An interesting mathematical result, but is there a simpler way?

I came across this article on the Math Plus Web site, which I recommend for those with an interest in math but who are not necessarily proficient in it. The article gives an introduction to group theory and then explains how the Klein 4 group can be used to give information about final outcomes in peg solitaire.

I thought that was a nice result, but it seems to me that the same conclusions, at least about possible outcomes with one marble left, can be arrived at with symbols 0 and 1 and adding mod 2. I tried it and had a 0 in the center and an even number of 1’s in the other positions, giving a sum of 0. That means that it is not possible to end up with a single marble left in a hole labeled 1. The same conclusion as in the article can be arrived at by using the same type of symmetry arguments.

If I am right, then this could easily be used at a high school level for those who are taught modular arithmetic.