Can someone explain the math behind how many combinations lego blocks can be assembled into?

Question

If you have two 8 stud lego blocks they can be assembled 24 ways.
If you have three they can be assembled 1,560 ways
If you have four they can be assembled 119,580 ways
If you have five they can be assembled 10,116,403 ways

I just can't figure it out.

Actually it is part of a wiki article but at one point they say 6 blocks can be arranged 915,103,765 and later they say 915,103,766 and I was wondering if there was some sort of mathematical formula behind it.

But I couldn't figure it out.

And if they don't count block A on top of block B as different than block B being on top of block A that makes it much more difficult.

Then do you count block A on block B's left three nubs as the same as block A on block B's right three numbs or are they the same because they are a simple rotation?

*Sigh* I guess as I do not have all the details as to what they count it will be very hard to figure out the answer.

Answer 1

The only real way to get to this is to enumerate the possibilities. I wouldn't recommend doing it by hand however. The attached links have some details on how several people have written a couple of independent computer programs to figure out and double-check the different number of combinations.

Note: from what I can see there are some assumptions in the counting. For example, you say there are 24 ways to combine two 2x4 bricks. This actually assumes that you don't consider the bricks to be unique. In other words, if I stack brick 1 directly on top of brick 2, that is the same as if I stack brick 2 on top of brick 1. The resulting shape is the same. So for counting purposes, all the bricks are considered identical.

You can see how they arrived at 24 ways for 2 bricks. They ended up with 46 combinations, but all but 2 of these were exact duplicates of each other, when rotated 180 degrees. So that is 2 + 44/2 = 24 ways.

It gets more complicated from there... if you read through the second link, they go into a lot of the math. The gist of the paper is essentially that the number grows exponentially and that there doesn't seem to be a easy way to calculate the next number based on the prior number, given all the combinations of rotations, symmetries, etc.

Answer 2

I have been thinking about, and here is what I have so far



The number of surfaces keep increasing as you add each block
With 2 blocks you have 1 surface

With 3 blocks you can have 2 & 3 surfaces to work with
[e.g. if the two blocks are stacked on top of each other there is a bottom surface and top surface for the 3rd block to attach too, each of these surfaces can have the 3rd block added in 24 different ways.

Now if the first two blocks are assembled staggered, there are 23 (24-1 the 1 being where they are stacked on each other) different configurations that you can add the third block to the assembly, and there are 3 surface areas where the block can be added, bottom (24 combinations), top (24 combinations), and whatever studs are exposed in the middle layer (with for me an unkown # of combinations of adding the 3rd block)]

With four blocks, I think there will be 2,3,4,5,6 possible surface areas depending whether the first three blocks are assembled on top of each other, or are staggered (the 2nd block can be added two ways to the first, directly on top, or staggered, the 3rd block can be added two ways, directly on top of 2nd block or staggered, and again we have the bottom and top surface with their combinations

good luck with it

Answer 3

That is definitely a toughy! You should be able to count the two block combos.

It's also not clear how symmetry might be involved.