Math Kangaroo Problem About a Cube Made of White and Black Blocks

This 5-point problem was offered to third and fourth graders on one of the Math Kangaroo tests.

Out of 125 small, white and black cubes, the big cube was formed (see the picture). Every two adjacent cubes have different colors. The vertices of the big cube are black. How many white cubes does the big cube contain?

Solution

Let's examine the front face of the big cube.

We can count 13 black and 12 white small cubes. Let's now imagine removing the front layer of blocks. This will reveals the layer behind it. (Behind each black small cube, there is a white one. Behind each white small cube, there is a black one.)

So the second layer has 12 black and 13 white small cubes. If we keep peeling one layer at a time revealing the layer behind it, we going to find out that the third and the fifth layer look exactly like the first one. Similarly, the fourth layer is an exact copy of the second layer.

Here is what we've got:

	Number of black cubes	Number of white cubes
Layer 1	13	12
Layer 2	12	13
Layer 3	13	12
Layer 4	12	13
Layer 5	13	12

The total number of small white cubes is $12 \times 3 + 13 \times 2 = 36 + 26 = 62$.