IN THE MATTER OF: The Pythagorean Path Puzzle — Report and Findings
Presiding: Homeschool Mathematics Review Panel
Student: Age 13 — Delivered an exemplary solution.
Statement of the Case
The task: connect five pre-marked points on a 7×7 grid to make one continuous path whose consecutive segment lengths, in order, are √10, 5, √10, √10. The order of the marked points was not given; the pupil located and connected them in a single continuous path that satisfies the length requirement.
Findings of Fact (Plain-English)
- The pupil recognized that a distance of √10 arises when the change in x and y are 1 and 3 (in either order), because 1^2 + 3^2 = 10.
- The pupil recognized that a distance of 5 arises when the change in x and y are 3 and 4 (in either order), because 3^2 + 4^2 = 25.
- She used the coordinate plane and computed Δx and Δy for each connected pair, then checked Δx^2 + Δy^2 to confirm the squared distance matched 10 or 25 as required.
- When the order of points was not given, she reasoned through permutations, eliminated impossible links, and found one continuous order that satisfied the required distances.
Demonstration of Method (Step-by-step Verification)
The following is a clear sample path on a 7×7 grid (coordinates from 0 to 6) that matches the required lengths. The pupil produced a path of the same nature; this example mirrors the verification method used.
Chosen points (example): A(0,0) → B(1,3) → C(5,0) → D(6,3) → E(3,4).
Compute each segment:
- Segment AB: from (0,0) to (1,3)
Δx = 1, Δy = 3
Δx^2 + Δy^2 = 1^2 + 3^2 = 1 + 9 = 10 → length = √10 - Segment BC: from (1,3) to (5,0)
Δx = 4, Δy = -3
Δx^2 + Δy^2 = 4^2 + (-3)^2 = 16 + 9 = 25 → length = 5 - Segment CD: from (5,0) to (6,3)
Δx = 1, Δy = 3
Δx^2 + Δy^2 = 1 + 9 = 10 → length = √10 - Segment DE: from (6,3) to (3,4)
Δx = -3, Δy = 1
Δx^2 + Δy^2 = (-3)^2 + 1^2 = 9 + 1 = 10 → length = √10
Each squared distance matches the required 10 or 25 in the given order, so the path is valid. This direct computation — Δx, Δy, then Δx^2 + Δy^2 — is the Pythagorean verification used by the pupil for every segment.
Reasoning and Problem-Solving Notes
- The student showed systematic thinking by cataloging allowed moves for √10 (1,3) permutations and for 5 (3,4) permutations, then using those allowed steps to build candidate chains.
- When the sequence of pre-marked points was unknown, she used logical elimination: if a given point cannot be reached from any remaining candidate by an allowed move, that order is discarded — reducing the search efficiently.
- Work shows spatial reasoning and procedural fluency: she moved between algebra (squared sums) and geometry (grid moves) smoothly, and checked arithmetic carefully.
Judgment (Conclusion)
Exemplary. The pupil produced a correct, single continuous path on the 7×7 grid whose consecutive segment lengths are √10, 5, √10, √10. Her reasoning was explicit and well-documented; computations were accurate; strategies for permutation and elimination were efficient and appropriate for her level.
Recommendations and Enrichment
- Generalize: Ask how many distinct 4-step sequences of moves (with the same length pattern) exist on a 7×7 grid. This leads to combinatorics and counting strategies.
- Graph-theory view: Represent grid points as vertices and allowed Pythagorean moves as edges. Find all Hamiltonian paths of a given pattern — a neat step toward discrete math.
- Algebraic challenge: Characterize all integer solutions to x^2 + y^2 = 10 and x^2 + y^2 = 25 (small Diophantine practice), and then use those to generate families of paths on larger grids.
- Creative application: Design puzzles with longer sequences or require closed loops (polygons) whose sides are drawn from a set of Pythagorean lengths.
Closing Remark (With Due Gravity and a Touch of Cadence)
The pupil did not merely find a path. She argued it. She verified it. She reflected on it. In short, she prosecuted mathematics with rigor and charm — and the court of mathematical reasoning finds her work exemplary.
Signed,
Homeschool Mathematics Review Panel