Selected Problem- A Marching Strip

PROBLEM SUMMARY:

A King has a rectangular courtyard with the dimensions of 63 rows and 90 columns. All tiles are the same, with the exception of the diagonal path from one corner of the courtyard to the other corner (for example; the front left corner to the back right corner). The tiles that are along the diagonal path are more expensive. He wants to know how many tiles he will need that will cover the diagonal path of his courtyard.

PROCESS:

In finding the solution to this problem, I simply started drawing out smaller courtyard with small dimensions to see if I could find a pattern of the amount of tiles on the diagonal path. First, I looked at a courtyard that has a dimension of 2 x 2:





When looking at this small courtyard, I noticed that the area of this courtyard is 4, after multiplying the rows (2) by the columns (2).

I then tried a larger courtyard. I tried using the dimensions of 6 columns and 4 rows. When finding the area of this courtyard, I got a total area of 24. I couldn't see any type of pattern in how the areas are increasing, so I decided to look back at my dimensions:






When looking at 4 and 6, the greatest common divisor of both numbers is 2:








The total amount of expensive tiles it took to create the diagonal path is 8, as counted from the diagram above. Then I realized that you can add the number of rows and columns, and subtract the greatest common divisor to get your number of expensive tiles needed for the diagonal path: 4 (rows) + 6 (columns)= 10... 10 - 2 (greatest common divisor of 4 & 6)= 8. I tried this same method for the smaller courtyard above:
2(rows) + 2(columns)= 4. 4- 2 (greatest common divisor of 2)= 2. This number matched the amount of expensive tiles needed to create the diagonal path since there were only two tiles on the path.


The last courtyard measurement I tried was 12 columns and 10 rows, since I could easily draw it to check my findings:







After drawing the diagram, I confirmed that there were 20 tiles needed to create the diagonal path. When looking at 10 and 12, the greatest common factor between the two of them is 2:








After finding the greatest common divisor of 2, I plugged the values into the function I found above:

10 (rows) + 12 (columns)= 22. 22- 2 (greatest common divisor)= 20.
Alas, the function worked because 20 (as taken from the drawing)= 20 (as found from the function).


I then realized that if a King were to have a courtyard with 63 rows and 90 columns, then I can use my function to find how many expensive tiles are needed for the diagonal. When factoring out 63 and 90, I got 9 as the greatest common divisor:








I then added 63 (rows) + 90 (columns)= 153. 153- 9 (greatest common divisor) = 144. This means that the king needs 144 expensive tiles to create his diagonal path.



If you were given any amount of rows or columns, you can find the tiles needed for the diagonal path from the following: ((rows)+ (columns))- (greatest common divisor of the rows and columns)= total amount of tiles needed.