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Difference between revisions of "2000 PMWC Problems/Problem T2"

(Created page with "==Problem== A <math>5 \times 5 \times 5</math> cube is formed using <math>1 \times 1 \times 1</math> cubes. A number of the smaller cubes are removed by punching out the 15 desig...")
 
(→‎Problem)
 
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==Problem==
 
==Problem==
 
A <math>5 \times 5 \times 5</math> cube is formed using <math>1 \times 1 \times 1</math> cubes. A number of the smaller cubes are removed by punching out the 15 designated columns from front to back, top to bottom, and side to side. Find the number of smaller cubes that remain.
 
A <math>5 \times 5 \times 5</math> cube is formed using <math>1 \times 1 \times 1</math> cubes. A number of the smaller cubes are removed by punching out the 15 designated columns from front to back, top to bottom, and side to side. Find the number of smaller cubes that remain.
 +
 +
<asy>
 +
import three;
 +
currentprojection=orthographic(1,1/2,0.4);
 +
draw(surface((5,1,1)--(5,1,2)--(5,2,2)--(5,2,1)--cycle),black,nolight);
 +
draw(surface((5,1,3)--(5,1,4)--(5,2,4)--(5,2,3)--cycle),black,nolight);
 +
draw(surface((5,3,1)--(5,3,2)--(5,4,2)--(5,4,1)--cycle),black,nolight);
 +
draw(surface((5,3,3)--(5,3,4)--(5,4,4)--(5,4,3)--cycle),black,nolight);
 +
draw(surface((5,2,2)--(5,2,3)--(5,3,3)--(5,3,2)--cycle),black,nolight);
 +
draw(surface((1,5,1)--(1,5,2)--(2,5,2)--(2,5,1)--cycle),black,nolight);
 +
draw(surface((1,5,3)--(1,5,4)--(2,5,4)--(2,5,3)--cycle),black,nolight);
 +
draw(surface((3,5,1)--(3,5,2)--(4,5,2)--(4,5,1)--cycle),black,nolight);
 +
draw(surface((3,5,3)--(3,5,4)--(4,5,4)--(4,5,3)--cycle),black,nolight);
 +
draw(surface((2,5,2)--(2,5,3)--(3,5,3)--(3,5,2)--cycle),black,nolight);
 +
draw(surface((1,1,5)--(1,2,5)--(2,2,5)--(2,1,5)--cycle),black,nolight);
 +
draw(surface((1,3,5)--(1,4,5)--(2,4,5)--(2,3,5)--cycle),black,nolight);
 +
draw(surface((3,1,5)--(3,2,5)--(4,2,5)--(4,1,5)--cycle),black,nolight);
 +
draw(surface((3,3,5)--(3,4,5)--(4,4,5)--(4,3,5)--cycle),black,nolight);
 +
draw(surface((2,2,5)--(2,3,5)--(3,3,5)--(3,2,5)--cycle),black,nolight);
 +
for(int i = 0; i <= 5; ++i) draw((i,0,5)--(i,5,5));
 +
for(int i = 0; i <= 5; ++i) draw((0,i,5)--(5,i,5));
 +
for(int i = 0; i <= 5; ++i) draw((5,i,0)--(5,i,5));
 +
for(int i = 0; i <= 5; ++i) draw((5,0,i)--(5,5,i));
 +
for(int i = 0; i <= 5; ++i) draw((i,5,0)--(i,5,5));
 +
for(int i = 0; i <= 5; ++i) draw((0,5,i)--(5,5,i));
 +
//Credit to chezbgone2 for the diagram</asy>
  
 
==Solution==
 
==Solution==
  
 
==See Also==
 
==See Also==

Latest revision as of 21:30, 21 April 2014

Problem

A $5 \times 5 \times 5$ cube is formed using $1 \times 1 \times 1$ cubes. A number of the smaller cubes are removed by punching out the 15 designated columns from front to back, top to bottom, and side to side. Find the number of smaller cubes that remain.

[asy] import three; currentprojection=orthographic(1,1/2,0.4); draw(surface((5,1,1)--(5,1,2)--(5,2,2)--(5,2,1)--cycle),black,nolight); draw(surface((5,1,3)--(5,1,4)--(5,2,4)--(5,2,3)--cycle),black,nolight); draw(surface((5,3,1)--(5,3,2)--(5,4,2)--(5,4,1)--cycle),black,nolight); draw(surface((5,3,3)--(5,3,4)--(5,4,4)--(5,4,3)--cycle),black,nolight); draw(surface((5,2,2)--(5,2,3)--(5,3,3)--(5,3,2)--cycle),black,nolight); draw(surface((1,5,1)--(1,5,2)--(2,5,2)--(2,5,1)--cycle),black,nolight); draw(surface((1,5,3)--(1,5,4)--(2,5,4)--(2,5,3)--cycle),black,nolight); draw(surface((3,5,1)--(3,5,2)--(4,5,2)--(4,5,1)--cycle),black,nolight); draw(surface((3,5,3)--(3,5,4)--(4,5,4)--(4,5,3)--cycle),black,nolight); draw(surface((2,5,2)--(2,5,3)--(3,5,3)--(3,5,2)--cycle),black,nolight); draw(surface((1,1,5)--(1,2,5)--(2,2,5)--(2,1,5)--cycle),black,nolight); draw(surface((1,3,5)--(1,4,5)--(2,4,5)--(2,3,5)--cycle),black,nolight); draw(surface((3,1,5)--(3,2,5)--(4,2,5)--(4,1,5)--cycle),black,nolight); draw(surface((3,3,5)--(3,4,5)--(4,4,5)--(4,3,5)--cycle),black,nolight); draw(surface((2,2,5)--(2,3,5)--(3,3,5)--(3,2,5)--cycle),black,nolight); for(int i = 0; i <= 5; ++i) draw((i,0,5)--(i,5,5)); for(int i = 0; i <= 5; ++i) draw((0,i,5)--(5,i,5)); for(int i = 0; i <= 5; ++i) draw((5,i,0)--(5,i,5)); for(int i = 0; i <= 5; ++i) draw((5,0,i)--(5,5,i)); for(int i = 0; i <= 5; ++i) draw((i,5,0)--(i,5,5)); for(int i = 0; i <= 5; ++i) draw((0,5,i)--(5,5,i)); //Credit to chezbgone2 for the diagram[/asy]

Solution

See Also

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