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A wooden block is 4 inches long, 4 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into sixteen 1 inch cubes. How many of the cubes each have a total number of red faces that is an even number?

Jun 16, 2023

#1
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The block has a total of 24 faces, and each cube has 6 faces. If a cube has an even number of red faces, then it must have 2, 4, or 6 red faces. There are 8 cubes with 2 red faces, 4 cubes with 4 red faces, and 4 cubes with 6 red faces. Therefore, there are 16 cubes with an even number of red faces.

Jun 16, 2023
#2
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A wooden block is 4 inches long, 4 inches wide, and 1 inch high. The block is painted red on all six sides and then cut into sixteen 1 inch cubes. How many of the cubes each have a total number of red faces that is an even number?

Draw a 4 x 4 grid on a sheet of paper, as a guide.

You can mentally visualize the third dimension.

Here's what I count:

The 4 corner blocks                      4 red sides each

The 8 outside edge blocks

excluding the corner blocks          3 red sides each

The 4 inside blocks                       2 red sides each

So, it looks like the cubes that have an even number

of red sides are the 4 on the corners and the 4 inside.

That comes to a total of 8 cubes that have an even number of red sides.

No cube can ever have all 6 sides red, because the side(s) where a

cube was butted up against another cube wouldn't have got painted.

.

Jun 16, 2023