SOLUTION: Pls help me with this qn: One thousand cubes are arranged to form a large cube. All the surfaces of the large cubes are then painted. What fraction of the cubes are not painted at

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Question 117081: Pls help me with this qn: One thousand cubes are arranged to form a large cube. All the surfaces of the large cubes are then painted. What fraction of the cubes are not painted at all?
Answer by checkley71(8403) About Me  (Show Source):
You can put this solution on YOUR website!
WHAT YOU HAVE HERE IS AN INTERESTING PUZZLE.
1000 CUBES FORM A 10x10x10 LARGE CUBE.
THEREFORE THERE ARE 100 CUBES FORMING THE TOP & THE BOTTOM.
THIS ACOUNTS FOR 200 PAINTED CUBES.
NOW FOR THE 4 SIDES WE HAVE ALREADY ACCOUNTED FOR 20 (10 AT THE TOP ROW & ANOTHER 10 ON THE BOTTOM ROW) THAT LEAVES US WITH 80 ON THE 4 SIDES.
HOWEVER THERE ARE 8 EDGE CUBES AS PART OF THE ADJOINING SIDE.
NOW WE HAVE TO SUBTRACT 8 FROM THE 80=72 FOR EACH SIDE BECAUSE OF THE OVERLAPPINGCORNERS.
THUS WE SHOULD NOW HAVE A TOTAL OF 2*100+4*72=488 CUBES THAT HAVE AT LEAST ONE SIDE PAINTED.
THIS LEAVES 1000-488=512 CUBES UNPAINTED.
OR 512/1000=51.2% UNPAINTED CUBES.
WISH I COULD DEVELOPE A SIMPLE FORMULA FOR THIS TYPE OF PROBLEM BUT I'M A PICTORIAL TYPE OF MATHEMATICIAN AND FIND A DIAGRAM MOST EFFECTIVE.
I JUST HAD A BRAIN STORM.
RATHER THAN WORKING WITH THE PAINTED CUBES LET'S LOOK AT THE UNPAINTED CUBES.
THE INNER CORE THAT ISN'T PAINTED MEASURES 8x8x8=512.
IMMAGINE THAT 51.2% ARE CORE CUBES & UNPAINTED.
SHOULD HAVE THOUGHT OF THIS APPROACH FIRST.
OH WELL WHEN YOU'RE WORKING A NEW FIELD YOU DON'T ALWAYS FIND THE EASIEST SOLUTION ON THE FIRST TRY. THIS IS A PRIME EXAMPLE.