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Let us see what happens when we make a PascGalois triangle using the group *Z*_{2}* _{ }* x

Now we have to choose which elements to place down the sides of our triangle. Notice that if we chose to place (0,1) down both sides, combining (0,1) with itself gives (0,0) and combining again with (0,1) gives (0,1) again so the element (0,1) cannot *generate* the whole group. Unlike the case of modular addition, where repeated addition of 1's will generate all of the possible results, in this set there is no single element that generates the whole set. Thus we decide to put the element (1,0) down one side of the triangle and the element (0,1) down the other side. In this triangle, pink is (0,1), yellow is (1,0), purple is (1,1), and teal is (0,0).

Here is the same triangle with a few more rows:

Does this triangle remind you of any of the modular arithmetic triangles? In particular does it remind you of the mod 2 triangle, also known as the *Z*_{2} PascGalois triangle? Let's take another look at the *Z*_{2}* _{ }* x

The following animation shows the same identification occurring in the 128-row triangles.

Kathleen M. Shannon and Michael J. Bardzell, "Patterns in Pascal's Triangle - with a Twist - Cross Products of Cyclic Groups," *Loci* (December 2004)

Journal of Online Mathematics and its Applications