The variable bln represents the base line grid number- i.e. the grid in which the base line passes through on that hour. Since we know that each grid size increase causes x to jump by 3, we must calculate the difference between the base line and x, in order to work out how many ‘jumps’ we need. This is done by taking the bln(base line grid no) from g, our grid number. We multiply this by three in order to get our number. Finally, we must take into account that the x number on the base line varies on the same three-stage cycle. To do this, we use the modulo function again, to counteract the difference made by the varying numbers.
When n mod 3 is 1, the start number is one, and the jumps of three work form this. Therefore, we must add on one each time to take account of this. The same applies for the other stages. In fact, we can shorten the last phase of the operation to (3(g-bln))+((n mod 3)+2) Extension When the bad tomato starts in a corner Here we have a 20×20 grid showing results for when the original bad tomato starts in the corner of the grid. The results for this situation are not nearly as complicated as those for when the tomato starts in the middle of a side, and there are only two regions to discuss. ng.
This is the results table for the proceeding grid. As can be clearly seen, there are two distinct regions, red and pink, separated along the line n=g. We can easily see expressions to calculate these: If g? n (red) then x = n If g<n (pink) then x = g+(g-n) For example, consider g as being 9, and n as being 12.
We see that n>g, therefore x is pink.9 – 12 is -3, and when you add this to 9, you get 6. If we look across the table, we find this is the answer. We will also try an example out of the range of this table: Grid size 20 and hour 25. Again, n>g, so we do 20-25. The answer, -5, yields 15 when added to 20, the grid size. By checking this against data extracted by my excel macro, we find this to be the correct answer. The formula can also be expressed more simply as x = 2g-n Conclusion Upon first seeing this investigation, I judged it to be rather uninteresting; my original impression of it was of an oversimplified ‘life genesis’ task.
While I still believe the latter to be the case, I have changed my opinion about the former. My original intention was to create a program capable of modelling the propagation of the bad tomatoes automatically, to give me the raw data I would require to find patterns in the data and to create formulae, as I have done. However, it did not take me long to realise I had neither the programming skills nor the tools to do this. I realised later this was not the case; however, by that time there was neither any need for new data or time to create the program. I had to resort to generating data manually, a tedious process.
The time it took to generate the data was, I believe, the biggest drawback in this investigation. It took both a lot of drawing and counting to do this, and each different position of the tomato had to be modelled independently. This was why I chose to primarily investigate the effects of changing the grid size; there was no way to effectively find data patterns when moving the first tomato, and it would have taken masses of diagrams. Later I managed to use features in excel to semi-automate the counting and drawing of grids, which helped me to create my final solution, but it would have been helpful for me to have achieved this earlier.
Had I been able, earlier in the project, to create a program to model the spread of bad tomatoes, this would have allowed me to analyse all the data on a much larger and more general scale. While I like to be as general as possible, the sheer amount of data that would have been needed to analyse patterns on a multi-encompassing scale (i. e. to have formulae including starting position and varying side lengths (I. e. rectangular shapes)) made it prohibitive and near impossible to do without an automatic data modelling system. Certainly, were I to have to improve on this project, that would be the first step.
Another drawback of the project was the lack of computer equipment while doing the project in lessons. As I have mentioned, without my automations for the drawing and counting of numbers in a grid, it would have been unlikely that I would have found a pattern. Had I had access to these facilities while doing the bulk of the project, I believe that I would have found the formulae much quicker, and would have gone on to further extend my project, doing such things as look at rectangular grids. As it was, a large amount of time was wasted due to the vast amount of time it would have taken to calculate things manually.
I feel, generally, that the project was a success. Despite a number of setbacks, I was able to find a formula encompassing everything I wished it to, and also did an extension upon the project. Both my main formulae have coped with any numbers I have fed through them, and I have thus far seen no faults in them that were not corrected upon re-examination of the data. 1 Show preview only The above preview is unformatted text This student written piece of work is one of many that can be found in our GCSE Bad Tomatoes section. Download this essay Print Save Not the one?