytical solution.
Key passages
“A Square has 20 doors on each side, which contains 21 equal parts. They are numbered all round, beginning at one corner. From which of the four, Nos. 9, 25, 52, 73, is the sum of the distances, to the other three, least?”
A square has 20 doors per side, with the sides divided into 21 equal segments. The doors are numbered consecutively around the square starting from a corner. Which of the four doors numbered 9, 25, 52, and 73 has the smallest total distance to the other three?
This is the central problem of the chapter.
“I used the words 'crossed to Number Seventy-three' for the special purpose of showing that shortcuts were possible.”
I deliberately wrote 'crossed to Number Seventy-three' to indicate that the lodgers could take diagonal paths across the square, not just walk along the pavement.
Clarifies a key assumption in the problem.