Pythagorean Paths — Beast Academy 5D Practice (Age 15): solve sequence 2,2,1,10,5,5,8,1,2,2,2

Step-by-step solution and ACARA v9–aligned teacher feedback (Nigella Lawson cadence) for the Beast Academy 5D Pythagorean Paths bonus problem. Clear strategy, a working lattice-path answer (coordinates), and teaching tips for a 15-year-old student.

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Quick idea

In Pythagorean Paths puzzles the numbers given are the sums of squares of the horizontal and vertical steps: each listed number = dx^2 + dy^2. So 1 means a step of (1,0) or (0,1), 2 means a diagonal (1,1), 5 means a (1,2) or (2,1) step, 8 means (2,2), and 10 means (1,3) or (3,1). We must pick a sequence of lattice-step vectors whose squared lengths match the list, and place them end-to-end so all visited dots are distinct.

The given sequence

2, 2, 1, 10, 5, 5, 8, 1, 2, 2, 2

One valid lattice path (start at (0,0))

  1. Step 1 (length 2): vector (1,1) — new point (1,1). Check: 1^2+1^2 = 2.
  2. Step 2 (length 2): vector (1,-1) — new point (2,0). Check: 1^2+(-1)^2 = 2.
  3. Step 3 (length 1): vector (1,0) — new point (3,0). Check: 1^2+0^2 = 1.
  4. Step 4 (length 10): vector (3,1) — new point (6,1). Check: 3^2+1^2 = 10.
  5. Step 5 (length 5): vector (2,1) — new point (8,2). Check: 2^2+1^2 = 5.
  6. Step 6 (length 5): vector (1,2) — new point (9,4). Check: 1^2+2^2 = 5.
  7. Step 7 (length 8): vector (2,2) — new point (11,6). Check: 2^2+2^2 = 8.
  8. Step 8 (length 1): vector (1,0) — new point (12,6). Check: 1^2+0^2 = 1.
  9. Step 9 (length 2): vector (1,1) — new point (13,7). Check: 1^2+1^2 = 2.
  10. Step 10 (length 2): vector (1,-1) — new point (14,6). Check: 1^2+(-1)^2 = 2.
  11. Step 11 (length 2): vector (1,1) — new point (15,7). Check: 1^2+1^2 = 2.

Full list of coordinates visited

(starting at (0,0)):

(0,0) → (1,1) → (2,0) → (3,0) → (6,1) → (8,2) → (9,4) → (11,6) → (12,6) → (13,7) → (14,6) → (15,7)

All 12 points are distinct and each step’s squared length matches the number in the sequence, so this is a correct Pythagorean path for the sequence given.

How I chose the moves — a short strategy you can reuse

  • Translate the numbers to possible step vectors: 1 → (1,0); 2 → (1,1); 5 → (1,2) or (2,1); 8 → (2,2); 10 → (1,3) or (3,1).
  • Choose small, varied vectors so the path doesn’t self-intersect — alternating diagonals with horizontal/vertical steps helps.
  • After each choice, check the new point isn’t already used. If it is, backtrack and pick the alternative orientation (flip sign or swap dx and dy).
  • Keep symmetry and compactness in mind when solving on a fixed-size puzzle board; the example above is one of many lattice-realizations of the sequence.

ACARA v9 aligned teacher feedback — in a Nigella Lawson cadence

Darling, what you have here is a geometry problem that asks you to taste the sweetness of Pythagoras — to measure the subtle, clean flavours of horizontal and vertical change and combine them so their squares sing. The mathematics is simple and elegant: dx squared plus dy squared gives you the dish’s total length. You have practised skilful selection of step vectors, examined the plate for repeats (no point may be visited twice), and adjusted your recipe when a choice threatened to overpower the composition.

Curriculum alignment: This exercise sits squarely in the Measurement and Geometry strand of the Australian Curriculum (v9). It develops the capacity to use Pythagoras’ theorem and coordinate geometry to determine distances between points, and to reason systematically when constructing paths and checking for self-intersections. Such tasks support the achievement standards for upper secondary middle years: applying geometric reasoning, using algebraic and spatial strategies, and persisting with multi-step problem solving.

What to praise: precision in translating numbers to (dx,dy) steps; careful checking that dx^2 + dy^2 equals the listed number; persistence through backtracking when a partial path blocks later moves.

Next steps to stretch gently and sweetly: try to draw the same path on a bounded grid (for example 8×8) and see whether your moves need reflecting or rotating to fit. Challenge yourself to find all distinct non-overlapping paths that realise the same sequence — count symmetries and consider reflections as different or the same, whichever you prefer.

A final whisper of advice: when a sequence has many small steps in a row (several 2s and 1s), think diagonal–horizontal–diagonal to weave through the lattice without revisiting points. When a large squared length appears (like 10), place it where it opens space rather than where it closes it. And as ever, take your time — savour the choices, and you’ll find the path as delicious as the solution.

If you want, I can also try to draw the path on a bounded grid (say 8×8 or 10×10) and adapt these coordinates to that board so it matches the puzzle layout in the book — tell me which grid size you need.

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