## Le Monde puzzle [#1120]

**A** board game as Le weekly Monde current mathematical puzzle:

11 players in a circle and 365 tokens first owned by a single player. Players with at least two tokens can either remove one token and give another one left or move two right and one left. How quickly does the game stall, how many tokens are left, and where are they?

The run of a R simulation like

od=function(i)(i-1)%%11+1 muv<-function(bob){ if (max(bob)>1){ i=sample(rep((1:11)[bob>1],2),1) dud=c(0,-2,1) if((runif(1)<.5)&(bob[i]>2))dud=c(2,-3,1) bob[c(od(i+10),i,od(i+1))]=bob[c(od(i+10),i,od(i+1))]+dud } bob}

always provides a solution

> bob [1] 1 0 1 1 0 1 1 0 1 0 0

with six ones at these locations. However the time it takes to reach this frozen configuration varies, depending on the sequence of random choices.

January 15, 2020 at 2:21 pm

If every player chooses the “move one left and remove one” option on every turn, the game ends in 363 moves (I may be off by one). Is this correct? The other extreme I can’t figure out: if nobody ever removes a token, is a steady-state condition reached, and if so how many configurations does it cycle thru?

January 15, 2020 at 8:13 am

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January 14, 2020 at 10:17 am

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