Here's a study I started on my own well over 25 years ago, but life got in the way. Essentially, I was interested in figuring out a mathematical description for how many ways cubical blocks could be attached to each other, where each block had at least one face attached to another. That is, to predict how many ways for each value of the number of cubes. So, no case where only an edge was the contact element between any two blocks. I was not interested in cases where there was symmetry (i.e., by rotation of the assembly or translation). For example, two blocks can be attached on any one of six faces, each. Boring- it is still just two blocks stuck together! I worked this most of the way through 6 blocks (it's a lot of shapes!) but life got in the way and it sat on a back burner. Then I lost the file containing the configurations. I recently found the file with all the pictures. I'll post page 1 of the set of drawings that I made to represent what I was interested in-
![[Image: ywReIhW.png]](https://i.imgur.com/ywReIhW.png)
One block has only one configuration
Two blocks have only one configuration
Three blocks have two configurations
Four blocks have eight configurations. That's an additional two pages of pictures that I didn't want to splatter this thread with, unless there was some interest or insight. I know how to do this with geometry for the number of line segments that connect some set of points in a plane, for example, because I've done that (made a proof of the number of line segments) when teaching high school geometry. For all I know, someone has already figured it out. After all, it's been over 25 years since I started looking at it.
BTW, as near as I can tell, 5 blocks have 27 configurations. Maybe there is a cubic equation in there somewhere- the count is
1, 1, 2, 8, 27... I don't think I've finished with the number of ways 6 blocks can be attached with the stated conditions. I made an earlier stab at using all rotational configurations, but it got out of hand pretty quickly, so I dropped that line of inquiry.
One block has only one configuration
Two blocks have only one configuration
Three blocks have two configurations
Four blocks have eight configurations. That's an additional two pages of pictures that I didn't want to splatter this thread with, unless there was some interest or insight. I know how to do this with geometry for the number of line segments that connect some set of points in a plane, for example, because I've done that (made a proof of the number of line segments) when teaching high school geometry. For all I know, someone has already figured it out. After all, it's been over 25 years since I started looking at it.
BTW, as near as I can tell, 5 blocks have 27 configurations. Maybe there is a cubic equation in there somewhere- the count is
1, 1, 2, 8, 27... I don't think I've finished with the number of ways 6 blocks can be attached with the stated conditions. I made an earlier stab at using all rotational configurations, but it got out of hand pretty quickly, so I dropped that line of inquiry.
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