Exercises #7

1. Show that the number of rhombi pointing at one side of a puzzle is the number of inversions on that side (counting counterclockwise).

2. Break a puzzle into regions by joining two pieces if they are of the same shape and share an edge, so there are 0-regions, 1-regions, and rhombus regions. Show that each region is convex, and that rhombus regions are parallelograms.

3. Orient the edges in a puzzle that separate regions; clockwise around 0-regions, counterclockwise around 1-regions. (Erase the other edges.) Classify the possible local pictures this directed graph can take around an interior or exterior vertex.

4. Let alpha,beta be two strings of k 1s and n-k 0s, labeling the SW and SE edges of a puzzle triangle read left to right.
a) If there is a puzzle, filling in this triangle, show that the last i entries of alpha have as many or more 1s as the first i entries of beta (for each i=1..n), a sort of dominance condition.
b) Show the converse -- if this dominance holds, then there is a filling.