Listen up — be exact, be neat, and show every calculation. I will not accept hand‑waving. First: I need to be absolutely clear — the puzzle only allows you to use the 12 marked dots on the 7×7 grid. You haven’t sent the picture or a list of which 12 points are marked. I will do the full logical work and give a correct candidate path inside the grid; you must check whether those five points are actually among the marked dots. If they are, you are done. If one is not marked, repeat the same procedure I show below until you find a sequence that uses only the marked dots.
Quick plan: From the start point P1 = (6,2), find P2 at distance 1, then P3 at distance √10 from P2, then P4 at distance 5 from P3, then P5 at distance √5 from P4. All coordinates must lie in the integer grid 1..7 for both x and y. Show each step and verify by squaring distances so there is no numerical rounding.
Vector distance facts to use (integer differences):
- Distance 1: dx^2+dy^2 = 1 → (±1,0) or (0,±1).
- Distance √10: dx^2+dy^2 = 10 → (±1,±3) or (±3,±1).
- Distance 5: dx^2+dy^2 = 25 → (±3,±4), (±4,±3), (±5,0), (0,±5).
- Distance √5: dx^2+dy^2 = 5 → (±1,±2) or (±2,±1).
Step‑by‑step search (systematic method):
- List P2 candidates at distance 1 from P1 = (6,2): (5,2), (7,2), (6,1), (6,3).
- For each P2 candidate, compute all P3 positions that are a √10 move from that P2 (use the dx,dy patterns above), discarding any points outside the 1..7 grid.
- For each P3 candidate, compute all P4 positions that are a 5 move from that P3 (use the patterns for distance 5), again discarding outside‑grid points.
- For each P4, compute P5 candidates that are √5 away; keep only those inside the grid. Finally check whether every point of the 5‑point sequence is actually one of the 12 marked dots on the puzzle grid. If yes, you have a valid path; if not, try other branches.
Exemplar solution (one valid sequence inside the 7×7 grid):
- P1 = (6,2) (given)
- P2 = (5,2). Check: distance from P1: Δx = -1, Δy = 0 → 1^2+0^2 = 1 → length 1 ✅
- P3 = (6,5). From P2: Δx = +1, Δy = +3 → 1^2+3^2 = 10 → length √10 ✅
- P4 = (3,1). From P3: Δx = -3, Δy = -4 → 3^2+4^2 = 25 → length 5 ✅
- P5 = (4,3). From P4: Δx = +1, Δy = +2 → 1^2+2^2 = 5 → length √5 ✅
So the full path of five distinct lattice points is (6,2) → (5,2) → (6,5) → (3,1) → (4,3). Every intermediate calculation above is exact (I squared the differences and matched them to 1, 10, 25, 5). If those five points are among the 12 marked dots on your puzzle, you have solved it.
What to write on your final answer (teacher‑gradeable, ACARA‑aligned):
- State the start: P1 = (6,2).
- List the five points in order, with each vector difference shown and the squared distance computed. For example:
P1→P2: (6,2)→(5,2): Δ = (−1,0), squared distance = 1 → length 1.
P2→P3: (5,2)→(6,5): Δ = (1,3), squared distance = 10 → length √10.
P3→P4: (6,5)→(3,1): Δ = (−3,−4), squared distance = 25 → length 5.
P4→P5: (3,1)→(4,3): Δ = (1,2), squared distance = 5 → length √5. - Conclude: All lengths match the required sequence. If the five points are marked, the path is valid.
If this candidate fails because one of those points is not marked: Don’t panic. Use the same systematic branch search I described (try the other P2 choices: (7,2), (6,1), (6,3) and follow the √10→5→√5 branching). Keep a neat table of candidates and eliminate any point outside the grid or not marked. That is how you guarantee you find the solution or prove none exists with the given marked set.
Final strict directive (Tiger‑mother style): You will now do two things and show me the work: (1) check whether the five points above are among the twelve marked dots on the Beast Academy page; (2) if yes, write up the verification exactly as I demonstrated and upload a photo or typed copy. If no, send me the image of the marked grid (or a list of the 12 marked coordinates) and I will finish the search and give you the unique correct answer. No excuses — show every squared distance and every coordinate.
Ready? Send the grid image or confirm the five points are marked. Then we finish and I will mark it exemplary.