The Painted Cube !!

Probability Level 2

You have a 10 inch by 10 inch cube that is made up of little 1 inch by 1 inch cubes... You paint the outside of the big cube red... How many of the little cubes get painted?


The answer is 488.

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Royal Tomar by Royal Tomar , 28, India

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4 solutions

Palash Som
Dec 27, 2014

see let the faces of the big cube be named as A, B ,C , D , E and F .

NOW no. 1 unit cubes on face A = 10 *10 = 100

face B = (let it be any adjacent face to A ) = 10 *10 -10(because 10 cubes of face A) = 90

face C = (let it be any adjacent face to B ) = 10 *10 -10 = 90

face D = (let the face be between A and C) = 10 *10 - 20 = 80

face E = ( let it be the top face) = 100 - 36 = 64

face F =(let it be the bottom face )= 100 - 36 = 64

NOTE : for better understanding refer to the image

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Saya Suka
May 28, 2021

Answer
= 10³ – (10 – 2)³
= 10³ – 8³
= 1000 – 512
= 488



0 Helpful 0 Interesting 0 Brilliant 0 Confused
Melissa Quail
Dec 30, 2014

For this you can use the principle of inclusion-exclusion. There are 100 cubes on each face and there are 6 faces. 6 x 100 = 600. Then you subtract all the cubes you have counted twice (the edge cubes) because these are on more than one face. There are 10 cubes on each edge and 12 edges so there are 10 x 12 = 120 such cubes. 600 - 120 = 480. However, the corner cubes have now been subtracted too many times because they are on three edges. Therefore you need to add the eight corner cubes back on : 480 + 8 = 488 cubes. For more information about how the principle of inclusion - exclusion works see this note:

https://brilliant.org/discussions/thread/principle-of-inclusion-and-exclusion/

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Royal Tomar
Dec 27, 2014

We can simply use the formula to determine the no. of small cubes in this cube to find out the no. of small cubes that have been painted ..... We can use the formula =

(n)^3-(n-2)^3

Here we have n=10,

(n)^3-(n-2)^ 3 = (10)^3-(10-2)^3 = 1000-512 = 488

0 Helpful 0 Interesting 0 Brilliant 0 Confused

it's not n^2 Here n^3

AMAN KUMAR - 6 years, 5 months ago

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Sorry, I now have corrected it. Thanks for notifying me.

Royal Tomar - 6 years, 5 months ago

n 3 − ( n − 2 ) 3 n^{3} - (n-2)^{3} = 1 0 3 − 8 3 10^{3} - 8^{3} =1000-512

=488

Ashwin Upadhyay - 6 years, 5 months ago

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10^2=100 and 8^2 is 64 now 100-64 =36 not 488

Muhammad Shahzad Khan - 6 years, 5 months ago

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hmmm... i've corrected it i was influenced by the solution given thats why i wrote 2

Ashwin Upadhyay - 6 years, 5 months ago

Now its all correct thanks for notifying :D

Royal Tomar - 6 years, 5 months ago

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