Math Is Fun Forum

rishabhsaxena

Member

Registered: 2009-11-25

Posts: 4

A tricky cube puzzle

Dear friends,

Kindly help me out with this puzzle

I have 27 identical cubes which are joined to form 1 single cube (3X3X3). Now my task is to start from any corner cube and reach the center cube such that I cross all cubes only once and also I cannot leave the larger cube.

I tried to solve it but every time I reach the center cube I find that one small cube has been left out.

Any suggestions or solutions are welcomed.

Thanx in advance

Rishabh Saxena

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: A tricky cube puzzle

I don't understand if you are allowed to just cross facets or squares like a bishop in chess.
Can you move like a rook in chess, or just diagonally around the faces?  Can you
go inside the little 27 cubes to the opposite corner?

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igloo myrtilles fourmis

rishabhsaxena

Member

Registered: 2009-11-25

Posts: 4

Re: A tricky cube puzzle

@Franklin: You are allowed to move across faces only i.e. like a rook........hope that answers ur question

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: A tricky cube puzzle

@rishabhsaxena
  Do you start in the middle of a face and move to the middle of another face on each move?

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igloo myrtilles fourmis

rishabhsaxena

Member

Registered: 2009-11-25

Posts: 4

Re: A tricky cube puzzle

ya u got dat right...:)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: A tricky cube puzzle

This can be proven impossible with a parity argument.

Imagine the cubes were painted black and white, like a 3D chessboard.
Each face of the cube would have a black X on a white background, and the centre would be white.

Counting them all up, the big cube is made up of 14 black cubes and 13 whites.

The path we want starts on a black cube and ends on a white, but since the path must change colour with each step, it must be made up of equal numbers of black and white. Therefore, one black cube will always get left out.

---

Why did the vector cross the road?
It wanted to be normal.

rishabhsaxena

Member

Registered: 2009-11-25

Posts: 4

Re: A tricky cube puzzle

thanx a lot mathsyperson...........the shortness of the method is stupendous..thanx a lot

soroban

Member

Registered: 2007-03-09

Posts: 452

Re: A tricky cube puzzle

Hello, rishabhsaxena!

I have 27 identical cubes which are joined to form 1 single cube (3x3x3).
Now my task is to start from any corner cube and reach the center cube
so that I cross all cubes only once and also I cannot leave the larger cube.

I tried to solve it, but every time I reach the center cube,
I find that one small cube has been left out.
This will always happen!

Paint 14 cubes black and 13 cubes white.
Assemble the large cube in "checkerboard" fashion.

            *-----*-----*-----*
           /:::::/     /:::::/|
          * - - * - - * - - *:|
         /     /:::::/     /|:*
        * - - * - - * - - * |/|
       /:::::/     /:::::/| * |
      *-----*-----*-----*:|/| *
      |:::::|     |:::::|:*:|/|
      |:::::|     |:::::|/|:*:|
      * - - * - - * - - * |/|:*
      |     |:::::|     | * |/
      |     |:::::|     |/| *
      * - - * - - * - - *:|/
      |:::::|     |:::::|:*
      |:::::|     |:::::|/
      *-----*-----*-----*

The three levels of the cube look like this:

           Top             Middle            Bottom
      *---*---*---*     *---*---*---*     *---*---*---*
      |:::|   |:::|     |   |:::|   |     |:::|   |:::|
      * - * - * - *     * - * - * - *     * - * - * - *
      |   |:::|   |     |:::|   |:::|     |   |:::|   |
      * - * - * - *     * - * - * - *     * - * - * - *
      |:::|   |:::|     |   |:::|   |     |:::|   |:::|
      *---*---*---*     *---*---*---*     *---*---*---*

Note that the centermost cube is White.

We start in a Black corner cube and (I assume)
we move orthogonally through the large cube.

Passing through each cube once,
we will pass through cubes of alternating color:
. . Black - White - Black - White - Black . . . etc.

Since there are 14 Black cubes and 13 White cubes,
. . our tour will end on a Black cube.

But the center cube is White.

Therefore, our tour cannot end at the center cube.

Edit: Darn . . . too slow again!

.

Last edited by soroban (2009-11-26 08:36:39)