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2020 AMC 10B Problem 13

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2020 AMC 10B Problem 13AMC 10 Step by Step
Name:                             Date: September 13, 2026
Coordinate Geometry · difficulty 3 of 5 · about 4 min · position P11-15

Andy the Ant lives on a coordinate plane and is currently at (−20,20)(-20, 20) facing east (that is, in the positive xx -direction). Andy moves 11 unit and then turns 90∘90^{\circ} left. From there, Andy moves 22 units (north) and then turns 90∘90^{\circ} left. He then moves 33 units (west) and again turns 90∘90^{\circ} left. Andy continues his progress, increasing his distance each time by 11 unit and always turning left. What is the location of the point which Andy makes the 20202020 left turn?

  1. A)

    (−1030,−994)(-1030, -994)

  2. B)

    (−1030,−990)(-1030, -990)

  3. C)

    (−1026,−994)(-1026, -994)

  4. D)

    (−1026,−990)(-1026, -990)

  5. E)

    (−1022,−994)(-1022, -994)

Answer

B

Key insight

Every block of four moves (east, north, west, south) shifts Andy by (-2, -2); 2020 turns is exactly 505 blocks.

Solution

Andy's kkth move has length kk, and the directions cycle east, north, west, south. The kkth left turn happens right after the kkth move, so we need his position after 20202020 moves.

Look at four consecutive moves of lengths m,m+1,m+2,m+3m, m+1, m+2, m+3 in the directions east, north, west, south. The net displacement is
(m−(m+2), (m+1)−(m+3))=(−2,−2),(m - (m+2),\ (m+1) - (m+3)) = (-2, -2),
independent of mm. For example the first four moves take Andy from (−20,20)(-20, 20) to (−22,18)(-22, 18).

Since 2020=4⋅5052020 = 4 \cdot 505, the first 20202020 moves form exactly 505505 such blocks (and the direction pattern restarts cleanly after each block). Total displacement: 505⋅(−2,−2)=(−1010,−1010)505 \cdot (-2, -2) = (-1010, -1010).

Position: (−20−1010, 20−1010)=(−1030,−990)(-20 - 1010,\ 20 - 1010) = (-1030, -990).

The answer is (B) (−1030,−990)\boxed{\textbf{(B)}\ (-1030, -990)}.

Why this works

A spiral walk with steadily increasing steps looks chaotic, but the four directions repeat, and opposite directions in the same block differ by a constant amount. Grouping the moves into full cycles turns the whole walk into a constant vector times the number of cycles. Whenever 20202020 (or any large index) is divisible by the cycle length, expect the answer to be "start plus a multiple of one block."

The trap

Counting the turn as happening before the move, so the position after 2019 or 2021 moves is used instead of 2020.

Common mistakes

  • Counting the turn as happening before the move, so the position after 2019 or 2021 moves is used instead of 2020.
  • Getting the sign of the block displacement wrong, e.g. (+2,−2)(+2, -2), by mixing up which moves are west versus east.

Techniques

Compute small cases, spot the pattern, generalize