Saw this amazing blog post from Lior Pachter earlier today:

Unsolved Problems with the Common Core

In the post he presents many unsolved problems in math that could be used as a fun examples for kids in grade levels ranging from kindergarten to high school. It is a sensational read and I was happy to see lots of ideas that I’d not considered. As a way to show how unsolved problems can be used to talk about math with younger kids, I wanted show some projects that we’ve done that illustrate some of Pachter’s ideas:

**Kindergarten: the 4 color theorem**

Somehow or other we haven’t talked about the 4 color theorem, but I love the idea. We have two projects, though, that where coloring could come into play. One involves a neat problem about tiling octagons that Colorado math professor Richard Green wrote about on his blog:

Coloring sheets from Math Munch

Using a Richard Green post to talk geometry with my son

**1st Grade – Ramsey theory**

We’ve touched on Ramsey theory with several projects on Graham’s number. These projects require a little bit of understanding of how powers work, so maybe not 1st grade but still a fun topic:

The last 4 digits of Graham’s number

An attempt to explain Graham’s number to kids

**2nd grade – Pascal’s triangle**

I was not aware of the unsolved problem in Pachter’s post – what a cool problem! Pascal’s triangle is full of fun projects for kids. Here’s just a few of ours:

The hockey stick theorem and some fun geometry in Pascal’s triangle

Talking through Dan Anderson’s mod 2 Pascal’s Triangle

Pascal’s triangle and powers of 11

3rd grade – the Collatz conjecture!!

The Collatz conjecture is a great problem for kids to play around with. We’ve had some really fun conversations about it. Our most recent was inspired by a John Conway paper:

The Collatz conjecture and John Conway’s “amusical” variation

And here’s an old conversation with my younger son about the conjecture:

**4th grade – The Goldbach Conjecture**

We haven’t done any projects about the Goldbach conjecture, but we have done a bunch about primes and some basic number theory. Here’s a few including a neat game involving prime numbers (with the 1st one being inspired by a question from a 4th grade teacher):

A review of Prime Climb by Math for Love

**5th grade – Convex Polygons**

Here Pachter presents another problem that I was not familiar with. While we haven’t discussed this problem in particular, I do think that a discussion of polygons can be really fun for kids. Our most recent polygon project was based on the new pentagon tiling pattern discovered last month:

Using Laura Taalman’s 3d printed pentagons to talk math with kids

and for graphing points in a plane, this was a really fun exercise:

Stuart Price and Joshua Bowman’s Pi-th roots of unity exercise

and Larry Guth’s “no rectangles” problem probably fits the bill for a project involving points in a plane:

Larry Guth’s “no rectangles” problem

**6th grade – polygon nets**

For me, the starting point with polygon nets was another Laura Taalman blog post:

3D printing and platonic solids

**7th grade – an unsolved problem about runners on a track**

The problem in Pachter’s blog here is another one that I’d not seen before and it is an amazing problem. We’ve never done anything like it. I’ve never seen anything like it ðŸ™‚

**8th grade – Beal’s conjecture**

Beal’s conjecture is a great math problem for kids to play around with.

The idea Pachter is using to introduce Beal’s conjecture is exponents. I mentioned our Graham’s number projects above, but we’ve done a few others, too:

A few projects for kids from Richard Evan Schwartz’s “Really Big Numbers”

and then a neat project related to exponents inspired by a Steven Strogatz tweet:

**High school**

Pachter’s first example about Gaussian integers is related to this project:

A really neat problem that Gauss solved

For the geometry example of the Euler Brick – this fun project inspired by Patrick Honner is related:

Mr. Honner’s 13-14-15 triangle and a surprising unsolved problem

Also it this paper was recently published claiming a solution to the perfect cuboid problem (I haven’t looked into it and do not know if the paper has been reviewed):

Has the perfect cuboid problem been solved?

For the function example he discusses Euler’s totient function. We’ve recently talked about a super fun problem posed on twitter by Christopher Long that sort relates to the totient function:

Talking through Christopher Long’s neat probability problem with kids part 1

Talking through Christopher Long’s neat probability problem with kids part 2

For the modelling example, I don’t have any similar project that we’ve done.

For the geometry example, we did a different unsolved geometry problem tweeted out by Steven Strogatz. This is another really neat 3d printing project, too:

Finally, for statistics and probability, though not unsolved problems, we’ve done some fun projects such as the Christopher Long-inspired project above. I also think that geometric probability is a neat subject that most people don’t get to see:

I can’t wait to talk about geometric probability with my kids

and for a more stats-related project, I’ll pick one from my own working life:

So . . . I loved seeing Pachter’s blog post this morning and love the idea of using unsolved math problems to inspire kids of all ages. Can’t wait to try out some of the new ideas that I learned from this post!