# The intersection of the same overlapping rectangles

A math challenge had the following statement: [[OVERLAP -Three identical rectangular frames are placed one on top of the other, as Shown (A frame on the left tilted over the center frame and symmetrically a frame on the right tilted over the center frame). The result of their intersections is seven regions. Can you work out a way to obtain twenty-five regions from the intersection of the same overlapping rectangles?]] I made a sketch, in an attempt to illustrate the situation, to see if I could identify the seven intersection regions, in order to try to solve the problem. The help I am requesting is on the possibility of a code that allows the identification of the seven regions.

(* Defining line 1 *)line1 = Line[{{-1, -4}, {10, 7}}];(* Defining line 2 *)line2 = Line[{{-1, 12}, {14, -3}}];

(* Defining the first rectangle *)point1 = {3, 0}; (* vertex on the line x = 0 *)point2 = point1 + {4, 4}; (* vertex along the line x *)point3 = point2 + {-2, 1}; (* vertex along the line y = x - 3 *)point4 = point1 + {-2, 1}; (* vertex along the line y = x - 3 *)rectangle1 = {EdgeForm[Thick], FaceForm[Yellow],
Polygon[{point1, point2, point3, point4}]};

(* Defining the second rectangle *)point5 = {5, 0}; (* vertex on the line x = 0 *)point6 = point5 + {3, 0}; (* vertex along the line x *)point7 = point6 + {0, 4}; (* vertex along the line y = x - 3 *)point8 = point5 + {0, 4}; (* vertex along the line y = x - 3 *)rectangle2 = {EdgeForm[Thick], FaceForm[Red],
Polygon[{point5, point6, point7, point8}]};

(* Defining the third rectangle *)point9 = {14, 1}; (* vertex on the line x = 0 *)point10 = point9 + {-3, -1}; (* vertex along the line y = -x +11 *)point11 = point10 + {-4, 4}; (* vertex along the line y = -x + 11 *)point12 = point9 + {-4, 4}; (* vertex on the line x = 0 *)rectangle3 = {EdgeForm[Thick], FaceForm[Green],
Polygon[{point9, point10, point11, point12}]};

(* Creating the plot *)g = Graphics[{line1, line2, rectangle1, rectangle2, rectangle3},   Axes -> True,
PlotRange -> {{-1, 15}, {-1, 6}}];(* Displaying the plot *)Show[g]


• We can rotate the the middle rectangle around its center with small degree.
t = 18 Degree;
{a, b} = {4, 5};
rect = Rectangle[{0, 0}, {a, b}];
Graphics[{FaceForm[], EdgeForm[Blue], rect,
GeometricTransformation[rect, RotationTransform[t, {a, b}/2]],
GeometricTransformation[rect, RotationTransform[-t, {a, b}/2]]}]


Clear["Global*"];
t = 18  Degree;
{a, b} = {4, 5};
rect0 = Rectangle[{0, 0}, {a, b}];
rect1 = RotationTransform[t, {a, b}/2]@rect0;
rect2 = RotationTransform[-t, {a, b}/2]@rect0;
lines = MeshPrimitives[#, 1] & /@ {rect0, rect1, rect2};
reg = DiscretizeGraphics /@ lines // RegionUnion;
g = Graph[MeshPrimitives[reg, 1] /. Line -> Apply@UndirectedEdge,
VertexCoordinates -> MeshCoordinates[reg]];
faces = PlanarFaceList[g];
faces = Select[faces, WindingCount[Line@#, Mean@#] == 1 &];
GraphicsRow[{Graphics@lines,
Graphics[{{EdgeForm[White], FaceForm[RandomColor[]], Polygon@#} & /@
faces}]}]
faces // Length


25`.