AMC 10 · 2012 · #15
Grade 8 geometry-2dPick an answer.
The turn never mixes corners with edges with the center, so the 9 squares split into three independent groups: the lone center, a loop of 4 corners, and a loop of 4 edges. Once the painting rule is restated as a simple ban on certain white pairs, each group can be counted on its own and the counts multiplied. Breaking the grid into these separate pieces is what turns a tangled picture into three tiny counting jobs.
See what the turn moves
The turn cycles the corners and the edges in loops.
A quarter turn just slides each ring of squares one step around its own circle.
A quarter turn just slides each ring of squares one step around its own circle.
▸ Why?
A rotation carries the board onto itself without stretching, so each square lands on another square.
▸ Why?
Four quarter turns make a full turn, so each ring is a closed loop of four positions.
Restate when a square ends white
A spot ends light only if both squares were light.
One white square alone always gets covered or paints over; only two whites that chase each other around the loop survive.
7.SP.C.7Change Focus Count The ComplementSplit into center, corners, edges
The centre, the corners, and the edges split apart.
Independent pieces multiply, so three small counts replace one giant count.
7.SP.C.8Identify SubproblemsCount one 4-square loop
Each four-square loop has 7 good colourings.
Around a loop of four, whites are only safe when they never sit side by side.
7.SP.C.8Make A Systematic ListMultiply and divide
Multiplying gives 49/512, choice (A).
Favorable grids over all grids gives the probability straight away.
7.SP.C.5Identify SubproblemsA quarter turn sorts the squares into a fixed center and two rings of four, so count each ring's safe patterns on its own and multiply.
- See what the turn moves
- Restate when a square ends white
- Split into center, corners, edges
- Count one 4-square loop
- Multiply and divide
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