In the rectangular grids below, the diagonal touches the interiors of some of the squares in the grid. For example, in the grid, the diagonal intersects the interiors of 6 squares. In the grid, the diagonal crosses through the interiors of 8 squares.

In general, in an rectangular grid of squares, a diagonal would pass through the interiors of how many squares in the grid?

Source: NCTM Problem to Ponder March 2011

**SOLUTION
**Consider the blown up rectangle in the figure below

**Observation 1**

By visual count, the number of interior crossings equals 8.

**The diagonal crosses every horizontal grid line beginning with the -axis. It crosses the interior of four squares namely 1, 2, 3, and 4 corresponding to the 4 rows of the rectangle.**

Observation 2

Observation 2

**Observation 2**

The diagonal also crosses every vertical grid line beginning with the -axis. It crosses the interior of four squares namely 5, 6, 7, and 8. At these squares the diagonal crosses the vertical lines at non-lattice points.

It is easy to find these non-lattice points in a non-visual way. A table of values for the diagonal’s equation reveals that the -coordinates of these points take on rational values as shown below

x |
y |
|||||||

0 |
0 |
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1 |
2/3 |
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2 |
4/3 |
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3 |
3 |
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4 |
8/3 |
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5 |
10/3 |
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6 |
4 |

**Summary for a ** **rectangle
**1. The diagonal crosses the interior of 4 squares corresponding to the 4 rows of the rectangle

2. The diagonal crosses the interior of 4 more squares corresponding to the 4 non-lattice points of intersection of the diagonal and the vertical grid lines. Notice the

*lattice*points occur at which are multiples of 3. So an alternative way to find the number of non-lattice points is where 7 is the number of values of .

3. Add the numbers found in step 1 and 2.

**Algorithm for a ** **rectangle
**1. The diagonal crosses the interior of squares, .

2. Simplify the slope of the diagonal . Find the multiples of starting from . Calculate .

3. Answer = .

**Example 1 ** **rectangle
**1.

2. because the diagonal crosses all the vertical grid lines at lattice points

3.

**Example 2 ** **rectangle
**1.

2. Slope = ; among the four values of there are two multiples of 3 namely 0 and 3. Thus,

3. .

**Example 3** **rectangle**

1.

2. Slope = ; among the seven values of there are three multiples of 3 namely 0, 3, and 6. Thus,

3. .

**Answer**: Given in Solution.