The simultaneous conditions x - y <= 6, x + y <= 6, and x >= 0 define a region R. What is the area of region R? How many lattice points are in region r? How many lattic points lie on the boundary of R ? What if 6 is replaced by 100?
+129850 See the graph here : https://www.desmos.com/calculator/qzfidyzqfw
We have two triangles whose base and height = 6....so...the area = 6 * 6 = 36
By Pick's Theorem
Area = Number of interior lattice points + Number of lattice points on the boundary/2 - 1
The number of lattice points on the boundary = 1 + 6 + 6 + 11 = 24
So
36 = I + 24 /2 - 1
36 = I + 12 - 1
36 = I + 11
I = 25 = 25 interior lattice points
If 6 is replaced by 100.....the area = 100^2 = 10000
The number of lattice points on the boundary = 1 + 100 + 100 + 199 = 400
By Pick's Theorem
10000 = I + 400/2 - 1
10000 = I + 200 - 1
10000 = I + 199
I = 9801 lattice points in the interior
EDIT TO CORRECT A PREVIOUS ERROR !!!!
