Problem

 

So now I have the choice of which problem to solve "the 165 way"...I chose the Diagonal Problem.

So now we start with the first step:

Step 1: Understand the problem

So the problem here is to find a a pattern to predict the number of squares crossed by a diagonal on a cxr grid.

Step 2: Devise a plan.

So the plan here is...

Draw out the squares and keep adding grids and diagonal lines across the grids. Record each of the findings in a chart and create an equation based on the results and recordings.

Step 3: Carry out the plan.

Carry out the plan with rows and columns, keeping track of how many grids the diagonal line crosses each time grids are added.

 

rows           1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 |

colums       1 | 2 | 3 | 4 | 3 | 4 | 5 | 4 | 5 | 6 | 5 | 6 | 7 |

diagonal     1 | 2 | 4 | 4 | 3 | 6 | 7 | 4 | 8 | 8 | 5 |10|11|

Step 4: Look back, and draw a conclusion.

The conclusion I can draw from this chart is that when the number of grids of rows and columns are equal, the diagonal line goes through the same amount of grids that there are (eg. row = column = diagonal); when either the row or column is even, then the diagonal goes through as many grids as 2 multiplied by the smallest column or diagonal; and when both the row and column are odd, then the diagonal goes through the number of grids 2 multiplied by the smallest of the row or column plus 1.