Running from Samsquanch - Part 3 [PARTIAL SOLUTION]
Solution for the week of 9/15/24
Our problem from 9/15/24 was
You’re thinking about doing a backpacking trip through Rainier National Park but there’s one big concern; that’s Samsquanch territory! After thinking it over for a moment you realize you have one big advantage; Samsquanch’s broad shoulders make it more difficult to navigate dense trees while your small, nerdy frame can fit quite easily. In order to make sure you have a chance to escape, you want to make sure the forest you’re hiking through has sufficient tree density.
Let’s model our forest as a 10x10 square. The top will be the entrance and the bottom will be the exit. For simplicity, we will assume the sides are impassable barriers. Each tree will be modeled by a point and Samsquanch will be modeled by a circle of radius 1. Trees will be distributed uniformly in the 10x10 square. Let’s further assume that Samsquanch can’t fit between any two trees that are less than 2R apart. We need to figure out what density the trees need to be in order for you to evade Samsquanch.Option 1
What’s the probability that Samsquanch can traverse a 10x10 forest from top to bottom (the sides are blocked) if 35 trees are uniformly placed in the forest and Samsquanch is modeled as a circle of radius 1?
Option 2
What’s the probability that Samsquanch can traverse a forest from the top to bottom (the sides are blocked) if the forest is modeled as a 10x10 grid with 30 trees uniformly placed on the grid line intersections and Samsquanch is modeled as a circle of radius 1? To clarify, there are 11 horizontal grid lines that the trees could be placed one and 9 vertical grid lines (because the sides are blocked so trees won’t be placed on the leftmost and rightmost grid line). Below is an illustration.
Give the difficulty of these problems I have not yet been able to find an analytic approach for either problem. I’ve been looking into percolation theory but its not clear to me yet whether this will yield results. I suspect the best that I can expect from myself will be to find some reasonably accurate analytic approximations. While I continue to work on this, I thought I’d provide the solutions I obtained via simulation. I calculated a probability of about 45% for option 1 and 41.3% for option 2. I’ll update this post when I get some reasonably interesting or accurate analytic results. I’ll also leave the solution form open to anyone who is continuing to work this problem. If you have any ideas for analytic approaches, please comment—god knows I could use the help :)
To determine the probability that Samsquanch can traverse a 10x10 forest from top to bottom with 35 uniformly placed trees, we need to approach this problem using principles from percolation theory or similar probabilistic methods.
Key Assumptions:
• Forest dimensions: 10x10 square.
• Samsquanch radius (R): 1.
• Impassable tree spacing: Samsquanch cannot fit between two trees that are less than 2R = 2 units apart.
• Number of trees: 35, uniformly distributed.
• Boundary conditions: The sides of the forest are barriers, so Samsquanch can only move vertically from the top to the bottom.
Strategy:
1. Define Tree Distribution: The trees are distributed randomly over the 10x10 area. For simplicity, we can treat this as a 2D point process, where each tree occupies a random position (x, y) in the square.
2. Path Obstruction Criteria: Samsquanch needs to move through gaps between trees. Any two trees that are closer than 2 units apart will block passage. To find the probability that Samsquanch is blocked, we need to estimate the likelihood of forming a continuous path of gaps larger than 2 units between trees that extend from the top to the bottom of the square.
3. Monte Carlo Simulation Approach:
• Step 1: Randomly place 35 trees in the 10x10 grid.
• Step 2: Check if a continuous vertical path can be traced from the top of the grid to the bottom, avoiding any trees that are spaced closer than 2 units apart.
• Step 3: Repeat the process for many trials to estimate the probability of Samsquanch being able to traverse the forest.
A rough analytical estimate can be challenging due to the complexity of the tree arrangement, but a Monte Carlo simulation would give a good approximation of the probability.
To determine the probability that Samsquanch can traverse a 10x10 forest from top to bottom with 35 uniformly placed trees, we need to approach this problem using principles from percolation theory or similar probabilistic methods.
Key Assumptions:
• Forest dimensions: 10x10 square.
• Samsquanch radius (R): 1.
• Impassable tree spacing: Samsquanch cannot fit between two trees that are less than 2R = 2 units apart.
• Number of trees: 35, uniformly distributed.
• Boundary conditions: The sides of the forest are barriers, so Samsquanch can only move vertically from the top to the bottom.
Strategy:
1. Define Tree Distribution: The trees are distributed randomly over the 10x10 area. For simplicity, we can treat this as a 2D point process, where each tree occupies a random position (x, y) in the square.
2. Path Obstruction Criteria: Samsquanch needs to move through gaps between trees. Any two trees that are closer than 2 units apart will block passage. To find the probability that Samsquanch is blocked, we need to estimate the likelihood of forming a continuous path of gaps larger than 2 units between trees that extend from the top to the bottom of the square.
3. Monte Carlo Simulation Approach:
• Step 1: Randomly place 35 trees in the 10x10 grid.
• Step 2: Check if a continuous vertical path can be traced from the top of the grid to the bottom, avoiding any trees that are spaced closer than 2 units apart.
• Step 3: Repeat the process for many trials to estimate the probability of Samsquanch being able to traverse the forest.
A rough analytical estimate can be challenging due to the complexity of the tree arrangement, but a Monte Carlo simulation would give a good approximation of the probability.
Based on the simulation, the estimated probability that Samsquanch can traverse the 10x10 forest from top to bottom with 35 uniformly placed trees is approximately 99.9%. This suggests that, with this number of trees and Samsquanch’s size constraints, it’s highly likely that Samsquanch can find a path through the forest.