The Inner Frame

Let’s change the paradigm. Here are the eight tripods that will inhabit our universe today. Note that each tripod uses exactly four different colors.

At the beginning, in year 0,  there was only one tripod. Feeling lonely, it sprouted three more, augmenting each leg with two more legs. 

We notice that there are choices: Each existing leg uses two of the four colors, and there are two possibilities to place the remaining two onto a newly sprouted tripod. So, in year 1, there are already 8 different possible universes. Free will is a good thing.

But there is also symmetry. At each node of the tree, we can exchange the branches, giving us lots of commuting involutions. We call trees that are obtained by these exchanges isometric. In particular, except for the coloring of the first tripod, all subsequent choices lead to isometric trees. Free will was an illusion. This insight should be hint enough to answer last time’s questions.

It also helps to go backwards in time, to uncover our past. For instance, the color at the center of the tree occurs most often at the rim of the universe, at year 2. It is easy to find more laws that allow us to understand the entire universe if we just know what it looks like in the leaves of year two. Even if the available information is only partial, we can often say a lot.

For instance, the partial information on the left universe above allows for two different pasts, one of which is shown on the right. Find the other one.

Another year has passed. Are the same laws still valid? Can you reconstruct the history? Is there more than one? Time is a complicated thing…

With all the emerging wildflowers, I have been using my tripod a lot lately, and this has led to today’s puzzle. We are going to color perfectly height balanced trivalent trees, like so:

This is of course too easy as it stands, so we have to impose restrictions. Today, I will insist that all tripods in your coloring look like this:

So a tripod is a perfectly height balanced trivalent tree of height 2, technically speaking. A quick inspection shows that the example above is of this sort. There still are many many such colorings, given the symmetries of the tree, and we’ll need further constraints for today’s puzzle. Before we get there, I have a few questions:

• The coloring above uses one color 21 times and the other color 25 times. Is this always the case?
• Is there always a single colored path from leaf to leaf through the center of the tree?
• How many different colorings exist if you disregard symmetries?

Above is a simple version of today’s puzzle. On the left, you see a partially colored tree. On the right there is a completed coloring, following the rules that all tripods in the tree must be colored as the two tripods above. In this case the solution is unique, as also in the puzzle below:

Enjoy. More tripods puzzles next week…