# Nonconvexity, and playing indoor paintball

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Following the two previous posts (here and there), on the number of people that don’t get wet while playing with water pistols, consider now an indoor version, in a non-convex room (i.e. player behind wall are now, somehow, protected). In the previous posts, players where playing on a square field, and I briefly mentioned that if the field was a disk, results would have been (roughly) the same: so far, the shape of the field was not an issue. But what if the field is no longer convex,

library(sp) plot(0:2,0:2,col="white",xlab="",ylab="") MAP=Polygon(cbind(c(0,0,1,1,2,2,0), c(0,2,2,1,1,0,0))) polygon([email protected],col="light blue")

Once again, it is possible to look more closely are those supposed-to-be “safe zones”, i.e. by looking at the distribution of the location of players that were dry at the end of the game. With 11 players, we obtain

What about the distribution of the number of dry players, over a game ?

touch=function(x1,y1,x2,y2,n=251){ X=seq(x1,x2,length=n) Y=seq(y1,y2,length=n) sum(point.in.polygon(X,Y,[email protected][,1], [email protected][,2], mode.checked=FALSE)==0)==0 } NOTWETnc=function(n,p){ sx=runif(50)*2;sy=runif(50)*2 IN=which(point.in.polygon(sx,sy,[email protected][,1], [email protected][,2], mode.checked=FALSE)==1) Sx=sx[IN];Sy=sy[IN] Sx=Sx[1:n];Sy=Sy[1:n] IN=IN[1:n] MI=matrix(NA,n,n) for(i in 1:(n-1)){ for(j in (i+1):(n)){ MI[j,i]=MI[i,j]=touch(Sx[i],Sy[i],Sx[j],Sy[j]) }} (d=as.matrix(dist(cbind(Sx,Sy), method = "euclidean",upper=TRUE))) diag(d)=999999 dpossible=d dpossible[MI==FALSE]=999999 dmin=apply(dpossible,2,which.min) #whonotwet=( (1:n) %notin% names(table(dmin)) ) notwet=n-length(table(dmin)) return(notwet)} NOTWET=function(n){ x=runif(n) y=runif(n) (d=as.matrix(dist(cbind(x,y), method = "euclidean",upper=TRUE))) diag(d)=999999 dmin=apply(d,2,which.min) notwet=n-length(table(dmin)) return(notwet)} NSim=10000 Nnc=Vectorize(NOTWETnc)(n=rep(11,NSim)) Nc=Vectorize(NOTWET)(n=rep(11,NSim)) T=table(Nc) Tn=table(Nnc) plot(as.numeric(names(Tn)), Tn/NSim,type="b",col="blue") lines(as.numeric(names(T)), T/NSim,type="b",col="red",pch=4)

On 11 players, we have the same distribution as the one on a square field. So convexity is not a key issue here…

Strange isn’t it. And with an odd number of player, not only there is at least one dry player, but at least, half of the players (maybe minus one) have to be wet…

To

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