Listen carefully — we will do this properly and quickly. You’ve been given four step-lengths to connect five marked dots on a 5×5 grid in order: 1, √10, 5, √5. I can give you a guaranteed method so you solve it yourself every time, and I can produce the exact path if you send the five dot locations (a photo or their coordinates). First I’ll teach the method; then I’ll tell you exactly what I need to finish the job.
- Understand the allowed moves (convert lengths to vector differences).
On a square grid, every move corresponds to an integer vector (Δx, Δy) and the squared distance is Δx²+Δy². For this puzzle the useful moves are:
- Distance 1 → squared 1 → vector (±1,0) or (0,±1) (adjacent horizontally or vertically).
- Distance √10 → squared 10 → vectors (±1,±3) or (±3,±1).
- Distance 5 → squared 25 → vectors (±3,±4) or (±4,±3) (the classic 3–4–5 right triangle).
- Distance √5 → squared 5 → vectors (±1,±2) or (±2,±1).
Because the grid is 5×5, coordinates run from 0 to 4. So any candidate vector must keep both endpoints inside those bounds.
- Label the dots and compute pairwise squared distances.
Number the five marked dots A, B, C, D, E and record their coordinates (row, column). For every pair (for example A–B), compute Δx and Δy and then Δx²+Δy². That gives you which of the four lengths (1, 5, 10, 25) each pair could represent. Make a little table of pairs and squared distances.
- Match pairs to required steps in order (1 → 10 → 25 → 5).
We need a single continuous path visiting all five dots once, with the consecutive edge-lengths in this exact order. So you need an ordering of the five dots P0→P1→P2→P3→P4 such that:
- distance(P0,P1)² = 1
- distance(P1,P2)² = 10
- distance(P2,P3)² = 25
- distance(P3,P4)² = 5
From your table of all pairwise squared distances, mark which pairs are candidates for each required squared distance.
- Use elimination and uniqueness to prune choices.
Look for pairs that are the only candidate for a particular length — that edge is forced. Also remember the path is linear (not a loop): the two end dots each occur in only one chosen edge, interior dots appear in two chosen edges. Use degree-count reasoning: if a dot only appears in candidate pairs of the same step position, that constrains whether it can be an endpoint or interior vertex.
- Backtrack systematically.
If elimination doesn’t finish it, do a small backtrack search by trying each possible P0 from the dots that can be the first endpoint of a length-1 step. Follow every candidate P1 linked by a length-1 edge, then from P1 try each candidate P2 at length √10 from P1, and so on. This search is short because most branches will fail quickly — you only need to try at most a handful of possibilities on a 5×5 grid.
- Geometric shortcuts to speed you up.
- Length 5 requires a (3,4) or (4,3) difference. Only pairs that are 3 apart in one coordinate and 4 in the other can be step #3 — that is rare on a small grid and easy to spot.
- Length √10 is (1,3) or (3,1): also uncommon. If there is only one pair at squared distance 10, that edge position is forced and narrows the whole path.
- Length 1 edges always attach orthogonally; they are the easiest to identify visually on the grid.
- Check the path visually and algebraically.
Once you find a sequence of five distinct dots that satisfies the four distances in order, check that each coordinate difference matches the vector types listed above. Then draw the path to make sure it’s continuous and doesn’t re-use dots.
Practical next step — I need the marked-dot positions to give the exact path. Send either:
- a clear photo/screenshot of the 5×5 grid with the five dots marked, or
- the coordinates of the five dots (use row, column with rows and columns numbered 0–4, or use A–E for columns and 1–5 for rows — whichever you prefer).
Once you send the dot locations I will:
- Compute all pairwise squared distances and list candidate pairs for each required step.
- Use elimination and backtracking exactly as above to find the unique path (or tell you if multiple paths exist).
- Return the final ordered path (for example: (0,2) → (0,3) → (3,0) → (4,4) → (3,2)) and a quick proof: the vector between each consecutive pair and its length.
Now get me the coordinates or the picture. I expect precision — label the dots and give coordinates, and I will give you the exemplary ACARA-aligned solution with a direct, rigorous explanation.
If you want to practise immediately without sending the grid: pick any 5 positions on paper, follow steps 1–6 exactly, and you will either find a path or prove none exists. Then show me your table of pairwise distances and I will mark it and correct any slips.
Do that. Send the dot coordinates now.