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AMC 10 · 2020 · #13

Grade 6 rate-ratio
Archetype Lattice / Grid Pattern from Small Cases
coordinate-geometrypattern-recognitionsequences-arithmetic pattern-recognitionidentify-subproblemsphysical-representation ↑ Prerequisites: coordinate-geometrysequences-arithmetic
📏 Medium solution 💡 2 insights

Problem

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 at which Andy makes the 20202020th left turn?

Pick an answer.

(A)
(-1030, -994)
(B)
(-1030, -990)
(C)
(-1026, -994)
(D)
(-1026, -990)
(E)
(-1022, -994)
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Toolkit + CCSS Solution

Understand

Restated: Andy the Ant starts at $(-20, 20)$ facing east. He moves $1$ unit, turns $90^\circ$ 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 $2020$th left turn?

Givens: Start point $(-20, 20)$, facing east; Move lengths are $1, 2, 3, 4, \ldots$; After each move, turn $90^\circ$ left; Direction cycle: east $\to$ north $\to$ west $\to$ south $\to$ east $\to \ldots$; Answer choices: $(-1030, -994),\; (-1030, -990),\; (-1026, -994),\; (-1026, -990),\; (-1022, -994)$

Unknowns: Coordinates of the point where Andy makes his $2020$th left turn

Understand

Restated: Andy the Ant starts at $(-20, 20)$ facing east. He moves $1$ unit, turns $90^\circ$ 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 $2020$th left turn?

Givens: Start point $(-20, 20)$, facing east; Move lengths are $1, 2, 3, 4, \ldots$; After each move, turn $90^\circ$ left; Direction cycle: east $\to$ north $\to$ west $\to$ south $\to$ east $\to \ldots$; Answer choices: $(-1030, -994),\; (-1030, -990),\; (-1026, -994),\; (-1026, -990),\; (-1022, -994)$

Plan

Primary tool: #5 Look for a Pattern

Secondary: #1 Draw a Diagram, #7 Identify Subproblems, #3 Eliminate Possibilities

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.

Execute — Answer: B

#1 Draw a Diagram 5.G.A.2 Step 1
  • Draw the first few moves on grid paper.
  • From $(-20, 20)$: move $1$ east to $(-19, 20)$; turn, move $2$ north to $(-19, 22)$; turn, move $3$ west to $(-22, 22)$; turn, move $4$ south to $(-22, 18)$.
  • Four moves later, Andy is $(-2, -2)$ relative to start — the picture shows an outward spiral.
$$(-20, 20) \to (-19, 20) \to (-19, 22) \to (-22, 22) \to (-22, 18)$$

💡 Grade 5 coordinate plane: drawing the first $4$ moves reveals the spiral and the direction pattern.

#5 Look for a Pattern 4.OA.C.5 Step 2
  • Spot the repeating block.
  • Moves $\{1,2,3,4\}$ go E,N,W,S with lengths $1,2,3,4$.
  • The next block $\{5,6,7,8\}$ goes E,N,W,S with lengths $5,6,7,8$, and so on.
  • So moves $4k+1, 4k+2, 4k+3, 4k+4$ form the $(k+1)$-st E,N,W,S block.
$$\text{block } k: \;\; (\text{E } 4k{+}1,\; \text{N } 4k{+}2,\; \text{W } 4k{+}3,\; \text{S } 4k{+}4)$$

💡 Grade 4 pattern rule: every $4$ moves cycle through E, N, W, S.

#7 Identify Subproblems 6.NS.C.5 Step 3
  • Compute the $x$-change of block $k$.
  • Block $k$'s east leg adds $(4k+1)$; its west leg subtracts $(4k+3)$.
  • Net $\Delta x = (4k+1) - (4k+3) = -2$.
  • Similarly $\Delta y = (4k+2) - (4k+4) = -2$.
  • Every block of $4$ moves shifts Andy by exactly $(-2, -2)$, regardless of $k$.
$$\Delta x_{\text{block}} = -2,\quad \Delta y_{\text{block}} = -2$$

💡 Grade 6 signed numbers: east minus west and north minus south each cancel to a constant $-2$.

