AMC 10 · 2020 · #13

Grade 6 rate-ratio

Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Coordinate Spiral Position

coordinate-geometrypattern-recognitionsequences-arithmetic pattern-recognitionidentify-subproblemsphysical-representation ↑ Prerequisites: coordinate-geometrysequences-arithmetic

📏 Medium solution 💡 2 insights

Problem

Andy the Ant starts at (-20, 20) facing east. He moves 1 unit, turns 90° left; moves 2 units, turns left; moves 3 units, turns left; and so on, always increasing the move length by 1. Where does he make his 2020th left turn?

Pick an answer.

(A)

(-1030, -994)

(B)

(-1030, -990)

(C)

(-1026, -994)

(D)

(-1026, -990)

(E)

(-1022, -994)

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

Tool #1 (Draw): sketch the first few moves of the outward spiral so you can see east/north/west/south alternation. Tool #5 (Pattern): every 4 consecutive moves form one repeating direction cycle E-N-W-S — look at the net displacement of one such block. Tool #7 (Subproblems): split the x-coordinate change and y-coordinate change into separate sums. Tool #3 (Eliminate): match the final coordinates to the five answer choices.

1STEP 1

Sketch moves 1–4 from (-20, 20): east, north, west, south — after 4 moves Andy sits at (-2, -2) relative to the start, an outward spiral.

(-20, 20) → (-19, 20) → (-19, 22) → (-22, 22) → (-22, 18)

2STEP 2

Every 4 consecutive moves repeat the cycle E, N, W, S: moves 4k+1, 4k+2, 4k+3, 4k+4 form the (k+1)-st block.

block k: (E 4k+1, N 4k+2, W 4k+3, S 4k+4)

3STEP 3

East (4k+1) minus west (4k+3) gives Δx = -2; north minus south gives Δy = -2. Every 4-move block shifts by exactly (-2, -2), whatever k is.

Δ x_block = -2, Δ y_block = -2

4STEP 4

The 2020th turn ends move 2020, and 2020 = 4 · 505, so 505 full blocks give a total shift of (-1010, -1010).

2020 ÷ 4 = 505 → shift = 505 · (-2, -2) = (-1010, -1010)

5STEP 5

Add the shift to the start: (-20, 20) + (-1010, -1010) = (-1030, -990).

(-20, 20) + (-1010, -1010) = (-1030, -990)

6STEP 6

(-1030, -990) is option (B) — the only choice with y - x = 40, the invariant preserved at every 4th corner.

(-1030, -990) → (B)

Answer

(-1030, -990)

Net effect of every 4 moves is (-2, -2), so the spiral drifts southwest at

\frac{1}{2}

unit per move on average. Over 2020 moves that's roughly 1010 units in each direction — landing Andy near (-1030, -990) is exactly the expected order of magnitude. The invariant y - x = 40 is preserved at every 4th turn (bottom-left corner of each loop), which kills four of the five choices instantly.

💡Key takeaway

This AMC 10 problem only needs Grade 6 signed coordinates you already know! Every 4 moves Andy goes east, north, west, south — and the east-west pair nets -2 in x, the north-south pair nets -2 in y, no matter how big the lengths get. With 2020 = 4 · 505 moves Andy completes 505 such blocks, shifting by (-1010, -1010). From (-20, 20) he lands at (-1030, -990), answer (B).

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