To find the minimum number of measurements required to get exactly $298$ litres, we can use a combination of drawing water from the tank (adding to our container) and pouring water back into the tank (subtracting from our container).

Approach 1: Only Adding Water If we only draw water, we would use the greedy approach (using the largest possible cylinders first):

• $2$ times $100$ litres = $200$ litres
• $3$ times $25$ litres = $75$ litres
• $3$ times $6$ litres = $18$ litres
• $5$ times $1$ litre = $5$ litres Total volume = $200 + 75 + 18 + 5 = 298$ litres. Total measurements = $2 + 3 + 3 + 5 = 13$ times.

Approach 2: Adding and Subtracting Water (Optimal) To minimize the number of measurements, we can overshoot the target volume and then remove the excess water by pouring it back into the tank.

• We can draw $100$ litres $3$ times: $3 \times 100 = 300$ litres. (This takes $3$ measurements)
• We have an excess of $2$ litres ($300 - 298 = 2$).
• We can remove this excess by using the $1$-litre cylinder to pour water back into the tank $2$ times: $2 \times 1 = 2$ litres. (This takes $2$ measurements)

Total volume = $300 - 2 = 298$ litres. Total measurements = $3$ (drawing) $+ 2$ (pouring back) = $5$ times.

Can we do it in $4$ measurements? To get close to $298$ with $4$ measurements, we must use the $100$-litre cylinder $3$ times ($300$ litres). We would then have only $1$ measurement left to subtract the excess. The closest volumes we could achieve are:

• $3 \times 100 - 1 \times 1 = 299$ litres
• $3 \times 100 - 1 \times 6 = 294$ litres Since neither gives exactly $298$ litres, $4$ measurements are not enough.

Thus, the minimum number of times one needs to measure is $5$.