# One that didn’t go so well

Last week my old son had a nice challenge problem in his weekend math program:

An 8x8x8 cube is painted black out the outside and then chopped into 1x1x1 cubes. How many of the 1x1x1 cubes have paint on 0, 1, 2, 3, 4, 5, and 6 faces?

This problem was even accessible to my younger son, in fact, we started the project to by discussing that problem:

So, for today’s project I thought it would be interesting to try out a similar problem in 4 dimensions.

It turned out to be far more confusing to the boys that I was expecting. So, whoops, I guess. I’ll go through this project again with them, but using a very different approach. But, for now, here are the 4 long discussions we had about how to attack this problem.

Mostly this post is a reminder to me that not everything goes as well as you’d hoped it would:

# Talking through Matt Enlow’s “truth” tweet with my older son

Saw this really neat tweet from Matt Enlow earlier today:

I thought it would be fun to talk through the list with each of my kids and ask the for examples of each on of the statements. My younger son was home this afternoon so he went first – the project with him is here:

Talking through Matt Enlow’s “truth” tweet with my younger son

Tonight I worked through the 7 parts with my older son

(1) The statement is true, here is a proof:

His ideas involved the area of a triangle and the Pythagorean theorem

(2) I believe the statement is true, here’s why

It took him a while to come up with something, but eventually he mentioned the quadratic formula, which seems like a great example:

(3) My gut tells me that this is true

Here he picked a postulate from Euclidean geometry – a single line passes through any two given points.

(4) I have no opinion as to whether or not the statement is true or false

OMG OMG OMG OMG – I’m not even going to give it away. The best!!

(5) My gut tells me the statement is false

He had a really hard time coming up with an example here.

Eventually he came up with a really interesting example from the quadratic formula – the quadratic equation $x^2 + 24x - 1 = 0$ has integer roots.

(6) I believe the statement is false – here’s why

He came up with a nice example here – a cube has integer sides and volume 7.

(7) The statement is false – here is a counter example.

He came up with a simple example first -> 1 + 1 = 7. I told him that was too cheap and asked for a second example.

The second example was $|x| = -2$. He also gave a really nice explanation.

So, a really fun project – as I said in my younger son’s project, I’d love to see lots and lots of kids come up with examples for each of Matt’s 7 statements.