AMC 10 · 2025 · #21
Grade 8 countingPick an answer.
AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question is a "how many colorings" count, so Tool #2 (Make a Systematic List) is the backbone: once the grid is mostly pinned down, we list every way to finish it. But listing all 3⁹ raw colorings is hopeless, so two ideas cut it down first. Tool #16 (Change Focus) exploits symmetry twice: rotations and reflections merge equal colorings, and relabeling colors along the loop red→blue→yellow→red is itself allowed, which lets us fix the center cell to red and multiply by 3 at the end. Tool #1 (Draw a Diagram) keeps the grid in front of us so each rule is visible, and Tool #3 (Eliminate Possibilities) forces most cells: once the center and one neighbor are set, the loop rule leaves only one legal color for cell after cell.
Read the rules as a color loop
Restate the three rules as one loop on the 3×3 grid: red must touch blue, blue must touch yellow, yellow must touch red.
Turning three separate rules into one repeating loop means you only have to remember a single arrow.
6.EE.B.5Draw A DiagramUse symmetry to fix the center as red
Shifting every color forward along the loop keeps a coloring legal, so count only red-center grids and multiply by 3.
Because relabeling colors keeps every rule true, the center is red exactly one-third of the time.
Because relabelling the colours keeps every rule true, each colour takes the centre equally often.
▸ Why?
The relabelling matches every valid picture with exactly one other valid picture.
▸ Why?
Matched pictures are equally likely, so the three colours must share the centre in equal thirds.
Place the forced first neighbors
Fix the center red; rotation puts its blue at the left-middle, and reflection puts that blue's yellow at the top-left corner.
Spending the rotations and reflections now fixes a corner, so later choices are real choices, not repeats.
8.G.A.1Draw A DiagramChase the loop around the top
The loop then forces the rest of the top: top-middle red, top-right blue, right-middle yellow — six cells settled.
With a neighbor's color known, the loop leaves only one legal color, so each cell falls like a domino.
6.EE.B.5Eliminate PossibilitiesList every legal bottom row
Only the bottom row is open, and checking every combination leaves exactly 4 legal rows: RRB, YRB, RYB, BYB.
With everything above fixed, the whole puzzle shrinks to filling three cells you can just list out.
7.SP.C.8Make A Systematic ListMultiply by the three center colors
Blue-center and yellow-center each give 4 as well, and the three groups never overlap, so the total is 12.
Each of the four red-center pictures has a twin in blue and in yellow, tripling the count.
8.G.A.2Make A Systematic ListPin down whatever the rules force, use the shape's symmetry to avoid counting the same picture twice, and then just list the few choices that are left.
- Read the rules as a color loop
- Use symmetry to fix the center as red
- Place the forced first neighbors
- Chase the loop around the top
- List every legal bottom row
- Multiply by the three center colors
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