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A chemist has an empty $7$ -litre flask and an empty $11$ -litre flask, a tap and a sink. One operation is: fill a flask to the brim, empty a flask completely, or pour from one flask into the other until the source is empty or the target is full. An amount $g$ is measured when some flask holds exactly $g$ litres. For how many whole amounts $g$ from $1$ to $11$ can $g$ be measured using at most $6$ operations?

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Hint

Find the fewest operations for each amount separately, and count the ones that need $6$ or fewer.

Solution

A breadth-first search from the empty pair gives the fewest operations for each amount from $1$ to $11$ .
Amounts $7$ and $11$ need $1$ operation each, $4$ needs $2$ , $3$ needs $4$ and $8$ needs $6$ .
The amounts $1$ , $2$ , $5$ , $6$ , $9$ and $10$ need $8$ , $14$ , $12$ , $10$ , $16$ and $8$ operations, which is more than $6$ .
So exactly the amounts $3$ , $4$ , $7$ , $8$ and $11$ qualify, and the count is $5$ .

Answer: $5$

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