My son is in a weekend enrichment math program and that program has been great for him. It comes to an end this week. The last problem on this week’s homework assignment gave him some trouble, so I thought it would be fun to see if we could work through it together.

I was a little worried because I’d not seen the problem until just before the project, but luckily things went ok.

Here’s the problem:

a, b, c, and d are positive integers less than 10. How many solutions are there to the equation a + bcd = ab + cd?

[post publication note: Originally the text presented the problem incorrectly. It is correct in the videos. Karen Carlson pointed out the typo to me. Sorry about that.]

Here’s how we got started – my son had found several cases, but not quite all of them:

After the introduction to the problem and my son’s work so far, we moved on to try to find more solutions. The main idea I gave my son involved writing the equation in a slightly different form:

Now that we had a plan, we moved on to counting the rest of the cases that we found in the last video:

Finally, we went to Mathematica to write a little program to count the solutions for us. This part of our project turned out to be more interesting than I was expecting. It was interesting to compare the brute force solution of the computer to the case by case counting technique that we’d just gone through.

So, a fun problem that definitely made my son think this week. It is