A playmat system comprising: at least eight right isosceles triangle shaped tiles (10, 10a, 10b), each tile having two triangle leg edges (12, 13) which form a right angle and a hypotenuse edge, opposite the right angle, each triangle tile has a front surface (11, 11a, 11b) and a bottom surface, each front surface having one of at least one of two patterns or colors, wherein at least one of the triangle tiles has a first color or pattern and at least one triangle tile has a second pattern or color, each triangle shaped tile having a series of projections and recesses along the two leg edges and the hypotenuse edge; wherein a first plurality of the at least eight triangles have their respective front surfaces in the first pattern or color and a second plurality of the at least eight triangles have their respective front surfaces in the second pattern or color; each leg edge of any one triangle tile can matingly engage with both leg edges of any other triangle tile completely along each respective tiles mated edge, thereby forming either a new isosceles triangle twice the size of each triangle tile or a parallelogram depending on orientation of one the mating triangle tiles relative to the other mating triangle tiles, hypotenuse edges of any two triangle tiles can matingly engage with each other, completely along each respective hypotenuse edges to thereby form a square; and a plurality of edge tiles (20, 30, 40), each edge tile having a non-mating straight edge (24, 34, 44) and a parallel opposite edge with a series of projections (25, 35, 45) and recesses (26, 36, 46) which are complementary to, and can matingly engage with the projections and recesses in the leg edges of the triangle tiles, each edge tile having a front surface having one of the at least one of two patterns or colors of the triangle shaped tiles; wherein a first plurality of the edge tiles have their respective front surfaces in the first pattern or color and a second plurality of the edge tiles have their respective front surfaces in the second pattern or color. at least eight right isosceles triangle shaped tiles (10, 10a, 10b), each tile having two triangle leg edges (12, 13) which form a right angle and a hypotenuse edge, opposite the right angle, each triangle tile has a front surface (11, 11a, 11b) and a bottom surface, each front surface having one of at least one of two patterns or colors, wherein at least one of the triangle tiles has a first color or pattern and at least one triangle tile has a second pattern or color, each triangle shaped tile having a series of projections and recesses along the two leg edges and the hypotenuse edge; wherein a first plurality of the at least eight triangles have their respective front surfaces in the first pattern or color and a second plurality of the at least eight triangles have their respective front surfaces in the second pattern or color; each leg edge of any one triangle tile can matingly engage with both leg edges of any other triangle tile completely along each respective tiles mated edge, thereby forming either a new isosceles triangle twice the size of each triangle tile or a parallelogram depending on orientation of one the mating triangle tiles relative to the other mating triangle tiles, hypotenuse edges of any two triangle tiles can matingly engage with each other, completely along each respective hypotenuse edges to thereby form a square; and a plurality of edge tiles (20, 30, 40), each edge tile having a non-mating straight edge (24, 34, 44) and a parallel opposite edge with a series of projections (25, 35, 45) and recesses (26, 36, 46) which are complementary to, and can matingly engage with the projections and recesses in the leg edges of the triangle tiles, each edge tile having a front surface having one of the at least one of two patterns or colors of the triangle shaped tiles; wherein a first plurality of the edge tiles have their respective front surfaces in the first pattern or color and a second plurality of the edge tiles have their respective front surfaces in the second pattern or color.