Competition · AMC preparation · step 4 of 4
AMC 8 · 2009 · #10
Grade 7 probability
Pick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The board is right there to look at, so Tool #1 (Draw a Diagram) is the natural way in: outline the squares that touch the outer edge — they form a one-square-thick "frame" around the board — and what's left in the middle is what we want. To make the framing rule crisp before counting on the 8 × 8 board, Tool #9 (Solve an Easier Problem) on a small 4 × 4 board shows the pattern: removing the border ring of a 4 × 4 leaves a 2 × 2 interior, i.e. an (n-2) × (n-2) inside an n × n. Then we apply that same picture to n = 8 and divide.
Count all 64 squares
Count the whole board first: 8 rows of 8 make 8 × 8 = 64 equally likely squares.
An 8 × 8 array is 8 equal groups of 8, which is the Grade 3 meaning of multiplication.
3.OA.A.1Draw A DiagramTry a smaller board
Test a smaller 4 × 4 board: peel its edge and a 2 × 2 center is left — so the interior is (n-2) × (n-2).
Peeling off the border of an n × n board takes one square from each side, leaving an (n - 2) × (n - 2) interior — easier to see on a tiny board first.
3.OA.A.1Solve An Easier Related ProblemApply it to the big board
Same picture on the 8 × 8: the squares off the edge fill a (8-2) × (8-2) = 6 × 6 block, or 36 squares.
Once the border ring is removed in the diagram, what remains is just another rectangle, so we can multiply rows by columns again.
On the 8 × 8 board, the squares that do not touch the outer edge fill a 6-by-6 block in the middle, which is 36 squares.
▸ Why?
A square touches the outer edge exactly when it sits in the outermost row or column, so in each row of 8 only the two end squares are on an edge and the 6 between them are safe — the safe squares run 6 across and, in the same way, 6 down.
▸ Why?
A full row of 8 splits with no gaps or overlaps into its left edge square, its right edge square, and the squares between them, so the middle count is 8 take away the 2 ends, which is 6.
▸ Why?
The columns behave just like the rows because the board is square: a quarter turn lays the board exactly onto itself and carries each row onto a column without changing any count, so 6 remain down the middle as well.
▸ Why?
That middle block is a rectangle of 6 rows with 6 squares in each row, and 6 equal groups of 6 make 6 × 6 = 36.
Simplify the probability
Make the probability favorable / total, , then divide both by 4 to reduce it to .
With every square equally likely, the probability of an event is just the fraction of squares that satisfy it — then reduce.
7.SP.C.7Draw A DiagramThis AMC 8 problem only needs Grade 7 probability — count favorable squares, divide by total, and reduce the fraction.
- Count all 64 squares
- Try a smaller board
- Apply it to the big board
- Simplify the probability
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