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Topic: Brain teasers
Replies: 2   Last Post: Jan 20, 1998 10:28 PM

 Messages: [ Previous | Next ]
 Alan Lipp Posts: 41 Registered: 12/6/04
Re: Brain teasers--SPOILER
Posted: Jan 16, 1998 1:56 PM

On Fri, 16 Jan 1998, Lilly Sun wrote:

> Some unit cubes are put together and form a larger cube, and then
> some of the faces of this cube are painted. After the paint dries,
> then the larger cube is disassembled into the unit cubes and there are
> 24 of these have no paint on any of their faces.
> Question: How many faces of the larger cube were painted?

If all the outer cubes of a 5x5x5 were removed there would be 27 unpainted
cubes inside . . . so clearly the large cube has side less than 5. If
none of the faces of a 2x2x2 cube were painted then only 8 cubes would be
unpainted. So the large cube has either side 4 or side 3. If even one
side of a 3x3x3 were painted then only 18 unpainted cubes would remain.

Therefore, some faces of a 4x4x4 cube were painted.

A 4x4x4 has a core of 8 unpainted cubes with 56 cubes lying on the
surface and we must color all but 16 of them. So the problem is: how to
paint 40 of the face cubes of a 4x4x4 cube.

The solution: paint any three faces which do not meet at a corner.

Alan

Date Subject Author
1/16/98 Lilly Sun
1/16/98 Alan Lipp
1/20/98 Joshua Zucker