Happy holidays, folks! And happy birthday to Topological Musings, which Vishal started just a little over a year ago — we’ve both had a really good time with it.

And in particular with the Problem of the Week series. Which, as we all know, doesn’t come out every week  — but then again, to pinch a line from John Baez’s This Week’s Finds, we never said it would: only that each time a new problem comes out, it’s always the problem for that week! Anyway, we’ve been very gratified by the response and by the ingenious solutions we routinely receive — please keep ‘em coming.

This problem comes courtesy of regular problem-solver Arin Chaudhuri and — I’ll freely admit — this one defeated me. I didn’t feel a bit bad though when Arin revealed his very elegant solution, and I’d like to see what our readers come up with. Here it is:

What is the maximum number of colors you could use to color the vertices of an 8-dimensional cube, so that starting from any vertex, each color occurs as the color of some neighbor of that vertex? (Call two vertices neighbors if they are the two endpoints of an edge of the cube.)

Arin, Vishal, and I would be interested and pleased if someone solved this problem for all dimensions $n$.

Please submit solutions to topological[dot]musings[At]gmail[dot]com by Friday, January 2, 2009, 11:59 pm (UTC); do not submit solutions in Comments. Everyone with a correct solution will be inducted into our Hall of Fame! We look forward to your response.