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AMC 8 · 2024 · #17

Grade 3 geometry-2d
Archetype Casework by Position / Vertex Type
systematic-enumerationcombinations-basic caseworksystematic-enumeration ↑ Prerequisites: multi-digit-arithmeticmental-arithmetic
📏 Long solution 💡 4 insights 📊 Diagram
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Problem

A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a 33 x 33 grid attacks all 88 other squares, as shown below. Suppose a white king and a black king are placed on different squares of a 33 x 33 grid so that they do not attack each other (in other words, not right next to each other). In how many ways can this be done?

Pick an answer.

(A)
20
(B)
24
(C)
27
(D)
28
(E)
32
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Toolkit + CCSS Solution

Understand

Restated: Place a distinct white king and black king on different squares of a $3 \times 3$ grid so that they do not attack each other (no horizontal, vertical, or diagonal neighbor). Count the number of such ordered placements.

Givens: Grid size: $3 \times 3$ (total $9$ squares); A king attacks every square one step away horizontally, vertically, or diagonally; The kings are different colors, so distinguishable (white-A·black-B differs from white-B·black-A); The two kings must occupy different squares and must not attack each other; Answer choices: (A) 20, (B) 24, (C) 27, (D) 28, (E) 32

Unknowns: The number of valid ordered (white king, black king) placements

Understand

Restated: Place a distinct white king and black king on different squares of a $3 \times 3$ grid so that they do not attack each other (no horizontal, vertical, or diagonal neighbor). Count the number of such ordered placements.

Givens: Grid size: $3 \times 3$ (total $9$ squares); A king attacks every square one step away horizontally, vertically, or diagonally; The kings are different colors, so distinguishable (white-A·black-B differs from white-B·black-A); The two kings must occupy different squares and must not attack each other; Answer choices: (A) 20, (B) 24, (C) 27, (D) 28, (E) 32

Plan

Primary tool: #1 Draw a Diagram

Secondary: #7 Identify Subproblems, #2 Make a Systematic List, #3 Eliminate Possibilities

Everything happens on a tiny $3 \times 3$ picture, so first use Tool #1 — **Draw the grid** and look directly at where the white king can sit. The diagram makes clear that the 9 squares naturally split into three types: **corner**, **edge**, and **center**, and the king attacks a different number of squares from each type. So Tool #7 — **Identify subproblems** — splits the big count into three smaller cases (white king on a corner / on an edge / in the center). Inside each case, Tool #2 — a **systematic listing** via the multiplication rule — gives the number of legal black-king squares. Finally Tool #3 — **eliminate possibilities** — checks our total against the answer choices. Since this is multiple choice on a tiny grid, simple visual + counting tools suffice; no algebra is needed.

Execute — Answer: E

#1 Draw a Diagram K.G.A.1 Step 1
  • Draw the $3 \times 3$ grid and color the 9 squares by position: **4 corner squares** at the four corners, **4 edge squares** in the middle of each side, and **1 center square** in the middle.
  • Together $4 + 4 + 1 = 9$, so every square is classified exactly once.
$$\text{corners}=4,\ \text{edges}=4,\ \text{center}=1,\ \text{total}=9$$

💡 Sorting squares using position words like "corner", "side", "middle" is exactly the Kindergarten position-vocabulary skill.

#1 Draw a Diagram K.G.A.1 Step 2
  • From the picture, count how many squares a king attacks from each type.
  • A king on a **corner** (e.g.
  • top-left) attacks the squares to the right, below, and diagonally — **3 squares**.
  • A king on an **edge** (e.g.
  • top-middle) attacks left, right, below-left, below, below-right — **5 squares**.
  • A king on the **center** attacks **all 8 surrounding squares**.
  • The three numbers $3, 5, 8$ are the key ingredients per case.
$$\text{corner attacks}=3,\ \text{edge attacks}=5,\ \text{center attacks}=8$$

💡 Counting the horizontal, vertical, and diagonal neighbors of one square is still a Kindergarten-level position/adjacency activity.

#7 Identify Subproblems 3.OA.A.1 Step 3
  • Break the big problem ("total number of placements") into **3 subproblems** by where the white king sits: ① corner, ② edge, ③ center.
  • In each subproblem the number of squares the black king may pick is $9 - 1 - (\text{attacks})$ (subtracting 1 for the white king's own square).
  • By the multiplication rule, the count for that case is $(\text{white-king squares}) \times (\text{black-king squares})$.
$$\text{count per case} = (\text{white-king squares}) \times (9 - 1 - \text{attacks})$$

💡 Reading "$A$ choices, each with $B$ choices, gives $A \times B$ total" as a product of equal-sized groups is the heart of Grade 3 multiplication meaning.

#2 Make a Systematic List 3.OA.C.7 Step 4
  • **Case ①: white king on a corner.** White-king squares: $4$.
  • The white king blocks its own square plus $3$ attacked squares, $4$ total, so the black king picks from $9 - 4 = 5$ squares.
  • Count for this case: $4 \times 5 = 20$.
$$4 \times (9 - 1 - 3) = 4 \times 5 = 20$$

💡 $4 \times 5 = 20$ is fluently within the Grade 3 multiplication-within-100 standard.

