hshiems wrote: Bucket $A$ contains 3 litres of syrup. Bucket $B$ contains $n$ litres of water. Bucket $C$ is empty. We can perform any combination of the following operations: - Pour away the entire amount in bucket $X$, - Pour the entire amount in bucket $X$ into bucket $Y$, - Pour from bucket $X$ into bucket $Y$ until buckets $Y$ and $Z$ contain the same amount. (a) How can we obtain 10 litres of 30% syrup if $n = 20$? (b) Determine all possible values of $n$ for which the task in (a) is possible. a) : Repeat operation "Pour B into C until A and C contain the same amount" then "Pour away entire bucket C" until $A=3$, $B=8$, $C=0$ Then "Pour B into C until A and C contain the same amount" and you get $A=3$, $B=5$, $C=3$ Then "Pour C into A until A and B contain the same amount" and you get $A=5$, $B=5$, $C=1$ Then "Pour B into A" you get your result in A b) : If $n\equiv 0\pmod 3$, any operation keeps any bucket content $\equiv 0\pmod 3$ and so no solution. If $n<7$ : obviously no solution. If $n\ge 8$ and $n\equiv 2\pmod 3$, method described in a) above always gives the result If $n\ge 7$ and $n\equiv 1\pmod 3$, use the following method to get the required result : Repeat operation "Pour B into C until A and C contain the same amount" then "Pour away entire bucket C" until $A=3$, $B=7$, $C=0$ Then "Pour B into A" you get your result in A. Hence the answer : $\boxed{n\in\bigcup_{k=2}^{+\infty}\left\{3k+1,3k+2\right\}}$

Since I don't see the requirement that $n$ is an integer, what happens if $n$ is not an integer?