If half of the earth is blown away by the impact of a comet, what happens to the orbit of the moon?
During impact
After impact
Details and Assumptions:
To turn a three-story pyramid of coins with coins upside down, we only need to move 2 coins: moving the bottom left and bottom right coins next to the top coin.
Now, what if there are coins instead? What is the minimum number of coins we have to move in order to turn this huge, 100-story pyramid upside down?
Bonus: Generalize this for a pyramid of coins.
Forest fires are a fact of life on a planet covered in trees. Some are minor occurrences, burning over a few acres before being snuffed out without event, while others take hold of hundreds of square miles, rage for months and destroy tremendous swaths of forest and property.
Knowing which course a wildfire will take is difficult and involves a number of factors that are hard to model, but there are fundamental questions we might expect to be insensitive to the details such as
Here, we're going to build a simple model that we can interrogate in a controlled way.
We start the fire by randomly lighting one tree at time zero.
As we increase the density of trees in the forest increases, as does the likelihood that trees form clusters. Therefore, we might expect that the extent and duration of the average forest fire varies as some function of
When find the tree density at which the expected duration of the forest fire is maximal.
Note: The code environment below has a runtime limit of so be judicious in how you measure the forest. There are three functions in the imported module available for your use: add_lists()
which takes two lists of length two and returns their vector sum, dedupe_list()
which takes a list of lists and returns its unique elements, and print_forest()
which takes a forest lattice and returns a pretty print representation of the state of the forest.
Diagram: At time the tree at is lit on fire. At time the first tree is burned out and the fire has spread to the trees at and .
Find the smallest positive integer for which there exist positive reals satisfying both
Find the closed form of the above limit to 3 decimal places.
Notation: denotes the floor function.
Bonus: Can you find a general formula for for