If you are a math teacher who hasn’t taken some time to get lost in the problems on Nrich, stop reading this and go there right now. You’ll need to finish reading this post tomorrow because that’s how long you will be immersed in its seemingly endless array of engaging problems.

Today, my intention was to do a little starter activity with my 9th graders to help support their number sense.

Here’s the basis of the problem:

For two out of three of my classes, it turned into a whole-class period problem-solving extravaganza. Seriously. 30 minutes later, the brain sweat is still palpable in the room. There were so many calculators in use, I think the smartphones were starting to get jelous.

Some chose to use whiteboards, some choose numbered cards 0 – 9 while some wanted to use paper. It was so interesting to me to see them figure things out that must be true about the different number places. A few remembered the divisibility rules for 3 and shared them. Then they were able to put the divisibility rules for 2 and 3 together to get divisibility for 6. I didn’t even know that there was a divisibility rule for 4 and 8!

Some student observations:

- The 2nd, 4th, 6th, and 8th numbers need to be even.
- The last number must be 0.
- The 5th number must be five, since the last number must be 0.
- The first three numbers have to add up to a multiple of 3.
- The first 9 numbers need to add up to a multiple of 9.

I even had a student say, “How much longer do we get to play this game?” Music to my ears.

It’s difficult to give students a task that you know most of them won’t solve which is why I’ve shied away from this one in the past. I made sure to praise the efforts of those that were able to get their numbers to work for *all* except *one* of the digits. (For example, their 2, 3, 4, 5, 6, 8, 9, and 10 digit numbers worked, but their 7 digit number didn’t).

Nrich gives another variation on this task by making it a game. Basically, students take turns creating 1, 2, 3…digit numbers by choosing from the 0 – 9 digit cards until someone can’t use any more of the cards. I think having them play this activity as a game would help alleviate some of the discontent of feeling like this problem was too difficult to solve.