Ant and Seeds

A laborious ant walks randomly on a $5 \times 5$ grid. The walk starts from the central square. At each step, the ant moves to an adjacent square at random, without leaving the grid; thus there are $2$, $3$ or $4$ possible moves at each step depending on the ant’s position.
At the start of the walk, a seed is placed on each square of the lower row. When the ant isn’t carrying a seed and reaches a square of the lower row containing a seed, it will start to carry the seed. The ant will drop the seed on the first empty square of the upper row it eventually reaches.
What’s the expected number of steps until all seeds have been dropped in the top row?
Give your answer rounded to $6$ decimal places.

This problem is pretty complex and demands quite deep mathematical knowledge, as well as computer programming expertise for exact numerical calculations. However, I will provide a simplified explanation and outline of the possible solution strategy.

The problem can be solved using the theory of Markov chains and Monte Carlo simulations or dynamic programming.

The key is to define the states meaningfully. In this case, a state can be defined by two components:

1. Position of the ant
2. State of the seeds (how many seeds were picked from the bottom row and how many were dropped in the top row)

Then, you need to identify all valid state transitions, i.e., moving from one state to another based on the ant’s movement and the picking-up/dropping-off of the seeds.

At each step, the ant can move to a neighboring cell with equal probability. The number of neighboring cells differs based on the ant’s position (2 for corner cells, 3 for border cells but not corners, and 4 for non-border cells). Additionally, if an ant is on a bottom row cell with an unpicked seed, it picks up the seed, and if an ant carrying a seed comes to a top row cell without a seed, it drops the seed.

The essential step is then to set up and solve a system of linear equations representing the expected number of steps needed to move from each state to the final state (all seeds moved to the top row) based on all possible transitions.

This is a pretty involved problem that’s not easy to solve analytically by hand due to its size. Instead, a software algorithm would be helpful to calculate the exact numerical answer.

For example, one can set up a Monte Carlo simulation, perform a large number of sample “walks”, calculate the average number of steps it takes, and round this number to six decimal places.

The mathematical exact solution would require a system of linear equations solver to handle the large system of equations defined by all possible transitions in the state space. Both these approaches require a good knowledge of algorithmic programming and possible usage of the mathematical software.

This problem is a typical example of Project Euler math tasks, which are intended to be solved with a help of computer programming. It’s beyond the reach of standard high school maths.

More Answers: