Maths Coursework Investigation

My aim is to investigate the mathematical propagation of ‘bad tomatoes’ This is essentially an investigation of patterns derived from a simple set of rules for this propagation, in the manner of a simplified life genesis program. The rules are as followed:

1. The first hour, any one of the tomatoes (depending on the investigation) turns ‘bad’
2. From that hour on, any tomato touched by a bad tomato will turn bad itself, on an hourly basis.
3. Tomatoes are constrained within an n*n grid, which restricts propagation of bad tomatoes.

As visible from the rules, this allows for creation of simple models to show the propagation of bad tomatoes. From these, I hope to derive formulae, or sets of rules if formulae are not possible, to make logical predictions.

We shall define the variables as will be used in the description of this investigation as follows:

Contents

This grid represents the propagation of bad tomatoes in an nxn square, covering grids up to size 24x24. Some of the results for this data are plotted on the table below:

While at first it seems the patterns in this table should be obvious, this is deceptive. Only by splitting the table into three regions do we see the separate patterns defining the table. These regions, as shown in the following table, allow patterns to emerge. These patterns do not, as you would expect, work down with different numbers in the same grids, but instead work across with the same number in different grids.

In the first region (yellow), we see that, in every case, x is equal to n+n-1. The latter two regions (green and purple) are substantially harder, and require a sequential approach. Naturally, the first step in devising a formula, to take n and g and return x, is determining which region the number lies in. This is a simple matter of comparing g with n. Once we know the region, we can use a set op steps to calculate the number x. The method for carrying out this operation will be described shortly.

The left grid shows an updated version of my results ...