This problem is dedicated to Leon the professional.
This question came to my mind when I was contemplating on the numbers 1-20 arranged interestingly around a regular dartboard:
QUESTION: Can we divide a circle with radius of $\sqrt{3}\sigma$ on a plain into 3 optimal pieces with equal areas assigned by $1,2,3$ as score ,which an ambitious dart player with density probability function of $f(r;\sigma )={\frac {r}{\sigma^{2}}}e^{-r^{2}/(2\sigma ^{2})}$ (Rayleigh distribution) as probability of dart hitting in distance of $r$ from his aim point, achieves least score from the designed dartboard plane in his throw (guaranty getting minimum equal score from each point on the board he may aim to shoot)?
Are the shapes of these pieces unique? what are they look like?
Note1: if dart goes out of the board player will get $0$ score. $\sigma$ is standard deviation in Rayleigh distribution and dart hits in circle of radius $\sigma$ around player's aim point by probability of about $0.39$.
Note2: At first I proposed a generalized form of this problem stating to find $n$ connected regions on a plane which they totally shape a connected closed board without any hole, assigned by $n$ natural numbers as score and the goal was to find optimal shape of each number ,But I found this simpler state of the problem as hard as enough to contemplate.
---Another Generalization can be considered: Setting a desired predefined probability score function over the dartboard plane domain(a function that you give a point of dartboard to it as input and it gives you the probable score achieves by a player who aim to hit that point as the output) and the challenge is to design a dartboard which gives us that predefined function as score probability in each point, like to design a deceiving dartboard which the score probability be least for points of region assigned by score $3$ and be highest for points of region assigned by score $1$. for easing the problem I have considered a constant probability score function with minimum value, which still finding its minimum value is challenging.
Note3: there can be other variants and generalizations of this problem which are more applied and even I think they maybe discussed earlier but it is great to discuss here too, for example in a combinatorics way a question arises where there are quantitative numbers of valuable sources in each country and a comet threaten the planet Earth with the same hitting probability for each of its points, the question here is how to divide these sources among different countries which we loose least number of sources when the comet hits (all of the sources become inaccessible in the whole country which has been hit). However I hope this does not happen until we become advance enough in technology and facilities to eliminate such kind of threats by solving such these problems and also we human being be wise and united enough to use and benefit solutions in order to truly share our valuable sources.
Ultimate Note: the song "shape of my heart" from the film leon the professional performed by Sting, which I like a lot, have also a very nice lyric. it says:
...
He deals the cards to find the answer
The sacred geometry of chance
The hidden law of a probable outcome
The numbers lead a dance
...
I am listening and singing the song while I'm still thinking about the problem: Is this life designed by God in a way which its "sacred geometry of chance" ,"the hidden law of its probable outcome",shapes our fate? what are the shapes look like?
would it be shape of my heart?...
We can clearly keep the average at $2$ or below. Make three Archimedean spirals that wind around the center and have width per turn much smaller than $\sigma$. Anywhere the player aims will have an equal mix of the three regions. I think we can do a little better by having an outer ring of $3$ area. If the player aims at that he has a fair chance of missing the board completely, and it allows the center of the board to be depleted in $3$ area. Under this thought the solution would be three functions, each from a radius $[0,1]$ and giving the fraction of the circumference at that radius that is in each region $1,2,3$