AMC 8 · 2024 · #17
Grade 3 geometry-2dProblem
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 3 x 3 grid attacks all 8 other squares, as shown below. Suppose a white king and a black king are placed on different squares of a 3 x 3 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.
AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
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
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.
💡 Sorting squares using position words like "corner", "side", "middle" is exactly the Kindergarten position-vocabulary skill.
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.
💡 Counting the horizontal, vertical, and diagonal neighbors of one square is still a Kindergarten-level position/adjacency activity.
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 $(\text{white-king squares}) \times (\text{black-king squares})$.
💡 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.
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: $4 \times 5 = 20$.
💡 $4 \times 5 = 20$ is fluently within the Grade 3 multiplication-within-100 standard.
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: $4 \times 3 = 12$.
💡 $4 \times 3 = 12$ is a basic Grade 3 multiplication fact.
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.
💡 "Anything times $0$ is $0$" is a basic Grade 3 multiplication property.
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.
💡 Adding two-digit numbers like $20 + 12 + 0$ is well within Grade 2 addition-within-100 fluency.
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)**.
💡 Picking the matching two-digit number from a short list is a Grade 1 two-digit comparison skill.
K.G.A.1 Draw the $3 \times 3$ grid and color the 9 squares by position: **4 corner squar K.G.A.1 From the picture, count how many squares a king attacks from each type. A king o 3.OA.A.1 Break the big problem ("total number of placements") into **3 subproblems** by w 3.OA.C.7 **Case ①: white king on a corner.** White-king squares: $4$. The white king bloc 3.OA.C.7 **Case ②: white king on an edge.** White-king squares: $4$. The white king block 3.OA.C.7 **Case ③: white king in the center.** White-king squares: $1$. The center king b 2.NBT.B.5 Add the three subproblem answers to get the total: $20 + 12 + 0 = 32$. The three 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.1Describe 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.3Compare 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.5Fluently add and subtract within 100 (Adding the three case totals $20 + 12 + 0 = 32$.)3.OA.A.1Interpret 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.7Fluently 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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