A neat problem from 5 Triangles and Dave Radcliffe

Saw a neat exchange on twitter last night:

Though my kids are far too young (4th and 6th grade) to find this solution on their own, I thought that going through this solution with them would be a useful and fun exercise. Each step in the solution is something that they can understand and Dave’s approach is also a great lesson in problem solving.

So, instead of our typical morning projects this morning we talked through this problem.

Here’s the introduction to the problem and a few initial thoughts. Right off the bat the kids have some nice thoughts about prime numbers:

The boys had some good thoughts about simplifying the problem in the last section. We looked at a few other simple examples – 3 consecutive integers in which two are prime (this led to a nice discussion about twin primes).

Next we moved to the computer to take a look at Dave Radcliffe’s idea. Luckily Mathematica has a function – PrimePi[n] – that counts primes less than or equal to n. We wrote a little program to count the number of primes occurring in a list of 2015 consecutive integers.

In this part of the project we began to use this program to explore the number of primes in various lists of 2015 consecutive integers.

At the end of the last section my younger son noticed the point that Dave Radcliffe had made last night -> as you go down the list the number of primes changes by +1, 0, or -1, but no other number.

In this part of the project we discussed why the changes were never greater than 1 and also how this property might help us solve the original question.

Finally we discussed how we could find a long list of consecutive integers with no primes. Unfortunately we were running a little longer than I expected so this part was a little more rushed than I would have liked. Still, though, they seemed to understand the idea.

So, I think walking through this problem with kids is a fantastic exercise. There are lots of interesting mathematical ideas from arithmetic and from number theory kids might find fascinating. Also, the idea that the proof shows a list of 2015 integers with exactly 15 primes exists without actually finding it is also an amazing idea (and likely one that is brand new to kids).

Finally, Dave Radcliffe’s idea to look at lists of 2015 consecutive integers to see how many primes they have is a fantastic problem solving idea. Seeing how a simple idea like Dave’s changes an almost unapproachable problem into a one that is now much easier to understand is an important example for kids to see.

Definitely a fun project.