Difference between revisions of "2017 AMC 8 Problems/Problem 11"

Latest revision as of 13:01, 27 January 2024

Problem 11

A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?

$\textbf{(A) }148\qquad\textbf{(B) }324\qquad\textbf{(C) }361\qquad\textbf{(D) }1296\qquad\textbf{(E) }1369$

Solution 1

Since the number of tiles lying on both diagonals is $37$ , counting one tile twice, there are $37=2x-1\implies x=19$ tiles on each side. Therefore, our answer is $19^2=361=\boxed{\textbf{(C)}\ 361}$ .

~AllezW

Solution 2

Visualize it as 4 separate diagonals connecting to one square in the middle. Each square on the diagonal corresponds to one square of horizontal/vertical distance (because it's a square). So, we figure out the length of each separate diagonal, multiply by two, and then add 1. (Realize that we can just join two of the separate diagonals on opposite sides together to save some time in calculations.) Therefore, the edge length is: $\frac{37-1}{4} \cdot 2 + 1 = 19$ Thus, our solution is $19^2 = 361 = \boxed{\textbf{(C)}\ 361}$ .

~Ligonmathkid2

Video Solution (CREATIVE THINKING!!!)

https://youtu.be/_XrCl3p--28

~Education, the Study of Everything

Video Solution

Associated video: https://youtu.be/QCWOZwYVJMg

https://youtu.be/8XtEOkP-AS0

~savannahsolver

2017 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 1: / Line 1: @@
 ==Problem 11==
 A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?
 <math>\textbf{(A) }148\qquad\textbf{(B) }324\qquad\textbf{(C) }361\qquad\textbf{(D) }1296\qquad\textbf{(E) }1369</math>
-==Solution==
+==Solution 1==
-Since the number of tiles lying on both diagonals is 37, counting one tile twice, there are <math>37=2x-1\implies x=19</math> tiles on each side. Hence, our answer is <math>19^2=361=\boxed{textbf{C}}</math>.
+Since the number of tiles lying on both diagonals is <math>37</math>, counting one tile twice, there are <math>37=2x-1\implies x=19</math> tiles on each side. Therefore, our answer is <math>19^2=361=\boxed{\textbf{(C)}\ 361}</math>.
+~AllezW
+==Solution 2==
+Visualize it as 4 separate diagonals connecting to one square in the middle. Each square on the diagonal corresponds to one square of horizontal/vertical distance (because it's a square). So, we figure out the length of each separate diagonal, multiply by two, and then add 1. (Realize that we can just join two of the separate diagonals on opposite sides together to save some time in calculations.) Therefore, the edge length is: <cmath>\frac{37-1}{4} \cdot 2 + 1 = 19</cmath>
+Thus, our solution is <math>19^2 = 361 = \boxed{\textbf{(C)}\ 361}</math>.
+~Ligonmathkid2
+==Video Solution (CREATIVE THINKING!!!)==
+https://youtu.be/_XrCl3p--28
+~Education, the Study of Everything
+==Video Solution==
+Associated video: https://youtu.be/QCWOZwYVJMg
+https://youtu.be/8XtEOkP-AS0
+~savannahsolver
-==See Also==
+==See Also:==
-{{AMC8 box|year=2017|num-b=8|num-a=10}}
+{{AMC8 box|year=2017|num-b=10|num-a=12}}
 {{MAA Notice}}

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