Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1990 AHSME Problems/Problem 10"

(→‎Problem)
(→‎Problem)
 
(5 intermediate revisions by 3 users not shown)
Line 3: Line 3:
 
An <math>11\times 11\times 11</math> wooden cube is formed by gluing together <math>11^3</math> unit cubes. What is the greatest number of unit cubes that can be seen from a single point?
 
An <math>11\times 11\times 11</math> wooden cube is formed by gluing together <math>11^3</math> unit cubes. What is the greatest number of unit cubes that can be seen from a single point?
  
−
<math>\text{(A) (Answer choices were lost)} \quad
+
<math>\text{(A) 328} \quad
−
\text{(B) (Answer choices were lost)} \quad
+
\text{(B) 329} \quad
−
\text{(C) (Answer choices were lost)} \quad
+
\text{(C) 330} \quad
−
\text{(D) 330} \quad
+
\text{(D) 331} \quad
−
\text{(E) (Answer choices were lost)} </math>
+
\text{(E) 332} </math>
  
 
== Solution ==
 
== Solution ==
Line 19: Line 19:
 
draw((0,1,1)--(1,1,1)--(1,0,1),red);
 
draw((0,1,1)--(1,1,1)--(1,0,1),red);
 
</asy>
 
</asy>
−
The best angle for cube viewing is centered on the corner. Meaning three of the six faces are visible. So therefore, the answer is just counting the number of cubes on the three faces. Which is 330 or <math>\fbox{D}</math>
+
The best angle for cube viewing is centered on the corner, meaning three of the six faces are visible. So, therefore, the answer is just counting the number of cubes on the three faces, which is <math>3 \times 10^2</math> for the inside parts of the faces, plus <math>3 \times 10</math> for the edges, plus <math>1</math> for the single shared cube in the corner, giving a total of <math>331</math> or <math>\fbox{D}.</math>
  
 
== See also ==
 
== See also ==

Latest revision as of 21:02, 2 July 2021

Problem

An $11\times 11\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?

$\text{(A) 328} \quad \text{(B) 329} \quad \text{(C) 330} \quad \text{(D) 331} \quad \text{(E) 332}$

Solution

[asy]import three; unitsize(1cm);size(100); draw((0,0,0)--(0,1,0),red); draw((0,0,1)--(0,0,0)--(1,0,0)--(1,0,1),red); draw((0,0,1)--(1,0,1),red); draw((0,0,1)--(0,1,1)--(0,1,0)--(1,1,0)--(1,0,0),red); draw((1,1,0)--(1,1,1),red); draw((0,1,1)--(1,1,1)--(1,0,1),red); [/asy] The best angle for cube viewing is centered on the corner, meaning three of the six faces are visible. So, therefore, the answer is just counting the number of cubes on the three faces, which is $3 \times 10^2$ for the inside parts of the faces, plus $3 \times 10$ for the edges, plus $1$ for the single shared cube in the corner, giving a total of $331$ or $\fbox{D}.$

See also

1990 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1990_AHSME_Problems/Problem_10&oldid=157266"