


Problem Source: Found at http://www.oise.utoronto.ca/~ggay/styl3121.htm. Page no longer accessible. Solution presentation copyright Howard C. McAllister, 1998.
A painted wooden cube, such as a child's block, is cut into twenty seven equal pieces. First the saw takes two parallel and vertical cuts through the cube, dividing it into equal thirds; then it takes two additional vertical cuts at 90 degrees to the first ones, dividing the cube into equal ninths. Finally, it takes two parallel and horizontal cuts through the cube, dividing it into twenty seven cubes. How many of these small cubes are painted on three sides? On two sides? On one side? How many cubes are unpainted?
SOLUTION: Drawing the cube showing the saw cut lines we can readily count the number of colored surfaces on each of the small cubes composing the front section of the large cube. The back section of the cube is no different than the front section so the number of colored surfaces on each of the small cubes is the same as at the front section. This leaves the middle section. Isolating this so we can look at it gives immediately the number of colored surfaces on each of the small cubes composing the middle section. The numbers on the individual small cubes denote the number of colored surfaces on the individual small cubes.
From the drawing there are 8 cubes with 3 colored sides, 12 cubes with 2 colored sides, 6 cubes with 1 colored side and 1 cube with no colored sides. We note that 8 + 12 + 6 +1 = 27. This is reassuring.
The analysis could just as well be done by using the sections topmiddlebottom or leftmiddleright. Note that the symmetry of the cube provides and automatic check on the validity of the number of colored faces.