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2007 AMC 8 Problems/Problem 11: Difference between revisions

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Created page with '== Problem == Tiles <math>I, II, III</math> and <math>IV</math> are translated so one tile coincides with each of the rectangles <math>A, B, C</math> and <math>D</math>. In the …'
 
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Tiles <math>I, II, III</math> and <math>IV</math> are translated so one tile coincides with each of the rectangles <math>A, B, C</math> and <math>D</math>. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle <math>C</math>?
Tiles <math>I, II, III</math> and <math>IV</math> are translated so one tile coincides with each of the rectangles <math>A, B, C</math> and <math>D</math>. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle <math>C</math>?


<center>[[Image:AMC8_2007_11.png]]</center>
<asy>
size(400);
defaultpen(linewidth(0.8));
path p=origin--(8,0)--(8,6)--(0,6)--cycle;
draw(p^^shift(8.5,0)*p^^shift(8.5,10)*p^^shift(0,10)*p);
draw(shift(20,2)*p^^shift(28,2)*p^^shift(20,8)*p^^shift(28,8)*p);
label("8", (4,6+10), S);
label("6", (4+8.5,6+10), S);
label("7", (4,6), S);
label("2", (4+8.5,6), S);
label("I", (4,6+10), N);
label("II", (4+8.5,6+10), N);
label("III", (4,6), N);
label("IV", (4+8.5,6), N);
label("3", (0,3+10), E);
label("4", (0+8.5,3+10), E);
label("1", (0,3), E);
label("9", (0+8.5,3), E);
label("7", (4,10), N);
label("2", (4+8.5,10), N);
label("0", (4,0), N);
label("6", (4+8.5,0), N);
label("9", (8,3+10), W);
label("3", (8+8.5,3+10), W);
label("5", (8,3), W);
label("1", (8+8.5,3), W);
label("A", (24,10), N);
label("B", (32,10), N);
label("C", (24,4), N);
label("D", (32,4), N);</asy>


<math>\mathrm{(A)}\ I \qquad \mathrm{(B)}\ II \qquad \mathrm{(C)}\ III \qquad \mathrm{(D)}\ IV \qquad \mathrm{(E)}</math> cannot be determined
<math>\mathrm{(A)}\ I \qquad \mathrm{(B)}\ II \qquad \mathrm{(C)}\ III \qquad \mathrm{(D)}\ IV \qquad \mathrm{(E)}</math> cannot be determined
Line 11: Line 40:
We first notice that tile III has a <math>0</math> on the bottom and a <math>5</math> on the right side. Since no other tile has a <math>0</math> or a <math>5</math>, Tile III must be in rectangle <math>D</math>. Tile III also has a <math>1</math> on the left, so Tile IV must be in Rectangle <math>C</math>.
We first notice that tile III has a <math>0</math> on the bottom and a <math>5</math> on the right side. Since no other tile has a <math>0</math> or a <math>5</math>, Tile III must be in rectangle <math>D</math>. Tile III also has a <math>1</math> on the left, so Tile IV must be in Rectangle <math>C</math>.


The answer is <math>\boxed{D}</math>
The answer is <math>\boxed{\textbf{(D)}}</math>


<center>[[Image:AMC8_2007_11S.png]]</center>
<center>[[Image:AMC8_2007_11S.png]]</center>
==Video Solution==
https://www.youtube.com/watch?v=66MVYKyetGc
==Video Solution by Why Math==
https://youtu.be/WobCYII7TRg
~savannahsolver
==Video Solution by SpreadTheMathLove==
https://www.youtube.com/watch?v=omFpSGMWhFc
==See Also==
{{AMC8 box|year=2007|num-b=10|num-a=12}}
{{MAA Notice}}

Latest revision as of 14:53, 2 July 2024

Problem

Tiles $I, II, III$ and $IV$ are translated so one tile coincides with each of the rectangles $A, B, C$ and $D$. In the final arrangement, the two numbers on any side common to two adjacent tiles must be the same. Which of the tiles is translated to Rectangle $C$?

[asy] size(400); defaultpen(linewidth(0.8)); path p=origin--(8,0)--(8,6)--(0,6)--cycle; draw(p^^shift(8.5,0)*p^^shift(8.5,10)*p^^shift(0,10)*p); draw(shift(20,2)*p^^shift(28,2)*p^^shift(20,8)*p^^shift(28,8)*p); label("8", (4,6+10), S); label("6", (4+8.5,6+10), S); label("7", (4,6), S); label("2", (4+8.5,6), S); label("I", (4,6+10), N); label("II", (4+8.5,6+10), N); label("III", (4,6), N); label("IV", (4+8.5,6), N); label("3", (0,3+10), E); label("4", (0+8.5,3+10), E); label("1", (0,3), E); label("9", (0+8.5,3), E); label("7", (4,10), N); label("2", (4+8.5,10), N); label("0", (4,0), N); label("6", (4+8.5,0), N); label("9", (8,3+10), W); label("3", (8+8.5,3+10), W); label("5", (8,3), W); label("1", (8+8.5,3), W); label("A", (24,10), N); label("B", (32,10), N); label("C", (24,4), N); label("D", (32,4), N);[/asy]

$\mathrm{(A)}\ I \qquad \mathrm{(B)}\ II \qquad \mathrm{(C)}\ III \qquad \mathrm{(D)}\ IV \qquad \mathrm{(E)}$ cannot be determined

Solution

We first notice that tile III has a $0$ on the bottom and a $5$ on the right side. Since no other tile has a $0$ or a $5$, Tile III must be in rectangle $D$. Tile III also has a $1$ on the left, so Tile IV must be in Rectangle $C$.

The answer is $\boxed{\textbf{(D)}}$

Video Solution

https://www.youtube.com/watch?v=66MVYKyetGc


Video Solution by Why Math

https://youtu.be/WobCYII7TRg

~savannahsolver

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=omFpSGMWhFc

See Also

2007 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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

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