# Combinatorial rule for a stable stack of bricks.

Suppose that you have a stack of identical, frictionless, uniform-density $$1 \times 2$$ bricks arranged in the ordinary configuration (each row is offset by $$1$$ relative to the row below.)

# Question

Is there a combinatorial rule that captures whether or not a given stack of bricks would be stable? In particular, that no small vertical force will cause the stack to move.

In other words, how can one determine whether or not a configuration is stable without doing a static force analysis.

(Admittedly, this question is a little hand-wavey, so let me know if I can clarify anything.)

# Examples

Clearly, a stack of bricks that does not have any "overhangs" should be stable:

But we can also allow a cantilever if there's a brick above:

# Non-examples

However, having a brick above is not sufficient, because a stack like this should not be stable.

Similarly, having a brick above is necessary because a configuration like this is in unstable equilibrium; in particular the upper-right brick in this example would fall if an arbitrarily small force were applied to the right side.

• The best I can do is enumerate total brick stacks of $n$ bricks, both stable and unstable. Surprisingly, the result is quite simple considering the difficulty of the problem. There are $3^{n-1}$ total brick stacks. Apr 11, 2018 at 1:20
• On the Code Golf Stack Exchange, there was a related puzzle—but it didn't obey Newtonian mechanics. codegolf.stackexchange.com/q/38548/53884 Apr 11, 2018 at 15:49
• @N.Shales, I enumerated a few cases by hand, and I believe you—but how do you prove this? May 4, 2018 at 20:57
• It is possible to construct a 'brick tower' from irreducible parts. The sequence of these parts can be formed using a generating function. It's quite difficult to explain without diagrams. The irreducible parts are called "pyramids" and "half-pyramids". A pyramid has a base of 1 brick and bricks on top to the left and right, half-pyramids only have bricks on top to the right. A 'brick tower' is always formed by 1 pyramid on the left and any number of subsequent half-pyramids successively to the right. May 4, 2018 at 22:32
• This OEIS sequence and the paper it links to, seem relevant here. Balanced configurations (let alone stable ones) are trickier than one might think. E.g. I was surprised that diamond configurations are only balanced if the number of bricks in the largest row is $\lt 5$ (see page 3 in the paper).
– Jens
May 6, 2018 at 0:25