AMC 10 · 2015 · #24

Grade 6 arithmetic
Archetype Lattice / Grid Pattern from Small Cases └ Sub-archetype Coordinate Spiral Position
pattern-recognitioncoordinate-geometryperfect-squaresspatial-visualization work-backwards ↑ Prerequisites: pattern-recognition
📏 Medium solution 💡 4 insights
Problem
An ant starts at (0,0) facing east and walks one unit at a time, building a counterclockwise square spiral. The rule: after each step, it turns left if the spot to its left is empty; otherwise it keeps going straight. Find the point p₂015 where it lands on step 2015.

Pick an answer.

(A)
(-22,-13)
(B)
(-13,-22)
(C)
(-13,22)
(D)
(13,-22)
(E)
(22,-13)

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

Walking out 2015 steps by hand is impossible, so the goal is to find where the spiral repeats. Tool #1 (Draw a Diagram) gets the spiral on paper so its shape is visible, and tool #5 (Look for a Pattern) is the engine: the spiral keeps hitting the same kind of landmark, the bottom-right corner of an odd-sided square. Tool #9 (Solve an Easier Related Problem) turns the giant number 2015 into the nearby clean square 45²=2025, a corner we can name exactly. Then tool #11 (Work Backwards) steps a few units back from that corner to reach step 2015.

1STEP 1

Trace the spiral and watch the corners

The path is a counterclockwise square spiral; the ant hits the bottom-right corner (1,-1) at step 8 and (2,-2) at step 24.

p₀=(0,0), p₈=(1,-1), p₂4=(2,-2)
2STEP 2

Write the corner pattern as a rule

Corners follow a rule: the odd square of side 2k+1 has bottom-right corner (k,-k), reached at step (2k+1)²-1, matching k=1→8 and k=2→24.

p_(2k+1)²-1=(k,-k)
3STEP 3

Find the corner nearest step 2015

The odd square just above 2015 is 45²=2025 with 45=2·22+1, so k=22 puts the ant at corner (22,-22) on step 2024 — just past 2015.

45²=2025→ p₂024=(22,-22)
4STEP 4

See which way the ant enters that corner

The ant slides into each bottom-right corner heading east, so just before (22,-22) it walks along y=-22 with x rising by one each step.

p₂024=(22,-22), arriving east: …,(20,-22),(21,-22),(22,-22)
5STEP 5

Step back to reach 2015

Step 2015 is 2024-2015=9 steps before the corner; backing up 9 eastward steps drops x by 9, giving x=22-9=13 and (13,-22), choice (D).

2024-2015=9→ p₂015=(22-9, -22)=(13,-22)
Answer
(13,-22)
The point (13,-22) sits below and right of the origin, which fits the bottom edge of the spiral the ant is on near step 2015. It also lands strictly inside the 45×45 square (corner (22,-22)) but past the 43×43 square (corner (21,-21), step 1848), so 1848 ＜ 2015 ＜ 2024 places it correctly on the final stretch. The signs (positive x, negative y) match a point reached while sliding east across the bottom, ruling out the choices with the signs flipped or the numbers swapped.
💡Key takeaway

Find the nearest clean landmark in a repeating pattern, name it exactly, then walk a few steps back to your target.

  • Trace the spiral and watch the corners
  • Write the corner pattern as a rule
  • Find the corner nearest step 2015
  • See which way the ant enters that corner
  • Step back to reach 2015

More like this

Same archetype — closest grade level first.