When I first started showing Tangle to people I noticed a common reaction. Level 1 and 2 were handled with ease but most people were afraid of Level 3. That led me to look at the difference in difficulty between Tangle’s levels.

There are multiple variables that determine the difficulty of a level. In general the graphs have an increasing number of vertices and edges, but the number of edges increases faster. But perhaps the biggest contributer to difficulty is the ratio of edges to vertices. It’s generally harder to untangle a vertex with four edges that is connected to three edges than a vertex with four edges connected to a vertex with two edges. The ratio of edges to vertices is what I ended up looking at most closely.

Consider the following graph:

The red line is Tangle’s the resulting ratio of edges to vertices for Tangle’s new level generation equations. The blue line is the ratio for the original. It became pretty apparent as to what was going on. Level 2 originally had much less complicated vertices because the vertices had fewer edges attached to them than even Level 1. While there were more vertices overall, making the level appear slightly harder, it wasn’t really a challenge to sort out the vertices. Everyone that I have introduced Tangle to after making the change to the red line has made a nice, smooth progression to and past Level 3.