2001 AMC 10 Problems/Problem 11
Contents
Problem
Consider the dark square in an array of unit squares, part of which is shown. The first ring of squares around this center square contains unit squares. The second ring contains unit squares. If we continue this process, the number of unit squares in the ring is
Solution
Solution 1
We can partition the ring into rectangles: two containing unit squares and two containing unit squares.
There are unit squares in the ring.
Thus, the ring has unit squares.
Solution 2
We can make the ring by removing a square of side length from a square of side length .
This ring contains unit squares.
Thus, the ring has unit squares.
Solution 3 (Less Rigorous)
Notice that the first ring around the center square contains unit squares, the second ring contains unit squares, the third contains unit squares, and so on. The number of squares in the ring is determined by the expression . Thus, the number of unit squares in the ring is equal to , which equals unit squares.
-Darth_Cadet
Video Solution by Daily Dose of Math
https://youtu.be/y52knpoCVYo?si=dYATo3Zxoj4obeMV
~Thesmartgreekmathdude
See Also
2001 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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 |
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