Last week Tim Gowers shared a great math problem on Twitter – here’s my retweet of it (again to help avoid the temptation to get hints in the original thread:

This is a really great math problem shared yesterday by @wtgowers. I'm sharing a pic rather than retweeting to take away the temptation to go to the original thread for hints. You'll know with 100% certainty when you've got it, but there's no rush – just ponder for a while! pic.twitter.com/F2rRp8wH7P

If you’ve not seen the problem before I’d definitely suggest spending some time thinking about it – it is really a terrific problem. The videos below give the solution, so fair warnng . . .

I’d talked about it a bit with my younger son on a car ride back and for to his (outdoor) karate class earlier this week. My older son hadn’t seen the problem until this morning. A discovery that my younger son had made in the car earlier in the week helped the boys solve the problem today, but even with that prior discovery the discussion was still really great.

Here’s how I introduced the problem – you’ll see that some of the elements in the statement of the problem that are pretty standard for mathematicians are a little confusing to the boys. This introduction clears up a bit of the confusion:

With the definition of a “repetitive” number now clear, we checked if 1/7th was a “repetitive” number – the boys were pretty sure that it was, though explaining exactly why that was true in a 100% precise way was a little challenging:

Now my younger son gave his explanation for why he thought was repetitive:

At the end of the last video the boys were starting to think that all numbers were repetitive. In this last video they finished the solution to the problem:

I really like this problem and think it is a great way for kids to have a fun – and non-computational – mathematical exploration. As the videos show, some of the ideas can be a little difficult for kids to make precise, but I think that’s just another nice reason to explore this problem with them!