Math Forum - Project of the Month, November 1996

A Math Forum Project

We have two winners this month and five (!) honorable mention. This is a neat problem because I knew that a lot of people would get it right, and I would get to concentrate on the explanations. That's exactly how it worked out.

Here's how I scored the solutions this month. There were four parts to the problem, and a point was given for each right equation, and a point for each good explanation. There was also a point given for pointing out any special cases or exceptions, and another point could be earned by writing something exceptional.

Five people scored a straight 8 points - they got all four equations and provided good explanations. These were the folks who won honorable mention. It is a thin line between honorable mention and the winners, but the winners did go just one step further.

Both the winners pointed out not only the four equations and provided good explanations, but they also pointed out that this only works if n is greater than 1. A small point, but it's important to look at all the parts of a problem and see if there is anything special going on with any of the cases.

There were a number of people who figured out the number of cubes of each sort for different size cubes, and then made a table and looked for patterns and figured out formulas for the different parts. That's okay, and most of those folks got the right answers, but this is a problem that can be attacked from a visual standpoint. There are always 8 cubes with three painted sides not because the tables of numbers said so, but because every cube has eight vertices.

Same thing goes for the other parts. The number of cubes with no painted faces is a little cube in the middle. That makes more sense from a visual standpoint than the way some people solved this part - they took n^3, which is the size of the whole thing, and subtracted their answers for the other parts. That answer is no less right, but it doesn't describe what's going on if you're simply looking at a cube. Just something to think about when you are solving problems like this.

Here are the winners, along with some of my comments. The honorable mentions will follow in a separate message, also with comments, along with a list of the students who also got all the right equations, but didn't provide as much, if any, explanation.