#2 Make a Systematic List 3.OA.C.7 Step 5
  • **Case ②: white king on an edge.** White-king squares: $4$.
  • The white king blocks $1 + 5 = 6$ squares, so the black king picks from $9 - 6 = 3$ squares.
  • Count for this case: $4 \times 3 = 12$.
$$4 \times (9 - 1 - 5) = 4 \times 3 = 12$$

💡 $4 \times 3 = 12$ is a basic Grade 3 multiplication fact.

#2 Make a Systematic List 3.OA.C.7 Step 6
  • **Case ③: white king in the center.** White-king squares: $1$.
  • The center king blocks $1 + 8 = 9$ squares — **every square** — so the black king has $9 - 9 = 0$ squares left.
  • Count for this case: $1 \times 0 = 0$.
  • Once the white king is in the middle, the black king has nowhere legal to go.
$$1 \times (9 - 1 - 8) = 1 \times 0 = 0$$

💡 "Anything times $0$ is $0$" is a basic Grade 3 multiplication property.

#7 Identify Subproblems 2.NBT.B.5 Step 7
  • Add the three subproblem answers to get the total: $20 + 12 + 0 = 32$.
  • The three white-king positions (corner / edge / center) partition the 9 squares with no overlap, so the addition rule applies cleanly.
$$20 + 12 + 0 = 32$$

💡 Adding two-digit numbers like $20 + 12 + 0$ is well within Grade 2 addition-within-100 fluency.

#3 Eliminate Possibilities 1.NBT.B.3 Step 8
  • Match $32$ against the answer choices.
  • (A) $20$ is just case ① alone, (B) $24$ and (C) $27$ never appear as one of our partial sums, and (D) $28$ is a near-miss trap that is $4$ short of the true total.
  • Our value $32$ is exactly choice **(E)**.
$$32 \;\Rightarrow\; \textbf{(E)}$$

💡 Picking the matching two-digit number from a short list is a Grade 1 two-digit comparison skill.

[1] #1 K.G.A.1 Draw the $3 \times 3$ grid and color the 9 squares by position: **4 corner squar
[2] #1 K.G.A.1 From the picture, count how many squares a king attacks from each type. A king o
[3] #7 3.OA.A.1 Break the big problem ("total number of placements") into **3 subproblems** by w
[4] #2 3.OA.C.7 **Case ①: white king on a corner.** White-king squares: $4$. The white king bloc
[5] #2 3.OA.C.7 **Case ②: white king on an edge.** White-king squares: $4$. The white king block
[6] #2 3.OA.C.7 **Case ③: white king in the center.** White-king squares: $1$. The center king b
[7] #7 2.NBT.B.5 Add the three subproblem answers to get the total: $20 + 12 + 0 = 32$. The three
[8] #3 1.NBT.B.3 Match $32$ against the answer choices. (A) $20$ is just case ① alone, (B) $24$ a

Review

Reasonableness: Sanity-check with the complement. Total ordered placements on different squares: $9 \times 8 = 72$. Adjacent pairs on a $3 \times 3$ grid: $3 \cdot 2 + 2 \cdot 3 = 12$ horizontal/vertical and $2 \cdot 2 \cdot 2 = 8$ diagonal, $20$ pairs in all. Since the two kings are distinguishable, attacking placements count as $20 \times 2 = 40$, and non-attacking ones are $72 - 40 = 32$ — matching our casework answer exactly. It also fits intuition that the center contributes $0$ and the corner case is the largest at $20$.

Alternative: An alternative is Tool #16 (Change Focus / Complement): subtract the $40$ attacking ordered placements from the $72$ total (used in the sanity check above). It produces the same answer $32$. For young students, though, "split into cases and add" is more concrete, which is why the main solution uses #7 plus #2.

CCSS standards used (min grade 3)

  • K.G.A.1 Describe positions of objects using above, below, beside, in front of (Classifying the 9 squares of the $3 \times 3$ grid into "corner / edge / center" and counting each square's horizontal, vertical, and diagonal neighbors.)
  • 1.NBT.B.3 Compare two two-digit numbers using symbols (Matching the computed total $32$ against the answer choices $20, 24, 27, 28, 32$ to identify (E).)
  • 2.NBT.B.5 Fluently add and subtract within 100 (Adding the three case totals $20 + 12 + 0 = 32$.)
  • 3.OA.A.1 Interpret products of whole numbers as total number of objects in groups (Reading "(white-king squares) × (black-king squares)" as a product of equal-sized groups (the multiplication rule per case).)
  • 3.OA.C.7 Fluently multiply and divide within 100 (Computing each case total directly: $4 \times 5 = 20$, $4 \times 3 = 12$, and $1 \times 0 = 0$.)

⭐ This AMC 8 problem only needs Grade 3 multiplication and the "split into cases, then add" idea you already know!

⭐ This AMC 8 problem only needs Grade 3 multiplication and the "split into cases, then add" idea you already know!

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