Pythagorean Paths — Age 14 solution help for Beast Academy Practice 5D Chapter 11 (start at x7,y3) — distances 1, √10, 5, √5

Step‑by‑step, ACARA‑aligned explanation for a Beast Academy Pythagorean Path (8×8 grid). Method, reasoning, and one explicit valid path starting at (7,3). If your puzzle uses a fixed set of 12 marked dots, send those coordinates or an image so I can verify the correct marked‑dot solution.

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Listen carefully — we will do this properly and precisely. A Pythagorean Path of five points P0→P1→P2→P3→P4 must satisfy the four step distances in order: 1, √10, 5, √5. You start at P0 = (7,3). I will show you the method and then give a clean, checked path that satisfies the distance constraints on a standard 8×8 integer grid.

  1. List the allowed displacement patterns for each required distance.
    • Distance 1: displacements (±1,0) or (0,±1).
    • Distance √10: squared distance 10 → integer displacements (±1,±3) or (±3,±1).
    • Distance 5: squared distance 25 → displacements (0,±5),(±5,0),(±3,±4),(±4,±3).
    • Distance √5: squared distance 5 → displacements (±1,±2) or (±2,±1).
  2. Apply the displacements step by step from the start (7,3).

    From (7,3) the distance‑1 neighbors (the only possible P1 candidates) are: (8,3), (6,3), (7,4), (7,2). For each candidate, apply the √10 displacement patterns to get P2 candidates, then apply the distance‑5 patterns to get P3 candidates, and finally the √5 patterns to get P4 candidates. At each stage enforce:

    • All coordinates must remain inside the 8×8 grid: 1 ≤ x,y ≤ 8.
    • For the Beast Academy puzzle you must also choose only from the 12 marked dots — so prune any candidate not on the marked list.
    • Usually the path must use distinct marked dots — avoid revisiting a dot (unless the puzzle text allows revisits, which is rare).
  3. Concrete, fully checked example path (valid on a full 8×8 integer grid):

    I found a clean, distinct 5‑point path that starts at (7,3) and meets the four distances in order. Check each step yourself:

    (7,3) → (8,3) → (7,6) → (7,1) → (6,3)

    Verify distances:

    • (7,3) to (8,3): dx=1, dy=0 ⇒ distance = 1 ✔
    • (8,3) to (7,6): dx=−1, dy=3 ⇒ dx²+dy²=1+9=10 ⇒ distance = √10 ✔
    • (7,6) to (7,1): dx=0, dy=−5 ⇒ distance = 5 ✔
    • (7,1) to (6,3): dx=−1, dy=2 ⇒ dx²+dy²=1+4=5 ⇒ distance = √5 ✔

    All coordinates are inside the 1..8 range, all five points are distinct, and the four step lengths match the required sequence (1, √10, 5, √5). If your Beast Academy sheet has these five marked dots among its twelve, this is a valid solution.

  4. If that path doesn’t match the marked dots on your specific puzzle:

    I cannot be certain which of the 12 dots are allowed because you haven’t sent the diagram or the list of the marked coordinates. Do one of these:

    1. Compare the five coordinates above to the 12 marked dots. If all five are marked, you are done.
    2. If any of those five are not marked, give me the 12 marked coordinates (or a clear photo of the grid). I will run the same stepwise search but restrict candidates to the marked set and return the unique (or all) valid path(s).

Final, strict advice: do the arithmetic — write down the displacement patterns, prune aggressively, and test each chain. Send the marked‑dot list or a picture now if you want the exact Beast Academy solution checked and marked to a high standard. I expect correct, neat work; show your steps and I will correct any errors.

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