Here is yet another domino variation. Below are the eight 3-binominoes:
You place them sideways so that colors of adjacent squares match. This is, alas, impossible, unless you use two identical binominoes. Therefore we are generous and allow the binominoes to be shifted up or down by one square, so that they have contact only along two squares. A chain using all eight might look like this:
The connectivity graph is surprisingly complicated. I have drawn the directed version, the target of an edge connects to the source by sliding it down one square.
A simple game for two players divides the eight binominoes in two sets of four. Each player gets one set, and they take turns placing binominoes in a chain so they match along two squares. If a player can’t play (either because their binominoes don’t fit, or because they are out of them), they have to take one from either end of the chain. The goal is for both players to finish simultaneously.
Above is a circular version with 4-binominoes, ragged so that not only the colors have to match, but the notches as well. I hope this looks appealing. The same rules apply, but now the goal is to create a closed chain, like so:
Here this is even done so that the binominoes are always shifted the same way. In other words, this solution represents a Hamiltonian cycle in the directed connectivity graph. I believe this graph is always Hamiltonian, for n-binominoes with arbitrary n.