2004 AMC 10A Problems/Problem 16
Problem
The grid shown contains a collection of squares with sizes from to . How many of these squares contain the black center square?
Solution
Solution 1
Since there are five types of squares: and We must find how many of each square contain the black shaded square in the center.
If we list them, we get that
- There is of all squares, containing the black square
- There are of all squares, containing the black square
- There are of all squares, containing the black square
- There are of all squares, containing the black square
- There is of all squares, containing the black square
Thus, the answer is .
Solution 2
We use complementary counting. There are only and squares that do not contain the black square. Counting, there are - squares, and squares that do not contain the black square. That gives squares that don't contain it. There are a total of squares possible - squares - squares - squares - squares and - square), therefore there are squares that contain the black square, which is .
See also
2004 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.