# Travel Each Edge of A 6 Dimensional Hypercube Once

I am trying to find a way to "travel" along each edge of a 6D hypercube once and only once. I know it is possible with a 2D square and a 4D tessarect. For the square, it is obvious how to accomplish this: Start at (0,0), move to (0,1), move to (1,1), move to (1,0), move to (0,0). Each edge has been traveled along only once. The tessarect is a bit more complicated, but it is possible (see the uploaded picture of the tessarect here). Start at 0000, and then move in the following order: 1000, 1100, 0100, 0000, 0001, 1001, 1101, 0101, 0001, 0011, 1011, 1111, 0111, 0011, 0010, 1010, 1000, 1001, 1011, 1010, 1110, 1100, 1101, 1111, 1110, 0110, 0100, 0101, 0111, 0110, 0010, 0000.

The 6-cube has stumped me. I am not sure how to move along every edge only once, but I assume that it is possible for all n-cubes where n is an even number. The following picture of a 6-cube may help. I have one colored so that each color represents the dimension that a certain edge is in. For example, an orange line represents an edge in the 1st dimension, between vertices 000000 and 000001. There is another picture of a 6-cube which in which all of the edges of a single 3D cube are colored similarly, and each cube is connected in higher dimensions through different colors.

Any help/insight is greatly appreciated, and please let me know if I need to elaborate. Thank you, Mitch

• You might want to have a look at this. – amd May 5 '17 at 20:43

This will let you solve the 6-dimensional case, but not produce a very nice description. Here's a recursive construction that takes a solution for the $n$-cube, and turns it into a solution for the $(n+2)$-cube. We write each coordinate as $(x,y_1,y_2)$, where $x$ is an $n$-dimensional coordinate (an element of $\{0,1\}^n$) and $y_1, y_2$ are each either $0$ or $1$.
1. Take a tour of all the points of the form $(x,0,0)$ by using the $n$-dimensional solution.
2. Modify it as follows: the first time a point $(x,0,0)$ is visited, splice in the cycle $$(x,0,0) \to (x,0,1) \to (x,1,1) \to (x,1,0) \to (x,0,0)$$ at that point in the tour.
3. Modify that as follows: when you visit $(0^n,0,1)$ for the first time, splice in a tour of all points of the form $(x,0,1)$, using the $n$-dimensional solution again.
4. Do step 3 for $(0^n,1,1)$ and $(0^n,1,0)$ as well.