Because of some unlucky travel issues we didn’t get a chance to talk about it until today. Our original talk through the problem happened this morning and the boys had a tough time with it. The whole conversation this morning probably took close to 45 minutes.

To reinforce some of the ideas from this morning I thought it would be useful to go through the problem again with them tonight. This is a tough problem, but it gives plenty of great opportunities to talk about both counting ideas and geometric ideas with kids.

We started the project tonight by just talking about the problem again. This morning the boys eventually settled on an approach that involves looking at lines on the surface of the shape, and we took that approach again tonight with snap cubes:

In the first part the boys counted 12 cubes that were “cut” by lines passing through them. Now we counted other cubes that were cut by the plane. I’m not sure we got the count right the first time (though I didn’t quite get what they got wrong, I just felt that something had gone astray), so we actually went through the whole counting process twice. There’s a lot to keep track of in this problem!

Next I showed them the short program in Mathematica that I wrote during the day. Having a 3D picture with piece sliced out but with grid lines makes counting the sliced cubes much easier. Having this picture of the sliced box on the screen gave me the idea to have them count the sliced cubes in a different way. This led to my older son noticing a few other geometric ideas in the picture:

Next, I showed them the bit that was sliced off. Despite one little inaccuracy in the way the picture was drawn (which may not even show up in the video) we were able to see the 18 sliced cubes again.

To wrap up I showed them that I’d 3D printed these two pieces and left those pieces for them to play around with.

So, a tough project but a fun project. I was surprised how difficult counting these sliced cubes was, but despite the difficult I’m glad we went through this problem because it has so many great mathematical ideas hiding in it.

One thought on “A neat counting problem shared by Five Triangles”

I couldn’t come up with a good way to count the cut cubes and it really bothered me. Watching your videos, though, i came up with an approach that you might like, if you’re interested in a quick revisit with the boys….

Basically, keep following the idea of drawing on the outside of the snap cubes the lines where the plane intersects, but do it by layer. Also, draw the line of intersection on the top of the layers and the bottom.

For me, this made it much easier to see the following:
(1) the bottom layer has 2 cubes cut, visible with the intersection on the top face of that layer
(2) the second layer has 4 cubes cut visible with the intersection on the top face. Also, it has 2 cubes cut, visible on the bottom face. These 2 correspond to the cubes we could see that were cut in the first layer.
(3) top layer, top face shows 6 cubes cut. Top layer, bottom face shows 4 more cubes cut; again, these correspond to the cubes we could see were cut when we looked at the middle layer, top face.

I couldn’t come up with a good way to count the cut cubes and it really bothered me. Watching your videos, though, i came up with an approach that you might like, if you’re interested in a quick revisit with the boys….

Basically, keep following the idea of drawing on the outside of the snap cubes the lines where the plane intersects, but do it by layer. Also, draw the line of intersection on the top of the layers and the bottom.

For me, this made it much easier to see the following:

(1) the bottom layer has 2 cubes cut, visible with the intersection on the top face of that layer

(2) the second layer has 4 cubes cut visible with the intersection on the top face. Also, it has 2 cubes cut, visible on the bottom face. These 2 correspond to the cubes we could see that were cut in the first layer.

(3) top layer, top face shows 6 cubes cut. Top layer, bottom face shows 4 more cubes cut; again, these correspond to the cubes we could see were cut when we looked at the middle layer, top face.