When that same question came up in my class, some of my students could visualize it, others were more "hands on" types. The night before I went down to my hobby woodshop and actually cut out a 10cm by 10cm by 10cm cube, found the midpoints the way you described and cut of the corners in the way you suggested.
(I put little dowels in so I could re-assemble the pieces)
The other students in my class were then able to see and understand the result
Can you visualize what will happen ?
Hint: consider what the results are on one of the sides.
My other part of the question was,
"What was the volume of the original cube, and the new shape?"
It sounds like it would look like a hexagon.
A hexagon is a 2-dimensional figure.
Would you not end up with a 3-D solid ?
Hint: See if you can a piece of styrofoam, it would be easy to see the results. It doesn't have to be precise.
Hint#2: Look at a die (one die, two dice)
Most dice used in games have their vertices slightly rounded. Isn't that the beginning of our cube process, except the "rounding" continues until we reach the midpoints.
Hint#3: What shape do you see at each of the original vertices?
What shape do you on each of the original sides ?
Ok, I found a box and cut off the corners as stated, but it looks like an octagon now. (and not a very good one) Am I missing something?