Difference between revisions of "2001 AMC 10 Problems/Problem 11"

Revision as of 11:35, 24 February 2023

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 $8$ unit squares. The second ring contains $16$ unit squares. If we continue this process, the number of unit squares in the $100^\text{th}$ ring is

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$\textbf{(A)}\ 396 \qquad \textbf{(B)}\ 404 \qquad \textbf{(C)}\ 800 \qquad \textbf{(D)}\ 10,\!000 \qquad \textbf{(E)}\ 10,\!404$

Solution

Solution 1

We can partition the $n^\text{th}$ ring into $4$ rectangles: two containing $2n+1$ unit squares and two containing $2n-1$ unit squares.

There are $2(2n+1)+2(2n-1)=4n+2+4n-2=8n$ unit squares in the $n^\text{th}$ ring.

Thus, the $100^\text{th}$ ring has $8 \times 100 = \boxed{\textbf{(C) }800}$ unit squares.

Solution 2

We can make the $n^\text{th}$ ring by removing a square of side length $2n-1$ from a square of side length $2n+1$ .

This ring contains $(2n+1)^2-(2n-1)^2=(4n^2+4n+1)-(4n^2-4n+1)=8n$ unit squares.

Thus, the $100^\text{th}$ ring has $8 \times 100 = \boxed{\textbf{(C)}\ 800}$ unit squares.

Solution 3 (Less Rigorous)

Notice that the first ring around the center square contains $8$ unit squares, the second ring contains $16$ unit squares, the third contains $24$ unit squares, and so on. The number of squares in the $n^\text{th}$ ring is determined by the expression $8 \times n$ . Thus, the number of unit squares in the $100^\text{th}$ ring is equal to $8 \times 100$ , which equals $\boxed{\textbf{(C) }800}$ unit squares.

-Darth_Cadet

@@ Line 34: / Line 34: @@
 ===Solution 3 (Less Rigorous)===
-Notice that the first ring around the center square contains 8 unit squares, the second ring contains 16 unit squares, the third contains 24 unit squares, and so on. The number of squares in the <math> n^\text{th} </math> ring is determined by the expression <math> 8 \times n </math>. Thus, the number of unit squares in the <math>100^\text{th}</math> ring is equal to <math> 8 \times 100 </math>, which equals <math> \boxed{\textbf{(C) }800} </math> unit squares.
+Notice that the first ring around the center square contains <math> 8 </math> unit squares, the second ring contains <math> 16 </math> unit squares, the third contains <math> 24 </math> unit squares, and so on. The number of squares in the <math> n^\text{th} </math> ring is determined by the expression <math> 8 \times n </math>. Thus, the number of unit squares in the <math>100^\text{th}</math> ring is equal to <math> 8 \times 100 </math>, which equals <math> \boxed{\textbf{(C) }800} </math> unit squares.
 -Darth_Cadet

2001 AMC 10 (Problems • Answer Key • Resources)
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