Every year, during Christmas baking, I chop almonds, which causes me to puzzle over the same question, and I don't quite know how to approach it.
I start out with N almonds. Let's assume they are all the same size, so if I drew a histogram of number over size, I'd get a single spike.
Now I start cutting. The first cut bisects some randomly selected almonds, at a random angle. The second and subsequent cuts may also bisect some of the pieces that result from a previous cut. Some rather irregular shapes result.
The experimental experience is that after some time, there are still surprisingly many large pieces, while some of the smaller pieces have become small enough to be hard to distinguish with the eye. So my histogram has "smeared out".
There are different ways to draw this histogram. I'm thinking volume of a particle may be best on the x-axis, and number (or perhaps number times volume) on the y-axis.
The question: how does this histogram evolve over time? It starts as a single spike, and then what? Does it have inflection points? If I want no more than M particles to be smaller than a certain minimum size, after how many cuts should I stop cutting?
P.S. I realize this post has a high likelihood of being marked as off-topic. Almonds? What was he thinking? But methinks it's an interesting problem to think of, and applicable by many people at this time of year who want their baking to work out just right.
One year later: I cut some more almonds, and I'm just as puzzled! This year I've been pondering the shapes the pieces end up having. It appears that without having a good idea of the shapes, the original question may not be answerable. Perhaps they pieces can be approximated as being round, in which case it would be straightforward to determine the size distribution of the resulting pieces after the piece is cut.