AMC 10 · 2018 · #15
Grade 8 countingPick an answer.
The whole question is about what the eight symmetries (four rotations and four reflections) do to the squares, so Tool #17 (Visualize Spatial Relationships) leads: mentally rotate and fold the grid to see which squares are locked to the same color. Tool #1 (Draw a Diagram) pins the squares to a coordinate grid so the folding is exact. Tool #7 (Identify Subproblems) splits the squares into a few position types (center, midline arm, diagonal arm, interior), and Tool #2 (Make a Systematic List) counts how many independent color choices that leaves. Finally the words 'at least one of each color' are a Tool #16 (Change Focus / Count the Complement) cue: count every coloring, then remove the two all-one-color cases.
Squares move in linked groups
The symmetries tie squares into groups.
If a move slides one square onto another, the picture only matches when both wear the same color.
If a move slides one square onto another, the picture only matches when both wear the same colour.
▸ Why?
A symmetry moves the grid onto itself without stretching, so squares are only relabelled, never changed.
▸ Why?
Each square is carried to exactly one square, so the colours must be passed along unchanged.
Fold into one-eighth
Fold the grid into one eighth.
Folding a symmetric picture along its mirror lines stacks every copy onto one master tile.
4.G.A.3Draw A DiagramCount the independent squares
There are ten independent groups.
Sorting the wedge by where a square sits makes sure every group is counted once and only once.
8.G.A.3Make A Systematic ListTwo colors for each group
Two colors each gives two to the tenth.
Ten yes-or-no color switches, each free, multiply to 2¹⁰ pictures.
6.EE.A.1Identify SubproblemsRemove the two banned grids
Removing the two solid grids gives 1022.
Count everything first, then subtract only the two pictures that use a single color.
4.OA.A.3Change Focus Count The ComplementFold the symmetric grid along its mirror lines until only 10 squares are truly free; each is black or white, giving 2¹⁰=1024 pictures, then drop the all-black and all-white ones for 1022, choice (B).
- Squares move in linked groups
- Fold into one-eighth
- Count the independent squares
- Two colors for each group
- Remove the two banned grids
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