Listen carefully. You are given the first three dots of a 5-dot path on a 7×7 grid: (6,2) → (6,3) → (5,0). The four required segment lengths are 1, √10, 5, √5 in exactly that order. Verify the first two segments immediately, then use the Pythagorean theorem to find every possible continuation on integer coordinates 0..6.
Step 1 — Verify the first two segments
- Distance from (6,2) to (6,3): dx = 0, dy = 1 → distance = √(0²+1²) = 1. Matches the first required length (1).
- Distance from (6,3) to (5,0): dx = −1, dy = −3 → distance = √(1+9) = √10. Matches the second required length (√10).
So the first two are correct. Good. Now we must choose two more points P4 and P5 so that:
- Distance P3=(5,0) to P4 = 5 (so dx²+dy² = 25).
- Distance P4 to P5 = √5 (so dx²+dy² = 5).
Step 2 — Find integer-coordinate points at distance 5 from (5,0)
All integer solutions of dx²+dy²=25 are (±5,0), (±4,±3), (±3,±4), (0,±5). Add these to (5,0) and keep only points inside the 0–6 grid:
- (5−5,0) = (0,0)
- (5−4,0+3) = (1,3)
- (5−3,0+4) = (2,4)
- (5,0+5) = (5,5)
So the possible P4 choices are: (0,0), (1,3), (2,4), (5,5).
Step 3 — For each candidate P4, find integer points P5 at distance √5 (dx²+dy²=5)
All integer solutions of dx²+dy²=5 are (±1,±2) and (±2,±1). Add these offsets to each P4 and keep points inside the 0–6 grid.
- P4 = (0,0): possible P5 are (1,2) or (2,1).
- P4 = (1,3): add offsets → possible P5 (within grid): (2,5), (2,1), (0,5), (0,1), (3,4), (3,2).
- P4 = (2,4): possible P5 (within grid): (3,6), (3,2), (1,6), (1,2), (4,5), (4,3), (0,5), (0,3).
- P4 = (5,5): possible P5 (within grid): (6,3), (4,3), (3,6), (3,4). Note: (6,3) is already used earlier in the path, so, if your rule forbids reusing a dot, exclude (6,3).
Step 4 — Write the full valid paths (grid-only constraint)
Each combination above yields a numeric-valid path (all coordinates integers inside the 7×7 grid) that satisfies segment lengths 1, √10, 5, √5. Examples (not exhaustive) of valid completed paths are:
- (6,2) → (6,3) → (5,0) → (0,0) → (1,2)
- (6,2) → (6,3) → (5,0) → (0,0) → (2,1)
- (6,2) → (6,3) → (5,0) → (1,3) → (2,5)
- (6,2) → (6,3) → (5,0) → (1,3) → (3,2)
- (6,2) → (6,3) → (5,0) → (2,4) → (4,3)
- (6,2) → (6,3) → (5,0) → (2,4) → (3,2)
- (6,2) → (6,3) → (5,0) → (5,5) → (4,3)
Each of those paths satisfies the required distances. For each path you should check every segment by computing dx and dy and confirming dx²+dy² equals the required square (1, 10, 25, 5 respectively).
Important note about uniqueness: The Beast Academy puzzle you referenced restricts you to 5 of the 12 marked dots. I do not have the exact positions of those 12 marked dots here. On the full integer grid there are multiple continuations (listed above). If the marked-dot pattern from the book excludes some of those coordinates, that will leave a unique path. To finish the puzzle exactly as in the book, pick only P4 and P5 that are among the 12 marked dots shown in the book’s diagram.
How to present your working (teacher expectations — strict, precise)
- Write the sequence of coordinates clearly, one per arrow.
- Under each arrow show dx and dy and compute dx²+dy² to justify the length (show the square root step for clarity).
- State any rule you used (for example: no repeated dots if that is a rule of your specific puzzle) and eliminate any candidate that breaks the rule.
- Box the final path and state why it is unique (or, if not unique given only grid constraints, explain which marked-dot constraint from the book makes it unique).
Teacher marking (Amy Chua cadence — high standards)
- Proficient (meets expectations): Student supplies a full coordinate path, checks each length by squaring dx and dy, and picks P4/P5 that are on the provided marked-dot set. Work is neat and complete. Minor arithmetic slips tolerated if reasoning is clear.
- Exemplary (exceeds expectations): Student gives the unique book path (using the book’s marked-dot map), shows all algebra (dx, dy, dx²+dy²) neatly, explains why other grid continuations are invalid because those points are not marked, and comments on why the Pythagorean theorem guarantees the distances are correct. Presentation is precise and efficient.
- Unacceptable: Student guesses points without checking dx²+dy² or repeats a dot when the rules forbid it, and presents no justification. Redo immediately until all steps are written and justified.
ACARA v9 alignment: This task practices coordinate geometry and the Pythagorean theorem—interpreting coordinates, calculating distances using dx and dy, and reasoning about integer solutions of dx²+dy² equals a constant. Those are directly aligned with the curriculum expectations for procedural fluency and problem solving.
If you paste the book’s marked-dot diagram here, I will immediately eliminate the impossible continuations and give you the one correct book answer, with the exact working you must write to earn exemplary marks. Now go — show me clean calculations for one candidate path and I will tell you whether you’ve earned full credit.