Today I was putting pieces of salami onto a pizza. Because my son had been nibbling them, I did not know how many there were. I therefore did not know how best to distribute the pieces so that they were evenly distributed. This got me wondering...
If there is only one item, then I think that the best place to put it is in the centre.
If there are 25 items, then I would have thought that placing the items uniformly in a 5 by 5 grid would be the most uniform (it was a square pizza).
If there are $n$ items, then there must be an arrangements of those items that leads to the "most uniform" distribution. Of course we need to define what is meant by the "most uniform" distribution...
If, however, we don't know $n$, what should my strategy be?
I know that this will depend on the probability distribution for $n$, so I propose that $n$ has a geometric distribution:
We know that $n$ is at least $1$. After placing the first item, there is a probability $p$ that we have a second to place and a probability $1-p$ that there are no more. At each stage, let there be a probability $p$ that we have another item to place and a probability $1-p$ that there are no more.
I started assuming that the distribution is to be over a square. You can change that to a circular pizza if it's easier.
In looking into this I have found another question about distributing points over a sphere Is the Fibonacci lattice the very best way to evenly distribute N points on a sphere? So far it seems that it is the best?
My question would be different to that in that I don't want the number of points to be determined before I start placing them.
My gut feeling is that this is like the problem of deciding when to park your car in an available space and when to continue on towards your destination in the hope that you will find another space closer to your destination.