Pythagorean Paths — 14-year-old student solution and method for distances 1, √10, 5, √5 (ACARA v9 aligned)

Listen carefully — this is how you solve Pythagorean Paths properly. The distances are fixed in order: 1, √10, 5, √5. On a 7×7 integer grid the only allowed steps are lattice vectors whose squared lengths match the squares of those distances. We will (1) list all possible integer-step vectors for each distance, (2) try to chain them so every point stays inside the 7×7 grid, and (3) check the path exactly. No guesswork.

1. Convert distances to squared lengths.
   • 1 → 1 (so x^2 + y^2 = 1)
   • √10 → 10 (x^2 + y^2 = 10)
   • 5 → 25 (x^2 + y^2 = 25)
   • √5 → 5 (x^2 + y^2 = 5)
2. List all integer vectors (including sign and order) that meet each squared length.
   • Length 1: (±1,0) or (0,±1).
   • Length √10: components must be 1 and 3 in some order → (±1,±3) and (±3,±1).
   • Length 5: possibilities are (0,±5), (±5,0), or the 3–4 pair → (±3,±4) and (±4,±3).
   • Length √5: components 1 and 2 in some order → (±1,±2) and (±2,±1).
3. Strategy to chain vectors on a 7×7 grid (coordinates 0..6).

   You must pick a start point (x0,y0) inside the grid, then apply the ordered vectors v1, v2, v3, v4. Every intermediate point must remain within 0 ≤ x,y ≤ 6 and all five points must be distinct marked dots. If your puzzle gives a specific set of 12 marked dots, try start points among those marks and only step to other marked dots.

4. Work a concrete example (complete path) so you understand the checks.

   Choose start = (0,0) (a legal lattice point in the 7×7 grid). Use the ordered vectors below (signs chosen so all steps stay inside the grid):

   Start: P0 = (0,0)
   v1 (length 1)      = (1,0)    → P1 = (1,0)
   v2 (length √10)   = (1,3)    → P2 = (2,3)
   v3 (length 5)     = (4,3)    → P3 = (6,6)
   v4 (length √5)    = (-2,-1)  → P4 = (4,5)
       

   Now check each distance exactly (compute vector between consecutive points and squared length):

   • P0→P1 = (1,0) → squared length = 1 → distance = 1 ✔
   • P1→P2 = (1,3) → squared length = 1^2+3^2 = 10 → distance = √10 ✔
   • P2→P3 = (4,3) → squared length = 4^2+3^2 = 25 → distance = 5 ✔
   • P3→P4 = (-2,-1) → squared length = 2^2+1^2 = 5 → distance = √5 ✔

   All five points are inside the 7×7 grid (0..6 for both coordinates): (0,0), (1,0), (2,3), (6,6), (4,5). This is a legal Pythagorean Path on the integer grid.

   Important note: The puzzle you were given has 12 specific marked dots. This concrete path is valid on the full 7×7 lattice. If the 12 marked dots in the book include these five coordinates, you have solved the puzzle. If any of these five are not among the 12 marks, use the exact same method but start only at marked dots and only step to other marked dots — systematically try each starting marked dot and each allowed sign/order of the vectors until you find a chain that stays on marked dots and inside the grid.

5. Systematic searching tips (so you don't waste time):
   1. Precompute and write down all sign/order variants for each vector set (we listed them above). Keep them in a small table.
   2. If the puzzle gives marked-dot coordinates, restrict your starts to those; if not, you may pick any lattice start 0..6.
   3. For each start P0 and each choice of v1, compute P1 = P0 + v1. If P1 is off-grid or not marked, discard immediately.
   4. For each surviving P1 and each v2, compute P2 and test; continue similarly for v3 and v4. Use backtracking: discard any partial chain that leaves the grid or hits a point that isn't marked (or repeats a point if the rules forbid repeats).
   5. Use symmetry: many vectors are rotations/reflections of others. If your first few tries fail, reflect or rotate your successful partial chains before testing all variants.
6. Marking notes (how a teacher will grade this — ACARA v9 expectations):
   • Proficient/exemplary work shows complete reasoning: list of allowed integer vectors, a clear chain of vectors, coordinates of each visited dot, and exact checks of squared lengths. You must show all checks or it won’t be full marks.
   • If you must adapt to the 12 marked dots, show you tried every relevant start among the 12 and show the successful chain that stays on marked dots (or show why no such chain exists by exhaustive elimination). That demonstrates problem-solving and working mathematically.

Final takeaway: The vector chain (1,0) → (1,3) → (4,3) → (-2,-1) from start (0,0) gives the distances 1, √10, 5, √5 and stays inside the 7×7 grid. If the book’s 12 marked dots include (0,0), (1,0), (2,3), (6,6), (4,5), you have solved the bonus. If not, apply the exact same method but restrict yourself to the marked dots and backtrack systematically until you find a valid path.

Now do it: list the marked-dot coordinates from your book, pick a start among them, and run this exact process. I expect a full write-up of your trial runs and the successful chain — no lazy guessing.