Rubik's Cube Destruction

Tony has become frustrated with his 2 × 2 × 2 2\times 2 \times 2 pocket cube and he's decided to destroy it by taking off all of the stickers.

How many possible orders are there in which Tony could take off the stickers?

24 ! 24! 6 ! 6! 4 ! × 6 ! 4! \times 6! 12 ! 12!

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16 solutions

Prasun Biswas
Feb 9, 2014

Since, there are 6 faces of the cube and each face has 4 stickers, so there are a total of ( 6 × 4 ) = 24 (6\times 4)=24 stickers in the cube.

Now, Tony can take off a sticker from any side in 24 24 ways and corresponding to the first sticker taken off, he can take off any one of the remaining stickers in 23 23 ways and corresponding to the second sticker taken off, he can take off any one of the remaining stickers in 22 22 ways and so on.

Thus, the total no. of ways to take off the stickers = ( 24 × 23 × 22 × . . . . . . . × 1 ) = 24 ! =(24\times 23\times 22\times.......\times 1)=\boxed{24!}

Also, we can say that the stickers can be taken off in 24 P 24 = 24 ! 24P24 = \boxed{24!} ways.

oh but there will be 4 of six types so ans must be 24!/4!^6.......cause in rubics cube we have each face with four squres of same color.......must reply...............

Suyash Gupta - 7 years, 4 months ago

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Actually, since here it is asked that in how many ways the stickers can be removed from the cube and there are 24 stickers in total, so removing any colour sticker from one side and then another of the same colour from the other side will not be considered the same as the reverse case. Thus, this is an example of permutation without repetitions.

Prasun Biswas - 7 years, 4 months ago

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salam g ap ko Prasun Biswas Prasun Biswas,

Abdul Rafiu - 7 years, 4 months ago

Thank you for your explanation

Valerian Pratama - 7 years, 4 months ago

we can also note that every individual sticker is unique as each of them have different adjacent stickers. Thus, it will make a difference, even if you are considering the stickers of the same faces.

Varun Suvarna - 7 years, 3 months ago

Yes, I was stuck for a moment because I took out the repetitions...these kind of problems require clarification

Jean Lille - 7 years, 3 months ago

Suyash you are absolutely right

Suman Saha - 7 years, 3 months ago

No. Four squares of same color will then have combinations of which corner first up-left, up-right, down-left and down-right....24! is the only answer.

Rahul Deshpande - 7 years, 3 months ago

i am with suyash gupta 24!/(4!)^6

Anirudha Nayak - 7 years, 3 months ago

good one!

Debasish Guha - 7 years, 3 months ago

Uurgh..my bad..i am confused of the question in first place

Trish Destiny - 7 years, 3 months ago

The question should specify that all the stickers are distinct, so that others would not mistake that removing stickers of the same colour of different order does not count as a new way

Julian Poon - 7 years, 2 months ago

Thanks for giving such nice solution

Adarsh Mahor - 5 years, 3 months ago
Finn Hulse
Feb 7, 2014

I'd like to note that it's spelled "Rubik's". But anyways, there are 6 sides to a cube, and 4 stickers on each side. Therefore, there are 24 stickers to take off. That's 24 times 23 times 22... etc., or 24!. I also posted a problem about 2x2x2 Rubik's Cubes.

What about the elements of symmetry this cube has ? That might reduce the answer to much greater extent. Plz Reply.

Rushi Rokad - 7 years, 4 months ago

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Actually, no two cubes (of the 4 cubes that make up the one larger cube), are the same, unless you peel of stickers. Because they are all corners, they will have 3 colors touching them, and no 3 are the same 3 for any of the 8 vertices, so there are no lines of symmetry. Hope this helps!

Finn Hulse - 7 years, 3 months ago

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OK..Got it..

Rushi Rokad - 7 years, 3 months ago

we have six sides and 24 stickers so possible is we can remove 24 stickes so answer is 24!

tamil arasan - 7 years, 4 months ago

I knew it

Mardokay Mosazghi - 7 years, 4 months ago
Amlan Mishra
Feb 10, 2014

There are 4 stickers on each face. So there are 24 stickers in all.

At first you chose any of the 24 stickers, then any one of the rest 23, then any one of the rest 22 , and so on......

So the total number of orders is 24 x 23 x 22 x ..... x 1 = 24!

Pj Valdez
Feb 10, 2014

So here how you do it, there 6 sides of the cube, and each side has 4 colors in it, so you multiply it. So it becomes 24 then add the "!" sign that shows permutation. Permutation is the arrangement of a group in a definite order. so if you were to use this method, your formula would be 24P24 (24x23x22x21.......x1), or you use your calculator to calculate 24P24

I did not think of that Thank u for the help

holy_ jet - 1 year, 1 month ago

each side contains 4 stickers and all 6 sides contains 4*6=24 Hence 24! different ways to take out the stickers.

Nikhil Agarwal
Feb 11, 2014

On every side of the cube there are four colored stripes. Therefore, in total there are 8 X 4=32 coloured plates to be removed. These can be removed in 24! because the permutations of these 24 plates are 24!.

Raiyun Razeen
Feb 10, 2014

There are 6×4=24 stickers in a 2×2×2 pocket cube.

So, the answer is 24!

Julio Reyes
Feb 9, 2014

Each sticker is on a specific side and on a specific corner. If you think of each of these positions as unique then you have: 6 x 4 = 24 positions.

You have 24 positions to choose from at the beginning. Once you take off one sticker, then you have 1 less position to choose from.

Each time you take off a sticker you multiply the possible positions to the previous possible positions to get the number possible orders you could have taken the stickers in. (e.g. If you want to know the possible orders to take off 2 stickers, then 24 x 23 = 552 possible orders).

In this case we want the possible orders to take off all the stickers:

24 × 23 × 22 × × 1 = 24 ! 24 \times 23 \times 22 \times \cdots \times 1= 24!

This was a great solution, thanks for writing it up.

Joe Roberts - 7 years, 3 months ago
Bryan Dellariarte
Mar 24, 2014

just 6*4=24

M Abdullah Chohan
Mar 21, 2014

As, there are total 6 sides and 4 blocks on each .Therefore, 6*4=24 . That's answer :)

Yumi Kim
Mar 18, 2014

There is 4 sticker on 1 side and there is 6 side in a cube so, 6 x 4 is equals to 24. As you can see there is 6x 4 in the choices, that is just to confuse you.

Saurav Banerjee
Feb 25, 2014

was tony's aim to destroy the rubik's cube or just remove all of the stickers . . . Because if tony wanted to destroy the cube then he could have done it by only striking off any two colours of sticker in the cube and then he would have 8! possible orders

Since it is a 2 x 2 x 2 cube it will have 24 distinct stickers. So to destroy it he can select any of the 24 stickers. So the total number of possibilities are 24 C 1 X 23 C 1 X .... X 1 C 1 = 24!

24 stickers so 24! ways to rip them off

Sanjay Raj
Feb 16, 2014

they is a boy playing with a100 coins .if he took 10 coin from total coin randomly ,so what is probability at 11th coin

Varun Suvarna
Feb 15, 2014

2 2 2 cube has 24 faces. therefore, first we can take out any of the 24 stickers, after that any of the remaining 23 stickers, after that any of the remaining 22 stickers, after that any of the remaining 21 stickers, after that any of the remaining 20 stickers, so on till we have to choose the last sticker. therefore, now we will multiply all the steps as they are occurring in the same time! i.e., 24 23 22 21 20 19... 2*1=24!

i am sorry, the last line should be 24 23 22 21 20 19... 2*1=24!

Varun Suvarna - 7 years, 3 months ago

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