AMC 8 · 2005 · #3
Easy mode Grade 5Problem
Picture a big square ABCD split into a 4×4 grid of small squares. A few of the small squares are already colored black.
We want the diagonal line from B to D to be a line of symmetry. That means if you fold the picture along BD, every black square should land on another black square.
To make this work, we may need to color a few more small squares black. What is the smallest number of extra small squares we have to color?
Pick an answer.
AMC 8 2005 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 $4 \times 4$ grid sits inside square $ABCD$, and five small squares are already shaded black. Find the smallest number of additional small squares that must be shaded so the diagonal $\overline{BD}$ becomes a line of symmetry of the black pattern.
Givens: Square $ABCD$ is divided into a $4 \times 4$ grid of small squares; Five small squares are already shaded black (top-left corner, two in the top row near $B$, one on the right edge, and one on the bottom row near $D$); We want $\overline{BD}$ to be a line of symmetry for the shaded pattern; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$
Unknowns: The minimum number of additional small squares that must be shaded black
Understand
Restated: A $4 \times 4$ grid sits inside square $ABCD$, and five small squares are already shaded black. Find the smallest number of additional small squares that must be shaded so the diagonal $\overline{BD}$ becomes a line of symmetry of the black pattern.
Givens: Square $ABCD$ is divided into a $4 \times 4$ grid of small squares; Five small squares are already shaded black (top-left corner, two in the top row near $B$, one on the right edge, and one on the bottom row near $D$); We want $\overline{BD}$ to be a line of symmetry for the shaded pattern; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$
Plan
Primary tool: #11 Find an Invariant
Secondary: #1 Draw a Diagram, #2 Make a Systematic List
A line of symmetry is exactly the statement that the black pattern is invariant under reflection across $\overline{BD}$ — tool #11. The reflection across $\overline{BD}$ swaps each square with its mirror partner, so the unknowns aren't numbers but pairings. Tool #1 (Draw a Diagram) lets us mark every black square right on the grid, and tool #2 (Make a Systematic List) walks through each black square one at a time to record its mirror image. Any mirror image that isn't already black is a square we must add — counting those new squares gives the minimum.
Execute — Answer: D
5.G.A.1 Step 1 - Label the grid with coordinates so we can describe each small square.
- Put $D$ at the origin so columns and rows run from $0$ to $3$.
- Then the diagonal $\overline{BD}$ becomes the line $y = x$, and reflecting a square at column $c$, row $r$ across that line lands it at column $r$, row $c$ — just swap the two numbers.
💡 Setting up perpendicular number lines for the grid is the Grade 5 "coordinate system" move, and it turns the mirror image into the easy rule "swap the two coordinates."
5.G.A.2 Step 2 List the five already-black squares by their $(\text{column}, \text{row})$ coordinates by reading them off the figure.
💡 Naming each black square with its coordinates is the Grade 5 "plot a point" step — now we can talk about them precisely.
4.G.A.3 Step 3 - For each black square, write down its mirror image across $\overline{BD}$ by swapping the two coordinates.
- The square $(3,3)$ sits on the diagonal, so its mirror image is itself.
💡 Symmetry across a line is the Grade 4 "line of symmetry" idea: every point has a partner on the other side, except points right on the line.
4.G.A.3 Step 4 - Check each mirror image against the original list.
- If it is already black, the symmetry condition is satisfied for that pair.
- If it is not, that mirror square must be added.
💡 The invariant is "every black square's twin is also black." Spot every place that invariant currently fails — those are the squares we have to fix.
4.G.A.3 Step 5 - Count the new squares we must shade.
- Four mirror images $(3,0), (0,1), (3,2), (1,3)$ are missing, so we must shade exactly those four.
- Adding them is enough: each new square's mirror image is already on the original list, so no further additions are needed.
💡 Once every black square is paired with a black twin, the invariant holds and the diagonal really is a line of symmetry.
5.G.A.1 Label the grid with coordinates so we can describe each small square. Put $D$ at 5.G.A.2 List the five already-black squares by their $(\text{column}, \text{row})$ coord 4.G.A.3 For each black square, write down its mirror image across $\overline{BD}$ by swa 4.G.A.3 Check each mirror image against the original list. If it is already black, the s 4.G.A.3 Count the new squares we must shade. Four mirror images $(3,0), (0,1), (3,2), (1 Review
Reasonableness: Sanity-check by pairing up all $9$ black squares after the fix: $\{(0,3),(3,0)\}$, $\{(1,0),(0,1)\}$, $\{(2,3),(3,2)\}$, $\{(3,1),(1,3)\}$, and the self-mirror $(3,3)$. Every pair is a mirror image across $y=x$, so the diagonal $\overline{BD}$ is indeed a line of symmetry. Could we do it with fewer? No — each of the four original off-diagonal black squares had a missing partner, and a single new square can fix at most one missing partner, so we need at least $4$. That rules out (A), (B), (C), and (E).
Alternative: Tool #1 (Draw a Diagram) on its own: fold the picture along the diagonal $\overline{BD}$ in your head (or on paper). Each of the five black squares lands on the spot where its twin should be. Mark every landing spot that is white — there are exactly four of them: the reflections of the top-left, top-row pair, right-edge, and bottom-row blacks. Coloring those four white spots makes the folded picture match itself, giving answer (D).
CCSS standards used (min grade 5)
4.G.A.3Recognize a line of symmetry for a two-dimensional figure (Treating $\overline{BD}$ as a line of symmetry: every black square must have a black mirror image, and squares on the line are their own mirror image.)5.G.A.1Use a pair of perpendicular number lines forming a coordinate system (Setting up column-row coordinates on the $4 \times 4$ grid so the diagonal becomes $y = x$ and reflection becomes the "swap coordinates" rule.)5.G.A.2Represent real-world and mathematical problems by graphing points (Recording each black square as an ordered pair $(c, r)$ and locating its reflected partner $(r, c)$ on the same grid.)
⭐ A line of symmetry just means every black square has a black mirror twin. Pair up the black squares, find the lonely ones, and color in their twins — four were missing, so the answer is (D).
⭐ A line of symmetry just means every black square has a black mirror twin. Pair up the black squares, find the lonely ones, and color in their twins — four were missing, so the answer is (D).
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