AMC 8 · 2009 · #18
Easy mode Grade 4Problem
Picture a square floor that is 7 feet on each side. It is covered with 1-foot square tiles, some black and some white.
In the picture, the tiles in the four corners are white. The rest of the pattern is shown in the diagram.
Now imagine a bigger floor — 15 feet on each side — tiled the same way, following the same pattern. How many white tiles would that bigger floor need?
Diagram
Try the same tiling rule on smaller odd-side floors and count the white tiles to spot the pattern.
Pick an answer.
AMC 8 2009 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: A $7$-by-$7$ floor is tiled with $1$-square-foot black and white tiles in a striped pattern: the corner tiles are white, and black stripes (rows and columns) sit one position in from each edge and continue every other position. The same rule is applied to a $15$-by-$15$ floor. Count the white tiles needed for the $15$-by-$15$ floor.
Givens: Each tile is $1$ square foot; The $7$-by-$7$ picture shows black tiles forming full rows at positions $2, 4, 6$ and full columns at positions $2, 4, 6$ (counting from one edge); All four corner tiles are white; The new floor is $15$ by $15$, tiled by the same rule; Answer choices: (A) $49$, (B) $57$, (C) $64$, (D) $96$, (E) $126$
Unknowns: The number of white tiles in the $15$-by-$15$ floor
Understand
Restated: A $7$-by-$7$ floor is tiled with $1$-square-foot black and white tiles in a striped pattern: the corner tiles are white, and black stripes (rows and columns) sit one position in from each edge and continue every other position. The same rule is applied to a $15$-by-$15$ floor. Count the white tiles needed for the $15$-by-$15$ floor.
Givens: Each tile is $1$ square foot; The $7$-by-$7$ picture shows black tiles forming full rows at positions $2, 4, 6$ and full columns at positions $2, 4, 6$ (counting from one edge); All four corner tiles are white; The new floor is $15$ by $15$, tiled by the same rule; Answer choices: (A) $49$, (B) $57$, (C) $64$, (D) $96$, (E) $126$
Plan
Primary tool: #5 Look for a Pattern
Secondary: #9 Solve an Easier Related Problem
Jumping straight to a $15$-by-$15$ count is messy. Tool #9 (Easier Related Problem) says: try the same tiling on the smallest odd-side floors first — $1 \times 1$, $3 \times 3$, $5 \times 5$, $7 \times 7$ — and count whites by hand. Tool #5 (Look for a Pattern) then turns the four counts into a rule we can apply to side $15$. The picture already does the $7 \times 7$ case for us, so most of the work is just counting and matching.
Execute — Answer: C
2.G.A.2 Step 1 - Look at where the white tiles sit on the $7 \times 7$ picture.
- The black stripes are the rows and columns at positions $2, 4, 6$, so white tiles are exactly the squares whose row AND column are in $\{1, 3, 5, 7\}$ — the odd positions.
- That's $4$ odd rows and $4$ odd columns.
💡 Partitioning a square into rows and columns of unit tiles and counting an intersection pattern is a Grade 2 array idea.
2.G.A.2 Step 2 - Repeat the same counting for the smaller odd-side floors.
- On a $1 \times 1$ floor the single tile is a corner, so it is white: $1$ white.
- On $3 \times 3$ the odd-position rows and columns are $\{1, 3\}$, giving $2 \times 2 = 4$ whites.
- On $5 \times 5$ the odd positions are $\{1, 3, 5\}$, giving $3 \times 3 = 9$ whites.
💡 Smaller, hand-countable versions of the same tiling let us see the structure without algebra.
4.OA.C.5 Step 3 - The four counts $1, 4, 9, 16$ are exactly the perfect squares $1^2, 2^2, 3^2, 4^2$.
- The side lengths $1, 3, 5, 7$ are the $1$st, $2$nd, $3$rd, $4$th odd numbers.
- So for the $k$-th odd-side floor, the number of white tiles is $k^2$.
💡 Generating a square-number pattern from a tiling rule is exactly the Grade 4 "generate a number or shape pattern" standard.
3.OA.D.9 Step 4 - Find which odd number $15$ is.
- The odd numbers go $1, 3, 5, 7, 9, 11, 13, 15$, so $15$ is the $8$th odd number, meaning $k = 8$.
💡 Recognizing odd numbers as $1, 3, 5, \dots$ and locating $15$ in the sequence is a Grade 3 arithmetic-pattern move.
3.OA.C.7 Step 5 Apply the pattern at $k = 8$.
💡 Computing $8 \times 8 = 64$ is exactly the Grade 3 "multiply fluently within $100$" standard.
2.G.A.2 Look at where the white tiles sit on the $7 \times 7$ picture. The black stripes 2.G.A.2 Repeat the same counting for the smaller odd-side floors. On a $1 \times 1$ floo 4.OA.C.5 The four counts $1, 4, 9, 16$ are exactly the perfect squares $1^2, 2^2, 3^2, 4^ 3.OA.D.9 Find which odd number $15$ is. The odd numbers go $1, 3, 5, 7, 9, 11, 13, 15$, s 3.OA.C.7 Apply the pattern at $k = 8$. Review
Reasonableness: Sanity check the pattern from a different angle. On any odd-side floor of size $s = 2k - 1$, the white tiles form a $k$-by-$k$ grid (one white tile at every odd-row, odd-column intersection), so the white count is $k^2$. Plug in $s = 7$: $k = 4$, $k^2 = 16$ — which matches the picture. Plug in $s = 15$: $k = 8$, $k^2 = 64$. The answer $64$ is also well below the $15 \times 15 = 225$ total tiles, which is the right ballpark since black stripes cover most of the floor.
Alternative: Tool #7 (Identify Subproblems) — count by complement. The $15 \times 15$ has $225$ tiles total. Black rows sit at positions $2, 4, 6, 8, 10, 12, 14$ ($7$ rows of $15$ = $105$ tiles). Black columns sit at the same positions ($7$ columns of $15$ = $105$ tiles). The overlap (cells in a black row AND a black column) is $7 \times 7 = 49$ tiles, counted twice. So black $= 105 + 105 - 49 = 161$, and white $= 225 - 161 = 64$. Same answer.
CCSS standards used (min grade 4)
2.G.A.2Partition a rectangle into rows and columns of same-size squares (Counting white tiles on the $7 \times 7$ picture and the smaller $3 \times 3$, $5 \times 5$ floors by reading off odd-position rows and columns.)3.OA.C.7Fluently multiply and divide within 100 (Computing $8 \times 8 = 64$ once the pattern $k^2$ is known.)3.OA.D.9Identify arithmetic patterns and explain using properties of operations (Identifying $1, 3, 5, 7, \dots$ as the odd-number sequence and locating $15$ as the $8$th odd number.)4.OA.C.5Generate a number or shape pattern following a given rule (Generalizing the $1, 4, 9, 16$ counts as $k^2$ for the $k$-th odd-side floor.)
⭐ This AMC 8 problem only needs Grade 4 pattern-finding with square numbers you already know!
⭐ This AMC 8 problem only needs Grade 4 pattern-finding with square numbers you already know!
More like this
Same archetype — closest grade level first.
