Sílvia goes to a fountain that has three taps to fill her ten barrels. One of the barrels takes one minute to fill, another two minutes, another three minutes, and so on. How should Sílvia distribute the barrels among the taps to spend the least amount of time possible? What is this time?
Solution 1: For simplicity, we number the water jugs according to the respective times they take to fill up. The idea is to use the "remaining time" of one jug to fill another jug, filling multiple jugs simultaneously. The figures illustrate the solution.
![](https://cdn.mathpix.com/cropped/2024_05_01_7edf59f8d27fef74e9deg-5.jpg?height=436&width=1390&top_left_y=1738&top_left_x=343)

In Figure I, the 3 taps take 10 minutes to fill jugs 10, 9, 8, 1, and 2. In Figure II, the 3 taps take 9 minutes to fill jugs 7, 6, 5, 2, 3, and 4. Therefore, the total time spent is 19 minutes.

Solution 2: If we had only one tap, the time spent to fill the 10 jugs is $1+2+\cdots+9+10=55$ minutes. Since $55=18 \times 3+1$, if we have 3 taps, we must spend at least 19 minutes. The following table shows how to do the work in 19 minutes.

| Tap 1 | 10 | 9 |  |  |
| :---: | :---: | :---: | :---: | :---: |
| Tap 2 | 8 | 7 | 3 |  |
| Tap 3 | 5 | 4 | 2 | 1 |