Yesterday Matt Enlow shared a list of his 100 favorite problems:
I flipped through the problems yesterday and problem #6 struck me as a terrific one to share with kids:
I mentioned the problem to the boys yesterday and today we dove into it. Here are their initial thoughts:
Next I asked them to see if they could say anything at all about what would have to be true if there were powers of 2 and 3 that met the conditions of the problem.
My older son noticed a pattern in the powers of 2 mod 3. That helped us understand some basic ideas about what would have to be true if powers of 2 and 3 differed by 1. We then moved on from that idea to see how the “difference of squares” idea from algebra could help us show that the equation in the problem would probably never be true for an even power of 2 greater than 4. Nice start – now we just had to get to the finish line:
The idea that we were missing at the end of the last video was that powers of 3 only had 3 as a prime factor. Once the boys noticed that, they were able to see that an even power of 2 could never satisfy the equation!
Now we had to look at odd powers of 2. They noticed that roughly the same idea works if the power of 3 was even. There was one little subtle difference in the argument, but luckily both boys were able to explain that bit!
Now we had to look at the case with odd powers of 3 and odd powers of 2. Here I showed them how polynomials like factor. I also shows how the numbers of the form factor when n is odd.
The interesting idea here was that the factorization was always a 2 and an odd number. That showed the product could never be a power of 2. It took a while for us to get to that via the polynomial factoring, but we did get there.
Which then solved the whole problem!
Finally – just to wrap things up, I went to the computer to find powers of 2 and 3 that were “close” together using continued fractions:
I was lucky to see Matt Enlow’s list of problems on twitter yesterday. It is going to be a great resource for me – can’t wait to share more of the problems with the boys.