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This problem is dedicated to Leon the professional.

first this question came to my mind when I was contemplating on the numbers 1-20 arranged interestingly around a regular dartboard, then I proposed it on mathexchange here about 15 days ago and after some tuning by helps of @Ross Millikian ,it finished in this way:

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 board he may aim to shoot)?

Are the shapes 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 in a plain which 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 score probability function over the dartboard plane domain and the challenge is to design a dartboard which give 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 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 I think they might be 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, 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 their rational solutions in order to share our valuale 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 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 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?...

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    $\begingroup$ Do you want this true for any $r$ in some range, or just for a specific $r$ to be supplied before you design the board? If I know $r$ when I design the board it seems I should put an outer ring of $3$ because aiming at that will give a good chance of scoring $0$ by missing the board entirely. $\endgroup$ – Ross Millikan Jun 30 '18 at 2:51
  • $\begingroup$ I considered $r$ be the distance between point of hit and goal point so $r$ can be any number respect to error function $\endgroup$ – MasM Jun 30 '18 at 3:02
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    $\begingroup$ As you present it in the question, $r$ is the standard deviation of the miss distance, not the miss distance on one throw. I was asking if we know that before we design the board or not. I don't know if it is possible, but designing a board for a specific $r$ has to be easier than designing one for any $r$. $\endgroup$ – Ross Millikan Jun 30 '18 at 4:14
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    $\begingroup$ I think we can prove that you can't have the probability of hitting $1$ greater than the probability of $2$ or $3$ for any aim point. What we can do is minimize the expected score by putting a lot of the $3$ area near the edge where he risks a $0$ for missing completely. If we know $r$ in advance we can choose the thickness of the outer band. You need to make the question precise. $\endgroup$ – Ross Millikan Jun 30 '18 at 4:18
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    $\begingroup$ I agree that this question needs more precision. It could be an interesting question if you work it out properly. In addition to the points raised by @RossMillikan, you should also clarify whether the overall shape of the board is given (and if so, what it is). Also, I don't understand what you mean by "the size of the board and the areas of the shapes are proper to the players distance range and so his error function" -- is this an answer to Ross's question? If so, I don't understand the answer. $\endgroup$ – joriki Jun 30 '18 at 7:29
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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 $1,2,3$

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  • $\begingroup$ Thanks for help, I assumed the proportion of $A=1/2 \pi\sigma^2$ between area $A$ of each shape and $\sigma$ but really I don't know what would differs in the problem by changin this proportion. Still I think the problem is not round enough to solve because of the ambiguity in asking minimal state of achieving scores probability in each board's point. $\endgroup$ – MasM Jul 2 '18 at 4:12
  • $\begingroup$ I was thinking one could make a simple model that chooses the outer ring of $3$ based on $\sigma$. You assume that the area inside the ring is equally populated with $1,2$ and the rest of the $3$ area. You choose the width of the $3$ ring so that the player can't improve over aiming at the center because too many shots fall outside the board. I don't know if that will work and would have to develop a number of tools to do that. $\endgroup$ – Ross Millikan Jul 2 '18 at 5:15

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