As you can see, after the 6x5 result comes up, the rest of the results are exactly the same as the ones at the top, because 6x5 is the same as 5x6 and etc. After a while of studying the table, I found out the rule was, say if you were trying to find out how many 1x1 squares were in a 6x5 grid, you would do 6-1=5 and 5-1=4, then, you times the answer ( 5 and 4 ) together, and that number will be the amount of 1x1 squares you will get in the grid. To find out how many 2x2 squares are in a grid, you do exactly the same thing, except you subtract 2 instead of 1 because now there are 2 boxes missing from each side. I found out that with 11 lines, the grid with the most squares is the 6x5 grid- this is what I will be calling the best combination of lines.
Here is a 4x7 grid
The second part of my task was to investigate the number of squares, which can be drawn on grids made from other numbers of lines. Here are my results;
This table of results is for 5 lines only
Total number of squares: 2
Best Combination: 3x2
This table of results is for 6 lines only
Total number of squares: 8
Best Combination 3x3
This table of results is for 7 lines only
Total number of squares: 12
Best Combination: 4x3
This table of results is for 8 lines only
Total number of squares: 30
Best Combination: 4x4
This table of results is for 9 lines only
Total number of squares: 40
Best Combination: 5x4
This table of results is for 10 lines only
Total number of squares: 80
Best Combination: 5x5
This table of results is for 12 lines only
Total number of squares: 174
Best Combination: 6x6
From these results, I found out some information. The larger number of lines you have, the more squares you are going to get. Also, the best combination is always the one before they start repeating, for example if I’m going up from 6x4 to 5x5 the next one I have to do would be 4x6 which is the same as 6x4. The best combination is also the one that repeats directly after it. Like if 5x4 is the best combination and 4x5 is right after it.
Conclusion
The more lines you have, the more squares you will get. The rules I found out are; if you were trying to find out how many 1x1 squares were in a 6x5 grid, you would do 6-1=5 and 5-1=4, then, you times the answer ( 5 and 4 ) together, and that number will be the amount of 1x1 squares you will get in the grid. To find out how many 2x2 squares are in a grid, you do exactly the same thing, except you subtract 2 instead of 1 because now there are 2 boxes missing from each side.
