Le Monde puzzle [#1149]

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The weekly puzzle from Le Monde is a leaking variant on an old puzzle:

Three buckets have capacities of 8, 5 and 3 litres, respectively. At the start of the game, the 8 litre bucket is full and both others are empty. Aiming at reaching exactly 4 litres in one bucket, water is transferred between buckets. However, a fraction 1/k is lost with each transfer. If k=9, it is possible to reach 4 litres in three operations? If k=7, is it at all possible to reach 4 litres?

By sheer random search

m<-function(s){ i=sample(1:3,2) s[i]=s[i]+ifelse( rep((a<-z[i[1]]-s[i[1]])<(b<-s[i[2]])*(1-k),2), a*c(1,-1-k),b*c(1-k,-1)) s}

I found that most fractions allow to reach 4 litres starting with k=2. (And am unsure the missing ones, like 18 or 21 are not due to a lack of luck… In particular, for k=9, the shortest path is

 8.000 0 0 2.375 5 0 0.000 5 2.11 0.000 4 3


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