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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.

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  • $\begingroup$ After each cut, do you move the almonds around and mix them up, or do you leave them alone? The former case might be easier to analyze. In the latter case, it might be interesting to imagine that the almonds are embedded in a big cake, and ask about the areas of the cake pieces after you've chopped it up with lots of random lines. $\endgroup$ – Rahul Dec 9 '13 at 1:43
  • $\begingroup$ Good question. I think we should assume that after each cut, the almonds are mixed randomly and put back out on the cutting board. $\endgroup$ – Johannes Ernst Dec 9 '13 at 2:09
  • $\begingroup$ Do we understand the case of $N=1$ almonds at all? And can the case of $N$ almonds be thought of as $N\times$(the case of one almond)? These are actual (not rhetorical) questions. $\endgroup$ – paw88789 Dec 23 '14 at 23:18
  • $\begingroup$ Not with any kind of precision. $\endgroup$ – Johannes Ernst Dec 24 '14 at 19:17
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To get a good answer you have to specify how you choose where to cut and what the objective is. If you want no more than $M$ small particles, you should make less than $\frac Mn$ cuts, where $n$ is the largest number of almonds you can cut at once. I am imagining the almonds on a cutting board and you bring the knife down. To really assure it, don't cut at all. A more reasonable goal is to get rid of all the big pieces. Are you allowed to survey the field and choose a place to cut? Then you can always cut the largest remaining piece.

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  • $\begingroup$ Of course I can always try to cut the biggest pieces, and I can accomplish any result I want if I just pay more attention and select/aim better. So assume the opposite: the almonds are mixed randomly after each cut, and the location and direction of the cut is random. $\endgroup$ – Johannes Ernst Dec 9 '13 at 2:11
  • $\begingroup$ Actually, in practice I do not succed in always targeting the largest piece. $\endgroup$ – Ross Millikan Dec 9 '13 at 2:14

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