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Difference between revisions of "2002 AMC 10A Problems/Problem 18"

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== Problem  ==
 
== Problem  ==
  
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A 3x3x3 cube is made of <math>27</math> normal dice. Each die's opposite sides sum to <math>7</math>. What is the smallest possible sum of all of the values visible on the <math>6</math> faces of the large cube?
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A <math>3</math>x<math>3</math>x<math>3</math> cube is made of <math>27</math> normal dice. Each die's opposite sides sum to <math>7</math>. What is the smallest possible sum of all of the values visible on the <math>6</math> faces of the large cube?
  
 
<math>\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)} 96</math>
 
<math>\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)} 96</math>
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https://www.youtube.com/watch?v=1sdXHKW6sqA  ~David
 
https://www.youtube.com/watch?v=1sdXHKW6sqA  ~David
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==Video Solution by Daily Dose of Math==
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https://youtu.be/QSLlfaf50Is
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~Thesmartgreekmathdude
  
 
==See Also==
 
==See Also==

Latest revision as of 17:40, 15 October 2024

Problem

A $3$x$3$x$3$ cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?

$\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)} 96$

Solution

In a 3x3x3 cube, there are $8$ cubes with three faces showing, $12$ with two faces showing and $6$ with one face showing. The smallest sum with three faces showing is $1+2+3=6$, with two faces showing is $1+2=3$, and with one face showing is $1$. Hence, the smallest possible sum is $8(6)+12(3)+6(1)=48+36+6=90$. Our answer is thus $\boxed{\text{(D)}\ 90}$.

Video Solution

https://www.youtube.com/watch?v=1sdXHKW6sqA ~David

Video Solution by Daily Dose of Math

https://youtu.be/QSLlfaf50Is

~Thesmartgreekmathdude

See Also

2002 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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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