#7 Identify Subproblems 5.NBT.B.5 Step 4
  • Count blocks.
  • The $2020$th left turn happens at the end of the $2020$th move.
  • Since $2020 = 4 \cdot 505$, Andy completes exactly $505$ full E-N-W-S blocks.
  • Total shift: $505 \cdot (-2, -2) = (-1010, -1010)$.
$$2020 \div 4 = 505 \;\Rightarrow\; \text{shift} = 505 \cdot (-2, -2) = (-1010, -1010)$$

💡 Grade 5 multiplication: $505$ identical blocks multiply directly.

#7 Identify Subproblems 6.NS.C.6 Step 5
  • Add the shift to the start point.
  • Start $(-20, 20)$ + shift $(-1010, -1010)$ = $(-1030, -990)$.
$$(-20, 20) + (-1010, -1010) = (-1030, -990)$$

💡 Grade 6 coordinates: add the displacement components separately.

#3 Eliminate Possibilities 6.NS.C.6 Step 6
  • Match $(-1030, -990)$ to the choices: option (B).
  • Quick sanity check from the diagram: starting at $(-20, 20)$ with $y - x = 40$ and each block preserves $y - x$ (both drop by $2$), so the answer must also satisfy $y - x = 40$.
  • Only (B) has $-990 - (-1030) = 40$.
$$(-1030, -990) \;\Rightarrow\; \textbf{(B)}$$

💡 Grade 6: only one option keeps $y - x = 40$.

[1] #1 5.G.A.2 Draw the first few moves on grid paper. From $(-20, 20)$: move $1$ east to $(-19
[2] #5 4.OA.C.5 Spot the repeating block. Moves $\{1,2,3,4\}$ go E,N,W,S with lengths $1,2,3,4$.
[3] #7 6.NS.C.5 Compute the $x$-change of block $k$. Block $k$'s east leg adds $(4k+1)$; its wes
[4] #7 5.NBT.B.5 Count blocks. The $2020$th left turn happens at the end of the $2020$th move. Si
[5] #7 6.NS.C.6 Add the shift to the start point. Start $(-20, 20)$ + shift $(-1010, -1010)$ = $
[6] #3 6.NS.C.6 Match $(-1030, -990)$ to the choices: option (B). Quick sanity check from the di

Review

Reasonableness: Net effect of every $4$ moves is $(-2, -2)$, so the spiral drifts southwest at $\tfrac{1}{2}$ 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 $4$th turn (bottom-left corner of each loop), which kills four of the five choices instantly.

Alternative: Tool #6 (Guess and Check) on the answer choices. Compute $y - x$ for each: (A) $36$, (B) $40$, (C) $32$, (D) $36$, (E) $28$. Since the start $(-20, 20)$ has $y - x = 40$ and every block of $4$ moves leaves $y - x$ unchanged ($\Delta y - \Delta x = -2 - (-2) = 0$), the answer must also satisfy $y - x = 40$ — only (B). This skips most of the arithmetic.

CCSS standards used (min grade 6)

  • 4.OA.C.5 Generate a number or shape pattern following a given rule (Identifying the repeating E-N-W-S direction cycle every $4$ moves.)
  • 5.G.A.2 Represent real-world and mathematical problems by graphing points (Sketching the first few moves on the coordinate plane to see the spiral.)
  • 5.NBT.B.5 Fluently multiply multi-digit whole numbers (Multiplying $505$ blocks by the per-block shift $(-2, -2)$.)
  • 6.NS.C.5 Understand that positive and negative numbers describe quantities (Treating east/west and north/south as signed contributions that cancel to $-2$.)
  • 6.NS.C.6 Understand a rational number as a point on the number line (Adding signed displacements $(-1010, -1010)$ to the start coordinates.)

⭐ 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 $2020 = 4 \cdot 505$ moves Andy completes $505$ such blocks, shifting by $(-1010, -1010)$. From $(-20, 20)$ he lands at $\mathbf{(-1030, -990)}$, answer (B).

⭐ 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 $2020 = 4 \cdot 505$ moves Andy completes $505$ such blocks, shifting by $(-1010, -1010)$. From $(-20, 20)$ he lands at $\mathbf{(-1030, -990)}$, answer (B).

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