# Grow, Contact, and Trinity (Cooperation Games VII)

Let’s bring back a bit of color. Today we get a trio of games, played with similar cards and similar mechanics. The simplest version I call Grow, and it’s a game for two players One and Two, to be played with triangular cards on a triangular board. The pieces are called (top to bottom) root, branch, and twig.

We will need a single root-piece, 12 twigs for player One, and 12 branches for player Two. We begin by placing the root somewhere on the triangular board. Then we take turns by adding twigs and branches so that adjacent edges match and we have a connected growing tree.

There are two more rules: We don’t allow loops, and the branches must branch away from the root. The layout above on the left is still correct, but in the middle we illegally have closed a loop, and on the right we closed a loop and are also branching in the wrong direction.

Above are two solutions. The players need to collaborate in order to fill the entire board. Can we find a solution for every position of the root-tile? Can we solve this on a triangle with edge length 7 instead of 5?

Contact uses tiles with two colors, and adds leaf tiles. We begin by placing two socially distanced roots, one of each color, inside a regular hexagon of edge length 6, like so, for instance.

Again this game is played with two players, Orange and Green, but this time a player controls all pieces of the same color. There is only one root tile of each color, and sufficiently many leaves, twigs, and branches. The goal is to tile the entire hexagon by taking turns, so that all twigs and branches end in leaves, as shown below. Note in particular that no loose ends of twigs or branches are allowed on the boundary of the hexagon.

However, this solution has a flaw: We have used 27 orange and only 20 green tiles. Can we do it with the same number of tiles for Orange and Green?

Finally, Trinity is played with pieces in three colors, and two chiral leaf tiles. Again, each of the three players controls one color. This time, multiple roots are allowed, but watch out that different trees of the same color don’t grow together.

At the beginning, each player places one root tile on the board, again socially distanced. Then we take turns either growing our own tree, placing a leaf tile that fits at least our own tree, or placing a root for a new tree. Below are two solutions on a hexagon of edge length 6. Enjoy.