Andy the Ant lives on a coordinate plane and is currently at (−20,20) facing east (that is, in the positive x-direction). Andy moves 1 unit and then turns 90∘ left. From there, Andy moves 2 units (north) and then turns 90∘ left. He then moves 3 units (west) and again turns 90∘ left. Andy continues his progress, increasing his distance each time by 1 unit and always turning left. What is the location of the point at which Andy makes the 2020th left turn?
Try it yourself first — the explanation is most useful after you’ve attempted it.
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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: $(-1030, -994),\; (-1030, -990),\; (-1026, -994),\; (-1026, -990),\; (-1022, -994)$
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.2Step 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.
💡 Grade 6 coordinates: add the displacement components separately.
#3 Eliminate Possibilities 6.NS.C.6Step 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.2Draw the first few moves on grid paper. From $(-20, 20)$: move $1$ east to $(-19
[2]
#5 4.OA.C.5Spot the repeating block. Moves $\{1,2,3,4\}$ go E,N,W,S with lengths $1,2,3,4$.
[3]
#7 6.NS.C.5Compute the $x$-change of block $k$. Block $k$'s east leg adds $(4k+1)$; its wes
[4]
#7 5.NBT.B.5Count blocks. The $2020$th left turn happens at the end of the $2020$th move. Si
[5]
#7 6.NS.C.6Add the shift to the start point. Start $(-20, 20)$ + shift $(-1010, -1010)$ = $
[6]
#3 6.NS.C.6Match $(-1030, -990)$ 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).
