Twenty buckets, each with a volume of $6 \mathrm{~L}$, are stacked in a pyramid as shown, with ten buckets in the bottom layer, six in the second layer, three in the third layer, and one in the top layer. Each layer is arranged in an equilateral triangle and the buckets are stacked so that each bucket, other than those in the bottom layer, rests on exactly three buckets in the layer below.

Water is poured into the top bucket one litre at a time. After a litre is poured, the water is allowed to settle before another litre is poured. If water is poured into a bucket that is full, the water spills over the rim of the bucket in such a way that exactly one third of the runoff goes into each of the three buckets on which it rests. For example, once the top bucket is full, additional water spills equally into the three buckets in the second-highest layer.

At the first instant when at least one bucket on the bottom layer is full, what is the total amount of water that has been poured into the top bucket?

![](https://cdn.mathpix.com/cropped/2024_04_17_d0cbae19e125c7a52478g-5.jpg?height=363&width=352&top_left_y=1464&top_left_x=949)
By symmetry, the buckets in the second layer from the top will become full at the same time. Thus, there will be a point in time at which the four buckets in the top two layers will be full and all other buckets will be empty.

Each bucket holds $6 \mathrm{~L}$, so this means that after $4 \times 6 \mathrm{~L}=24 \mathrm{~L}$ are poured, the top four buckets will be full and the other 16 will be empty.

There are six buckets in the third row from the top. We will divide them into three "corner" buckets and three "edge" buckets.

The corner buckets each receive water from exactly one bucket in the second layer.

The edge buckets each receive water from exactly two buckets in the second layer.

Every litre poured into the top bucket after the $24^{\text {th }}$ litre ends up in the third layer.

The corner buckets in the third layer from the top will receive one third of the water that spills into one bucket of the second layer.

Thus, for every litre poured into the top bucket after the $24^{\text {th }}$ litre, the corner buckets in the third layer receive $\frac{1}{9} \mathrm{~L}$. The edge buckets each receive $\frac{2}{9} \mathrm{~L}$. Notice that there are three corner buckets and three edge buckets, and $3\left(\frac{1}{9}\right)+3\left(\frac{2}{3}\right)=1$, so all of the water is accounted for. This means the three edge buckets in the third layer will fill twice as fast as the corner buckets. Since they receive $\frac{2}{9} \mathrm{~L}$ for every $1 \mathrm{~L}$ poured into the top bucket and their volume is $6 \mathrm{~L}$, the edge buckets will fill after $6 \div \frac{2}{9}=6 \times \frac{9}{2}=27$ additional litres have been poured into the top bucket.

This means that after $24+27=51$ litres are poured into the top bucket, we will have

- all buckets in the top two layers full,
- the three edge buckets in the third layer will be full,
- the three corner buckets in the third layer will have $27 \times \frac{1}{9} \mathrm{~L}=3 \mathrm{~L}$ in them, and
- all buckets in the bottom layer will be empty.

The diagram below shows the buckets in each of the top three layers with the amount of water in each bucket.

![](https://cdn.mathpix.com/cropped/2024_04_17_698c2a84b151b65f3cc1g-18.jpg?height=420&width=1038&top_left_y=2202&top_left_x=606)

After another $27 \mathrm{~L}$ are poured into the top bucket, the three corner buckets in the third layer will be full for the first time.

As well, $27 \mathrm{~L}-3(3 \mathrm{~L})=18 \mathrm{~L}$ will have spilled out of the edge buckets in the third layer and into buckets in the bottom layer.

We will categorize the ten buckets in the bottom layer as three corner buckets, one centre bucket, and six edge buckets. The diagram below shows this.

![](https://cdn.mathpix.com/cropped/2024_04_17_698c2a84b151b65f3cc1g-19.jpg?height=482&width=542&top_left_y=515&top_left_x=846)

For every litre poured into the top bucket after the $51^{\text {st }}$, we know that $\frac{2}{9} \mathrm{~L}$ ends up in each edge bucket of the third layer.

Of this $\frac{2}{9} \mathrm{~L}, \frac{1}{3} \times \frac{2}{9} \mathrm{~L}=\frac{2}{27} \mathrm{~L}$ ends up in each of two edge buckets of the fourth layer, and the remaining $\frac{2}{27} \mathrm{~L}$ ends up in the centre bucket.

Since there are three edge buckets in the third layer, this means that of each litre poured into the top bucket after the $51^{\text {st }}$ (up to the $51+27=78^{\text {th }}$ ), $3 \times \frac{2}{27} \mathrm{~L}=\frac{2}{9} \mathrm{~L}$ ends up in the centre bucket.

No buckets other than the edge buckets in the third layer spill into the centre bucket on the bottom layer.

For each litre poured after the first 51 litres are poured, we know that $\frac{2}{9} \mathrm{~L}$ ends up in each edge bucket in the third layer.

One third of the water spilling from each edge bucket in the third layer ends up in the centre bucket in the bottom layer.

Since there are three edge buckets in the third layer, this implies $3 \times \frac{1}{3} \times \frac{2}{9} \mathrm{~L}=\frac{2}{9} \mathrm{~L}$ ends up in the centre bucket for every litre poured after the first 51 litres.

Therefore, it takes at least $6 \div \frac{2}{9}=27$ litres poured into the top bucket after the first 51 litres are poured to fill the centre bucket in the bottom layer.

This means it takes at least $78 \mathrm{~L}$ poured into the top bucket to fill the centre bucket in the bottom layer, and that the centre bucket in the bottom layer will be full after $78 \mathrm{~L}$ have been poured into the top bucket.

To finish the argument, we must show that no other bucket in the bottom layer fills up before the centre bucket.

One way to argue this is to notice that after $78 \mathrm{~L}$ have been poured into the top bucket, all ten buckets in the top three layers will be full as well as the centre bucket in the bottom layer. This accounts for $11 \times 6 \mathrm{~L}=66 \mathrm{~L}$ of the $78 \mathrm{~L}$.

We also know that each edge bucket in the bottom layer will have received $\frac{2}{27} \mathrm{~L}$ from an edge bucket in the third layer for each of the $27 \mathrm{~L}$ poured into the top bucket after the $51^{\text {st }}$ litre.

This accounts for an additional $6 \times 27 \times \frac{2}{27} \mathrm{~L}=12 \mathrm{~L}$.

Since $66 \mathrm{~L}+12 \mathrm{~L}=78 \mathrm{~L}$, this accounts for all $78 \mathrm{~L}$.

Therefore, the centre bucket is the first in the bottom layer to fill, and this happens after $78 \mathrm{~L}$ of water are poured into the top bucket.

ANSWER: $78 \mathrm{~L}$