An interesting counting problem from an old AMC 10

My older son and I were driving down from Boston yesterday. Instead of our usual math lesson, I had him pick an old AMC 10 to talk through on the way down. No pencil or paper (we had to skip a couple of geometry problems) – just talking through how to approach the problems.

Problem #13 led to a great conversation:

Problem 13 from the 2005 AMC 10 b

Here’s the problem:

“How many numbers between 1 and 2005 are integer multiples of 3 or 4 but not 12?

This morning I had my son talk through his solution. His presentation here is obviously much more of a straight line to the answer than our conversation in the car, but I thought it was still interesting to hear his explanation:

I enjoyed talking through the problem so much that I decided to give it a try with my younger son, too. It goes without saying that he’s not the intended audience for this problem, but talking through it was still super fun. It is so great to hear his ideas.

He wants to break the problem into an easier problem, but his first instinct – looking at the integers from 1 to 10 rather than the integers from 1 to 2005 – doesn’t give a ton of insight into the problem. However, he’s on the right track – looking at the integers from 1 to 12 does help.

Our conversation here was interrupted by what sounded like an 18 wheeler coming down our driveway – still don’t know what the noise was.

After the mystery noise was gone, we picked up where we left off – looking at how we could chop up the problem into groups of 12. Once he understood this idea, getting to the end of the problem wasn’t too hard for him. What I was really happy about was that his solution used a completely different approach that his brother’s solution.

So, I’m just realizing now that I never showed my older son my younger son’s solution. I thought it would be nice to finish up the conversation with my younger son by showing him the other solution, though. If nothing else, the solution provides a great opportunity to get in some arithmetic practice.

So, a fun problem and a great conversation with both kids. Always love finding a great problem like this one.