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A 5x5x5 cube is formed by assembling 125 unit cubes. Nine unit squares are painted on each of the six faces of the cube according to the pattern shown. How many of the 125 unit cubes have no paint on them?

 

https://latex.artofproblemsolving.com/e/f/a/efa0cc439f48f4a24f83d140f98f9e6d1f4a624c.png

 Feb 18, 2018
 #1
avatar+17330 
+4

Another wrong answer!   See discussion.....and Tertre's CORRECT answer below.

(Geez)

 Feb 18, 2018
edited by ElectricPavlov  Feb 18, 2018
edited by ElectricPavlov  Feb 18, 2018
edited by ElectricPavlov  Feb 18, 2018
edited by ElectricPavlov  Feb 18, 2018
 #5
avatar+3994 
+3

EDIT: Wait, I'm getting 83... 

The number of painted faces is 9*6=54. However, the four edge faces painted on each face are also painted on the other face. Thus 4*6 of the painted faces are on cubes with two painted faces. These account for only 12 painted cubes, in addition to the 54-24=30 singly painted cubes. So there are 42 painted cubes, leaving 125-42=83 unpainted cubes.

 

@EP, I don't know how I'm getting 83.....Is this correct?

tertre  Feb 18, 2018
edited by tertre  Feb 18, 2018
edited by tertre  Feb 18, 2018
 #6
avatar+17330 
+2

Wow...this is really twisting the visualizations in my mind around!

I believe YOU have it CORRECT !     Well done...

I recounted....(yes, yet again)

two opposite faces   18

two faces with  7       14

two faces with  5        10       for a total of 42 painted little cubes!

 

125-42 = 83 without paint ....as you found!     Yikes...My batting average today is very low! cheeky

ElectricPavlov  Feb 18, 2018
 #7
avatar+3994 
+2

Thank you! It's okay, my mind gets confused a lot....

tertre  Feb 18, 2018
 #2
avatar+17330 
+3

See above edited answer.....   Thanx Tertre !  cheeky

 Feb 18, 2018
edited by ElectricPavlov  Feb 18, 2018
edited by ElectricPavlov  Feb 18, 2018
 #4
avatar+3994 
+3

No problem, EP! smiley

tertre  Feb 18, 2018
 #3
avatar+3994 
+2

I don't think 95 is correct, EP.

 Feb 18, 2018

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