This morning as I was making both Max and Nora breakfast, I noticed that while their meals were rather similar in content and form, they were quite different in their divisibility. Max’s French Toast (or “pain perdu,” as he prefers to call it) can be cut an infinite number of ways, as I could vary the cuts both in width/height, as well as angle. Nora’s waffle, with its rigid lines at right angles, can only be cut a finite number of different ways.
So this week’s GeekDad Puzzle of the Week came to me in a flash: just how many different ways could Nora’s waffle be cut?
Using one of her waffles, I traced the diagram, below. (Please excuse the syrup!) As you can see, and for the purposes of this puzzle, a waffle is a circular shape truncating a grid. Each of the middle rows/columns have 7 “segments,” along the edges of which a cut can be made. Each succeeding row or column has one fewer “segment,” with only 5 squares of syrup gathering goodness at the edge.
If we cut only at these edges, and if each cut traverses the waffle in a straight line from edge to edge, how many different ways can the waffle be cut? I will count rotations, horizontal flips, and vertical flips of cut sets as the same “set” of cuts. For example, each of the four cut sets below (H1V13, H13V6, H46V1, H6H46) and a few others will all count as just one set of cuts, as they produce the same set of shapes.