LEGO Mathematics

Figure 1 shows a 1 x 4 LEGO brick.

Figure 1

It turns out that if you have another three of these bricks and you stack them in every possible way on top of this base brick, then there are 27591 ways that you can do this. Today I'm 27591 days old and this number appeared in OEIS A123782:

A123782: number of ways to build a contiguous building with n LEGO blocks of size 1 x 4 on top of a fixed block of the same size. The base block is not counted among the \(n\) and must be the only block in the bottom layer of the building

The sequence begins 23, 679, 27591, 1227556, 59212274, 2990304700, 156407426393, ...

So there are 23 ways to stack a single 1 x 4 brick on top of another 1 x 4 brick. It took me a little sketching and visualisation but there are indeed 23 ways to do it. One way is to place the brick directly on top, then there are six more ways to do it with the long axes parallel and then sixteen ways to do with the long axes at right angles. Rotational symmetries are ignored here and only the strict physical orientation of the bricks are considered. Thus we have 1 + 6 + 16 = 23. So the famous number 23 pops up even in the context of LEGO.

There are other OEIS sequences relating to LEGO blocks and they are: 

• A007575: number of stable towers of 2 X 2 LEGO blocks.
  
  
• A007576: number of maximally stable towers of 2 X 2 LEGO blocks.
  
  
• A112389: Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 2 X 4.
  
  
• A112390: number of ways of building a building of height k using n 2 X 4 LEGO blocks, counted up to symmetry.
  
  
• A123762: number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 2
  
  ... right through to 
  
  A123827: number of ways to build a contiguous building with n LEGO blocks of size 2 X 3 on top of a fixed block of the same size so that the building is symmetric after a rotation by 180 degrees.
  
  
• A123829: number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 2x4 which is symmetric after a rotation by 180 degrees
  
  ... right through to
  
  
• A123849: number of ways to build a contiguous building with n LEGO blocks of size 5 X 5 on top of a fixed block of the same size so that the building is symmetric after a rotation by 90 degrees.
  
  
• A272690: number of buildings with n 2 X 4 LEGO bricks of maximal height.