Next, you need to find the largest basins so you know what areas are most important to avoid.

A basin is all locations that eventually flow downward to a single low point. Therefore, every low point has a basin, although some basins are very small. Locations of height 9 do not count as being in any basin, and all other locations will always be part of exactly one basin.

The size of a basin is the number of locations within the basin, including the low point. The example above has four basins.

The top-left basin, size 3:

The top-right basin, size 9:

The middle basin, size 14:

The bottom-right basin, size 9:

Find the three largest basins and multiply their sizes together. In the above example, this is \(9 \times 14 times 9 = 1134\).

Assignment

What do you get if you multiply together the sizes of the three largest basins? Determine this in the following way:

Write a function basin that takes the pathname (str) of a text file containing a heightmap of the floor of the nearby caves. The function must return the product (int) of the sizes of the three largest basins.

Example

In this interactive session we assume the text files caves01.txt¹ and caves02.txt`² to be located in the current directory.

>>> basins('displays01.txt')
1134
>>> basins('displays02.txt')
916688