Competition · AMC preparation · step 4 of 4
AMC 10 · 2002A · #8
Grade 6 geometry-2d
Pick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Comparing B, W, and P turns into three separate area jobs, so Tool #7 (Identify Subproblems) is the backbone: find the red area, then the white area, then the blue area, and compare. Tool #1 (Draw a Diagram) supplies the trick that makes every piece countable — overlay a unit grid on the flag so the flag is 64 little squares and each tilted white square is exactly half of a 2 × 2 box. Tool #16 (Count the Complement) then gets the blue for free: instead of adding up all the blue triangles, subtract red and white from the whole flag.
Grid the whole flag
Lay a unit grid on the flag: 8 × 8 = 64 unit squares, the budget all three colors share.
Covering a square with unit tiles turns 'area' into 'how many little squares,' which you can just count.
3.MD.C.7Draw A DiagramRed area P
The red square runs from -2 to 2 both ways, so its side is 4 and P = 4 × 4 = 16.
A square's area is just one side times itself, so a 4-wide square holds 16 unit tiles.
3.MD.C.7Identify SubproblemsWhite area W
Each tilted white square is half its 2 × 2 box, area 2; 3 per edge × 4 edges = 12, so W = 24.
A diamond drawn through the midpoints of a box is always half the box, so each white square is worth 2 tiles.
6.G.A.1Identify SubproblemsBlue area B, then compare
Blue is the leftover: B = 64 - 16 - 24 = 24, which equals W, so choice (A) is the only true one.
Since the colors tile the flag exactly, blue is just the whole minus the parts you already measured.
Since the colours tile the flag exactly, the last colour is the whole minus the parts already measured.
▸ Why?
The flag is exactly its coloured pieces put together, with no gaps and no overlaps.
▸ Why?
Every point is one colour or another, so subtracting the known colours leaves exactly the last one.
Lay a grid over a colored picture, count each color's squares, and get the last color for free by subtracting from the whole.
- Grid the whole flag
- Red area P
- White area W
- Blue area B, then compare
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